Infinity.J. Wallis (1655).

For the first time it is found in the treatise of the English mathematician John Valis "On Conic Sections".

Base of natural logarithms. L. Euler (1736).

Mathematical constant, transcendental number. This number is sometimes called non-Perov in honor of the Scottish scientist Napier, author of the work "Description of the amazing table of logarithms" (1614). For the first time, the constant is tacitly present in the appendix to the English translation of the aforementioned work by Napier, published in 1618. The very same constant was first calculated by the Swiss mathematician Jacob Bernoulli in the course of solving the problem of the limiting value of interest income.

2,71828182845904523...

The first known use of this constant, where it was denoted by the letter b, found in Leibniz's letters to Huygens, 1690-1691. letter e started using Euler in 1727, and the first publication with this letter was his Mechanics, or the Science of Motion, Stated Analytically, 1736. Respectively, e commonly called Euler number. Why was the letter chosen? e, is not exactly known. Perhaps this is due to the fact that the word begins with it exponential("exponential", "exponential"). Another assumption is that the letters a, b, c and d already widely used for other purposes, and e was the first "free" letter.

The ratio of the circumference of a circle to its diameter. W. Jones (1706), L. Euler (1736).

Mathematical constant, irrational number. The number "pi", the old name is Ludolf's number. Like any irrational number, π is represented by an infinite non-periodic decimal fraction:

π=3.141592653589793...

For the first time, the designation of this number with the Greek letter π was used by the British mathematician William Jones in the book A New Introduction to Mathematics, and it became generally accepted after the work of Leonhard Euler. This designation comes from the initial letter of the Greek words περιφερεια - circle, periphery and περιμετρος - perimeter. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrien Marie Legendre in 1774 proved the irrationality of π 2 . Legendre and Euler assumed that π could be transcendental, i.e. cannot satisfy any algebraic equation with integer coefficients, which was eventually proven in 1882 by Ferdinand von Lindemann.

imaginary unit. L. Euler (1777, in press - 1794).

It is known that the equation x 2 \u003d 1 has two roots: 1 and -1 . The imaginary unit is one of the two roots of the equation x 2 \u003d -1, denoted by the Latin letter i, another root: -i. This designation was proposed by Leonhard Euler, who took the first letter of the Latin word for this imaginarius(imaginary). He also extended all the standard functions to the complex domain, i.e. set of numbers representable in the form a+ib, where a and b are real numbers. The term "complex number" was introduced into wide use by the German mathematician Carl Gauss in 1831, although the term had previously been used in the same sense by the French mathematician Lazar Carnot in 1803.

Unit vectors. W. Hamilton (1853).

Unit vectors are often associated with the coordinate axes of the coordinate system (in particular, with the axes of the Cartesian coordinate system). Unit vector directed along the axis X, denoted i, a unit vector directed along the axis Y, denoted j, and the unit vector directed along the axis Z, denoted k. Vectors i, j, k are called orts, they have identity modules. The term "ort" was introduced by the English mathematician and engineer Oliver Heaviside (1892), and the notation i, j, k Irish mathematician William Hamilton.

The integer part of a number, antie. K. Gauss (1808).

The integer part of the number [x] of the number x is the largest integer not exceeding x. So, =5, [-3,6]=-4. The function [x] is also called "antier of x". The integer part function symbol was introduced by Carl Gauss in 1808. Some mathematicians prefer to use the notation E(x) proposed in 1798 by Legendre instead.

Angle of parallelism. N.I. Lobachevsky (1835).

On the Lobachevsky plane - the angle between the linebpassing through the pointOparallel to a straight linea, not containing a dotO, and perpendicular fromO on the a. α is the length of this perpendicular. As the point is removedO from straight athe angle of parallelism decreases from 90° to 0°. Lobachevsky gave a formula for the angle of parallelismP( α )=2arctg e - α /q , where q is some constant related to the curvature of the Lobachevsky space.

Unknown or variable quantities. R. Descartes (1637).

In mathematics, a variable is a quantity characterized by the set of values ​​that it can take. This can mean both a real physical quantity, temporarily considered in isolation from its physical context, and some abstract quantity that has no analogues in the real world. The concept of a variable arose in the 17th century. initially under the influence of the demands of natural science, which brought to the fore the study of movement, processes, and not just states. This concept required new forms for its expression. The literal algebra and analytic geometry of René Descartes were such new forms. For the first time, the rectangular coordinate system and the notation x, y were introduced by Rene Descartes in his work "Discourse on the method" in 1637. Pierre Fermat also contributed to the development of the coordinate method, but his work was first published after his death. Descartes and Fermat used the coordinate method only on the plane. The coordinate method for three-dimensional space was first applied by Leonhard Euler already in the 18th century.

Vector. O.Koshi (1853).

From the very beginning, a vector is understood as an object having a magnitude, a direction, and (optionally) an application point. The beginnings of vector calculus appeared along with the geometric model of complex numbers in Gauss (1831). Advanced operations on vectors were published by Hamilton as part of his quaternion calculus (the imaginary components of a quaternion formed a vector). Hamilton coined the term vector(from the Latin word vector, carrier) and described some vector analysis operations. This formalism was used by Maxwell in his works on electromagnetism, thereby drawing the attention of scientists to the new calculus. Gibbs' Elements of Vector Analysis (1880s) soon followed, and then Heaviside (1903) gave vector analysis its modern look. The vector sign itself was introduced by the French mathematician Augustin Louis Cauchy in 1853.

Addition, subtraction. J. Widman (1489).

The plus and minus signs were apparently invented in the German mathematical school of "kossists" (that is, algebraists). They are used in Jan (Johannes) Widmann's textbook A Quick and Pleasant Count for All Merchants, published in 1489. Prior to this, addition was denoted by the letter p(from Latin plus"more") or the Latin word et(conjunction "and"), and subtraction - by letter m(from Latin minus"less, less"). In Widman, the plus symbol replaces not only addition, but also the union "and". The origin of these symbols is unclear, but most likely they were previously used in trading as signs of profit and loss. Both symbols soon became common in Europe - with the exception of Italy, which used the old designations for about a century.

Multiplication. W. Outred (1631), G. Leibniz (1698).

The multiplication sign in the form of an oblique cross was introduced in 1631 by the Englishman William Outred. Before him, the most commonly used letter M, although other designations were also proposed: the symbol of a rectangle (French mathematician Erigon, 1634), an asterisk (Swiss mathematician Johann Rahn, 1659). Later, Gottfried Wilhelm Leibniz replaced the cross with a dot (end of the 17th century), so as not to be confused with the letter x; before him, such symbolism was found by the German astronomer and mathematician Regiomontanus (XV century) and the English scientist Thomas Harriot (1560 -1621).

Division. I.Ran (1659), G.Leibniz (1684).

William Outred used the slash / as the division sign. Colon division began to denote Gottfried Leibniz. Before them, the letter was also often used D. Starting from Fibonacci, the horizontal line of the fraction is also used, which was used by Heron, Diophantus and in Arabic writings. In England and the United States, the ÷ (obelus) symbol, which was proposed by Johann Rahn (possibly with the participation of John Pell) in 1659, became widespread. An attempt by the American National Committee on Mathematical Standards ( National Committee on Mathematical Requirements) to remove the obelus from practice (1923) was inconclusive.

Percent. M. de la Porte (1685).

One hundredth of a whole, taken as a unit. The word "percent" itself comes from the Latin "pro centum", which means "one hundred". In 1685, the book Manual of Commercial Arithmetic by Mathieu de la Porte was published in Paris. In one place, it was about percentages, which then meant "cto" (short for cento). However, the typesetter mistook that "cto" for a fraction and typed "%". So because of a typo, this sign came into use.

Degrees. R. Descartes (1637), I. Newton (1676).

The modern notation for the exponent was introduced by René Descartes in his " geometries"(1637), however, only for natural powers with exponents greater than 2. Later, Isaac Newton extended this form of notation to negative and fractional exponents (1676), the interpretation of which had already been proposed by this time: the Flemish mathematician and engineer Simon Stevin, the English mathematician John Vallis and French mathematician Albert Girard.

arithmetic root n th power of a real number a≥0, - non-negative number n-th degree of which is equal to a. The arithmetic root of the 2nd degree is called the square root and can be written without indicating the degree: √. The arithmetic root of the 3rd degree is called the cube root. Medieval mathematicians (for example, Cardano) denoted the square root with the symbol R x (from the Latin Radix, root). The modern designation was first used by the German mathematician Christoph Rudolf, from the Cossist school, in 1525. This symbol comes from the stylized first letter of the same word radix. The line above the radical expression was absent at first; it was later introduced by Descartes (1637) for a different purpose (instead of brackets), and this feature soon merged with the sign of the root. The cube root in the 16th century was designated as follows: R x .u.cu (from lat. Radix universalis cubica). Albert Girard (1629) began to use the usual notation for the root of an arbitrary degree. This format was established thanks to Isaac Newton and Gottfried Leibniz.

Logarithm, Decimal Logarithm, Natural Logarithm. I. Kepler (1624), B. Cavalieri (1632), A. Prinsheim (1893).

The term "logarithm" belongs to the Scottish mathematician John Napier ( "Description of the amazing table of logarithms", 1614); it arose from a combination of the Greek words λογος (word, relation) and αριθμος (number). J. Napier's logarithm is an auxiliary number for measuring the ratio of two numbers. The modern definition of the logarithm was first given by the English mathematician William Gardiner (1742). By definition, the logarithm of a number b by reason a (a ≠ 1, a > 0) - exponent m, to which the number should be raised a(called the base of the logarithm) to get b. Denoted log a b. So, m = log a b, if a m = b.

The first tables of decimal logarithms were published in 1617 by Oxford mathematics professor Henry Briggs. Therefore, abroad, decimal logarithms are often called brigs. The term "natural logarithm" was introduced by Pietro Mengoli (1659) and Nicholas Mercator (1668), although the London mathematics teacher John Spidell compiled a table of natural logarithms as early as 1619.

Until the end of the 19th century, there was no generally accepted notation for the logarithm, the base a indicated to the left and above the symbol log, then over it. Ultimately, mathematicians came to the conclusion that the most convenient place for the base is below the line, after the symbol log. The sign of the logarithm - the result of the reduction of the word "logarithm" - occurs in various forms almost simultaneously with the appearance of the first tables of logarithms, for example Log- I. Kepler (1624) and G. Briggs (1631), log- B. Cavalieri (1632). Designation ln for the natural logarithm was introduced by the German mathematician Alfred Pringsheim (1893).

Sine, cosine, tangent, cotangent. W. Outred (middle of the 17th century), I. Bernoulli (18th century), L. Euler (1748, 1753).

Shorthand notation for sine and cosine was introduced by William Outred in the middle of the 17th century. Abbreviations for tangent and cotangent: tg, ctg introduced by Johann Bernoulli in the 18th century, they became widespread in Germany and Russia. In other countries, the names of these functions are used. tan, cot proposed by Albert Girard even earlier, at the beginning of the 17th century. Leonard Euler (1748, 1753) brought the theory of trigonometric functions into its modern form, and we also owe him the consolidation of real symbolism.The term "trigonometric functions" was introduced by the German mathematician and physicist Georg Simon Klugel in 1770.

The sine line of Indian mathematicians was originally called "arha jiva"("semi-string", that is, half of the chord), then the word "archa" was discarded and the sine line began to be called simply "jiva". Arabic translators did not translate the word "jiva" Arabic word "vatar", denoting the bowstring and chord, and transcribed in Arabic letters and began to call the sine line "jiba". Since short vowels are not indicated in Arabic, and long "and" in the word "jiba" denoted in the same way as the semivowel "y", the Arabs began to pronounce the name of the sine line "jibe", which literally means "hollow", "bosom". When translating Arabic works into Latin, European translators translated the word "jibe" Latin word sinus, having the same meaning.The term "tangent" (from lat.tangents- touching) was introduced by the Danish mathematician Thomas Fincke in his Geometry of the Round (1583).

Arcsine. K.Scherfer (1772), J.Lagrange (1772).

Inverse trigonometric functions are mathematical functions that are the inverse of trigonometric functions. The name of the inverse trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix "arc" (from lat. arc- arc).Inverse trigonometric functions usually include six functions: arcsine (arcsin), arccosine (arccos), arctangent (arctg), arccotangent (arcctg), arcsecant (arcsec) and arccosecant (arccosec). For the first time, special symbols for inverse trigonometric functions were used by Daniel Bernoulli (1729, 1736).Manner of notating inverse trigonometric functions with a prefix arc(from lat. arcus, arc) appeared at the Austrian mathematician Karl Scherfer and gained a foothold thanks to the French mathematician, astronomer and mechanic Joseph Louis Lagrange. It was meant that, for example, the usual sine allows you to find the chord subtending it along the arc of a circle, and the inverse function solves the opposite problem. Until the end of the 19th century, the English and German mathematical schools offered other notation: sin -1 and 1/sin, but they are not widely used.

Hyperbolic sine, hyperbolic cosine. W. Riccati (1757).

Historians discovered the first appearance of hyperbolic functions in the writings of the English mathematician Abraham de Moivre (1707, 1722). The modern definition and detailed study of them was carried out by the Italian Vincenzo Riccati in 1757 in the work "Opusculorum", he also proposed their designations: sh,ch. Riccati proceeded from the consideration of a single hyperbola. An independent discovery and further study of the properties of hyperbolic functions was carried out by the German mathematician, physicist and philosopher Johann Lambert (1768), who established a wide parallelism between the formulas of ordinary and hyperbolic trigonometry. N.I. Lobachevsky subsequently used this parallelism, trying to prove the consistency of non-Euclidean geometry, in which ordinary trigonometry is replaced by hyperbolic.

Just as the trigonometric sine and cosine are the coordinates of a point on a coordinate circle, the hyperbolic sine and cosine are the coordinates of a point on a hyperbola. Hyperbolic functions are expressed in terms of an exponent and are closely related to trigonometric functions: sh(x)=0.5(e x-e-x) , ch(x)=0.5(e x +e -x). By analogy with trigonometric functions, hyperbolic tangent and cotangent are defined as ratios of hyperbolic sine and cosine, cosine and sine, respectively.

Differential. G. Leibniz (1675, in press 1684).

The main, linear part of the function increment.If the function y=f(x) one variable x has at x=x0derivative, and incrementΔy \u003d f (x 0 +? x)-f (x 0)functions f(x) can be represented asΔy \u003d f "(x 0) Δx + R (Δx) , where member R infinitely small compared toΔx. First Memberdy=f"(x 0 )Δxin this expansion is called the differential of the function f(x) at the pointx0. AT works of Gottfried Leibniz, Jacob and Johann Bernoulli word"differentia"was used in the sense of "increment", I. Bernoulli denoted it through Δ. G. Leibniz (1675, published in 1684) used the notation for "infinitely small difference"d- the first letter of the word"differential", formed by him from"differentia".

Indefinite integral. G. Leibniz (1675, in press 1686).

The word "integral" was first used in print by Jacob Bernoulli (1690). Perhaps the term is derived from the Latin integer- whole. According to another assumption, the basis was the Latin word integro- restore, restore. The sign ∫ is used to denote an integral in mathematics and is a stylized image of the first letter of a Latin word summa- sum. It was first used by the German mathematician Gottfried Leibniz, the founder of differential and integral calculus, at the end of the 17th century. Another of the founders of differential and integral calculus, Isaac Newton, did not offer an alternative symbolism of the integral in his works, although he tried various options: a vertical bar above a function or a square symbol that stands in front of a function or borders it. Indefinite integral for a function y=f(x) is the collection of all antiderivatives of the given function.

Definite integral. J. Fourier (1819-1822).

Definite integral of a function f(x) with lower limit a and upper limit b can be defined as the difference F(b) - F(a) = a ∫ b f(x)dx , where F(x)- some antiderivative function f(x) . Definite integral a ∫ b f(x)dx numerically equal to the area of ​​\u200b\u200bthe figure bounded by the x-axis, straight lines x=a and x=b and function graph f(x). The French mathematician and physicist Jean Baptiste Joseph Fourier proposed the design of a definite integral in the form we are used to at the beginning of the 19th century.

Derivative. G. Leibniz (1675), J. Lagrange (1770, 1779).

Derivative - the basic concept of differential calculus, characterizing the rate of change of a function f(x) when the argument changes x . It is defined as the limit of the ratio of the increment of a function to the increment of its argument as the increment of the argument tends to zero, if such a limit exists. A function that has a finite derivative at some point is called differentiable at that point. The process of calculating the derivative is called differentiation. The reverse process is integration. In classical differential calculus, the derivative is most often defined through the concepts of the theory of limits, however, historically, the theory of limits appeared later than differential calculus.

The term "derivative" was introduced by Joseph Louis Lagrange in 1797; dy/dx— Gottfried Leibniz in 1675. The manner of designating the derivative with respect to time with a dot above the letter comes from Newton (1691).The Russian term "derivative of a function" was first used by a Russian mathematicianVasily Ivanovich Viskovatov (1779-1812).

Private derivative. A. Legendre (1786), J. Lagrange (1797, 1801).

For functions of many variables, partial derivatives are defined - derivatives with respect to one of the arguments, calculated under the assumption that the remaining arguments are constant. Notation ∂f/ ∂ x, ∂ z/ ∂ y introduced by the French mathematician Adrien Marie Legendre in 1786; fx",zx"- Joseph Louis Lagrange (1797, 1801); ∂ 2z/ ∂ x2, ∂ 2z/ ∂ x ∂ y- second-order partial derivatives - German mathematician Carl Gustav Jacob Jacobi (1837).

Difference, increment. I. Bernoulli (late 17th century - first half of the 18th century), L. Euler (1755).

The designation of the increment by the letter Δ was first used by the Swiss mathematician Johann Bernoulli. The symbol "delta" entered into common practice after the work of Leonhard Euler in 1755.

Sum. L. Euler (1755).

The sum is the result of adding values ​​(numbers, functions, vectors, matrices, etc.). To denote the sum of n numbers a 1, a 2, ..., a n, the Greek letter "sigma" Σ is used: a 1 + a 2 + ... + a n = Σ n i=1 a i = Σ n 1 a i . The sign Σ for the sum was introduced by Leonhard Euler in 1755.

Work. K. Gauss (1812).

The product is the result of multiplication. To denote the product of n numbers a 1, a 2, ..., a n, the Greek letter "pi" Π is used: a 1 a 2 ... a n = Π n i=1 a i = Π n 1 a i . For example, 1 3 5 ... 97 99 = ? 50 1 (2i-1). The symbol Π for the product was introduced by the German mathematician Carl Gauss in 1812. In Russian mathematical literature, the term "work" was first encountered by Leonty Filippovich Magnitsky in 1703.

Factorial. K.Krump (1808).

The factorial of a number n (denoted n!, pronounced "en factorial") is the product of all natural numbers up to and including n: n! = 1 2 3 ... n. For example, 5! = 1 2 3 4 5 = 120. By definition, 0! = 1. The factorial is defined only for non-negative integers. The factorial of a number n is equal to the number of permutations of n elements. For example, 3! = 6, indeed,

♣ ♦

♣ ♦

♣ ♦

♦ ♣

♦ ♣

♦ ♣

All six and only six permutations of three elements.

The term "factorial" was introduced by the French mathematician and politician Louis Francois Antoine Arbogast (1800), the designation n! - French mathematician Christian Kramp (1808).

Module, absolute value. K. Weierstrass (1841).

Module, the absolute value of the real number x - a non-negative number defined as follows: |x| = x for x ≥ 0, and |x| = -x for x ≤ 0. For example, |7| = 7, |- 0.23| = -(-0.23) = 0.23. Modulus of a complex number z = a + ib is a real number equal to √(a 2 + b 2).

It is believed that the term "module" was proposed to be used by the English mathematician and philosopher, a student of Newton, Roger Cotes. Gottfried Leibniz also used this function, which he called "module" and denoted: mol x. The generally accepted notation for the absolute value was introduced in 1841 by the German mathematician Karl Weierstrass. For complex numbers, this concept was introduced by the French mathematicians Augustin Cauchy and Jean Robert Argan at the beginning of the 19th century. In 1903, the Austrian scientist Konrad Lorenz used the same symbolism for the length of a vector.

Norm. E. Schmidt (1908).

A norm is a functional defined on a vector space and generalizing the concept of the length of a vector or the modulus of a number. The sign "norm" (from the Latin word "norma" - "rule", "sample") was introduced by the German mathematician Erhard Schmidt in 1908.

Limit. S. Luillier (1786), W. Hamilton (1853), many mathematicians (until the beginning of the 20th century)

Limit - one of the basic concepts of mathematical analysis, meaning that some variable value in the process of its change under consideration approaches a certain constant value indefinitely. The concept of a limit was used intuitively as early as the second half of the 17th century by Isaac Newton, as well as by mathematicians of the 18th century, such as Leonhard Euler and Joseph Louis Lagrange. The first rigorous definitions of the limit of a sequence were given by Bernard Bolzano in 1816 and Augustin Cauchy in 1821. The symbol lim (the first 3 letters from the Latin word limes - border) appeared in 1787 with the Swiss mathematician Simon Antoine Jean Lhuillier, but its use did not yet resemble the modern one. The expression lim in a more familiar form for us was first used by the Irish mathematician William Hamilton in 1853.Weierstrass introduced a designation close to the modern one, but instead of the usual arrow, he used the equal sign. The arrow appeared at the beginning of the 20th century with several mathematicians at once - for example, with the English mathematician Godfried Hardy in 1908.

Zeta function, d Riemann zeta function. B. Riemann (1857).

Analytic function of the complex variable s = σ + it, for σ > 1, determined by the absolutely and uniformly convergent Dirichlet series:

ζ(s) = 1 -s + 2 -s + 3 -s + ... .

For σ > 1, the representation in the form of the Euler product is valid:

ζ(s) = Π p (1-p -s) -s ,

where the product is taken over all primes p. The zeta function plays a big role in number theory.As a function of a real variable, the zeta function was introduced in 1737 (published in 1744) by L. Euler, who indicated its decomposition into a product. Then this function was considered by the German mathematician L. Dirichlet and, especially successfully, by the Russian mathematician and mechanic P.L. Chebyshev in the study of the law of distribution of prime numbers. However, the most profound properties of the zeta function were discovered later, after the work of the German mathematician Georg Friedrich Bernhard Riemann (1859), where the zeta function was considered as a function of a complex variable; he also introduced the name "zeta function" and the notation ζ(s) in 1857.

Gamma function, Euler Γ-function. A. Legendre (1814).

The gamma function is a mathematical function that extends the notion of factorial to the field of complex numbers. Usually denoted by Γ(z). The z-function was first introduced by Leonhard Euler in 1729; it is defined by the formula:

Γ(z) = limn→∞ n! n z /z(z+1)...(z+n).

A large number of integrals, infinite products, and sums of series are expressed through the G-function. Widely used in analytic number theory. The name "Gamma function" and the notation Γ(z) were proposed by the French mathematician Adrien Marie Legendre in 1814.

Beta function, B function, Euler B function. J. Binet (1839).

A function of two variables p and q, defined for p>0, q>0 by the equality:

B(p, q) = 0 ∫ 1 x p-1 (1-x) q-1 dx.

The beta function can be expressed in terms of the Γ-function: В(p, q) = Γ(p)Г(q)/Г(p+q).Just as the gamma function for integers is a generalization of the factorial, the beta function is, in a sense, a generalization of the binomial coefficients.

Many properties are described using the beta function.elementary particles participating in strong interaction. This feature was noticed by the Italian theoretical physicistGabriele Veneziano in 1968. It started string theory.

The name "beta function" and the notation B(p, q) were introduced in 1839 by the French mathematician, mechanic and astronomer Jacques Philippe Marie Binet.

Laplace operator, Laplacian. R. Murphy (1833).

Linear differential operator Δ, which functions φ (x 1, x 2, ..., x n) from n variables x 1, x 2, ..., x n associates the function:

Δφ \u003d ∂ 2 φ / ∂x 1 2 + ∂ 2 φ / ∂x 2 2 + ... + ∂ 2 φ / ∂x n 2.

In particular, for a function φ(x) of one variable, the Laplace operator coincides with the operator of the 2nd derivative: Δφ = d 2 φ/dx 2 . The equation Δφ = 0 is usually called the Laplace equation; this is where the names "Laplace operator" or "Laplacian" come from. The notation Δ was introduced by the English physicist and mathematician Robert Murphy in 1833.

Hamiltonian operator, nabla operator, Hamiltonian. O. Heaviside (1892).

Vector differential operator of the form

∇ = ∂/∂x i+ ∂/∂y j+ ∂/∂z k,

where i, j, and k- coordinate vectors. Through the nabla operator, the basic operations of vector analysis, as well as the Laplace operator, are expressed in a natural way.

In 1853, the Irish mathematician William Rowan Hamilton introduced this operator and coined the symbol ∇ for it in the form of an inverted Greek letter Δ (delta). At Hamilton, the point of the symbol pointed to the left; later, in the works of the Scottish mathematician and physicist Peter Guthrie Tate, the symbol acquired a modern look. Hamilton called this symbol the word "atled" (the word "delta" read backwards). Later, English scholars, including Oliver Heaviside, began to call this symbol "nabla", after the name of the letter ∇ in the Phoenician alphabet, where it occurs. The origin of the letter is associated with a musical instrument such as the harp, ναβλα (nabla) in ancient Greek means "harp". The operator was called the Hamilton operator, or the nabla operator.

Function. I. Bernoulli (1718), L. Euler (1734).

A mathematical concept that reflects the relationship between elements of sets. We can say that a function is a "law", a "rule" according to which each element of one set (called the domain of definition) is associated with some element of another set (called the domain of values). The mathematical concept of a function expresses an intuitive idea of ​​how one quantity completely determines the value of another quantity. Often the term "function" means a numerical function; that is, a function that puts some numbers in line with others. For a long time, mathematicians gave arguments without brackets, for example, like this - φх. This notation was first used by the Swiss mathematician Johann Bernoulli in 1718.Parentheses were only used if there were many arguments, or if the argument was a complex expression. Echoes of those times are common and now recordssin x, lg xetc. But gradually the use of parentheses, f(x) , became the general rule. And the main merit in this belongs to Leonhard Euler.

Equality. R. Record (1557).

The equal sign was proposed by the Welsh physician and mathematician Robert Record in 1557; the character's outline was much longer than the current one, as it imitated the image of two parallel segments. The author explained that there is nothing more equal in the world than two parallel segments of the same length. Before that, in ancient and medieval mathematics, equality was denoted verbally (for example, est egale). Rene Descartes in the 17th century began to use æ (from lat. aequalis), and he used the modern equals sign to indicate that the coefficient could be negative. François Viète denoted subtraction with an equals sign. The symbol of the Record did not spread immediately. The spread of the Record symbol was hindered by the fact that since ancient times the same symbol has been used to indicate the parallelism of lines; in the end, it was decided to make the symbol of parallelism vertical. In continental Europe, the sign "=" was introduced by Gottfried Leibniz only at the turn of the 17th-18th centuries, that is, more than 100 years after the death of Robert Record, who first used it for this.

About the same, about the same. A. Günther (1882).

Sign " ≈" was introduced by German mathematician and physicist Adam Wilhelm Sigmund Günther in 1882 as a symbol for the relationship "about equal".

More less. T. Harriot (1631).

These two signs were introduced into use by the English astronomer, mathematician, ethnographer and translator Thomas Harriot in 1631, before that the words "more" and "less" were used.

Comparability. K. Gauss (1801).

Comparison - the ratio between two integers n and m, meaning that the difference n-m of these numbers is divided by a given integer a, called the modulus of comparison; it is written: n≡m(mod a) and reads "numbers n and m are comparable modulo a". For example, 3≡11(mod 4) since 3-11 is divisible by 4; the numbers 3 and 11 are congruent modulo 4. Comparisons have many properties similar to those of equalities. So, the term in one part of the comparison can be transferred with the opposite sign to another part, and comparisons with the same module can be added, subtracted, multiplied, both parts of the comparison can be multiplied by the same number, etc. For example,

3≡9+2(mod 4) and 3-2≡9(mod 4)

At the same time true comparisons. And from a pair of true comparisons 3≡11(mod 4) and 1≡5(mod 4) the correctness of the following follows:

3+1≡11+5(mod 4)

3-1≡11-5(mod 4)

3 1≡11 5(mod 4)

3 2 ≡11 2 (mod 4)

3 23≡11 23(mod 4)

In number theory, methods for solving various comparisons are considered, i.e. methods for finding integers that satisfy comparisons of one kind or another. Modulo comparisons were first used by the German mathematician Carl Gauss in his 1801 book Arithmetical Investigations. He also proposed the symbolism established in mathematics for comparison.

Identity. B. Riemann (1857).

Identity - the equality of two analytical expressions, valid for any admissible values ​​of the letters included in it. The equality a+b = b+a is valid for all numerical values ​​of a and b, and therefore is an identity. To record identities, in some cases, since 1857, the sign "≡" (read "identically equal") has been used, the author of which in this use is the German mathematician Georg Friedrich Bernhard Riemann. Can be written a+b ≡ b+a.

Perpendicularity. P.Erigon (1634).

Perpendicularity - the mutual arrangement of two straight lines, planes or a straight line and a plane, in which these figures make a right angle. The sign ⊥ to denote perpendicularity was introduced in 1634 by the French mathematician and astronomer Pierre Erigon. The concept of perpendicularity has a number of generalizations, but all of them, as a rule, are accompanied by the sign ⊥ .

Parallelism. W. Outred (1677 posthumous edition).

Parallelism - the relationship between some geometric shapes; for example, straight lines. Defined differently depending on different geometries; for example, in the geometry of Euclid and in the geometry of Lobachevsky. The sign of parallelism has been known since ancient times, it was used by Heron and Pappus of Alexandria. At first, the symbol was similar to the current equals sign (only more extended), but with the advent of the latter, to avoid confusion, the symbol was turned vertically ||. It appeared in this form for the first time in a posthumous edition of the works of the English mathematician William Outred in 1677.

Intersection, union. J. Peano (1888).

An intersection of sets is a set that contains those and only those elements that simultaneously belong to all given sets. The union of sets is a set that contains all the elements of the original sets. Intersection and union are also called operations on sets that assign new sets to certain sets according to the above rules. Denoted ∩ and ∪, respectively. For example, if

A= (♠ ♣ ) and B= (♣ ♦ ),

That

A∩B= {♣ }

A∪B= {♠ ♣ ♦ } .

Contains, contains. E. Schroeder (1890).

If A and B are two sets and there are no elements in A that do not belong to B, then they say that A is contained in B. They write A⊂B or B⊃A (B contains A). For example,

{♠}⊂{♠ ♣}⊂{♠ ♣ ♦ }

{♠ ♣ ♦ }⊃{ ♦ }⊃{♦ }

The symbols "contains" and "contains" appeared in 1890 with the German mathematician and logician Ernst Schroeder.

Affiliation. J. Peano (1895).

If a is an element of the set A, then write a∈A and read "a belongs to A". If a is not an element of A, write a∉A and read "a does not belong to A". Initially, the relations "contained" and "belongs" ("is an element") were not distinguished, but over time, these concepts required a distinction. The membership sign ∈ was first used by the Italian mathematician Giuseppe Peano in 1895. The symbol ∈ comes from the first letter of the Greek word εστι - to be.

The universal quantifier, the existential quantifier. G. Gentzen (1935), C. Pierce (1885).

A quantifier is a general name for logical operations that indicate the area of ​​truth of a predicate (mathematical statement). Philosophers have long paid attention to logical operations that limit the scope of the truth of a predicate, but did not single them out as a separate class of operations. Although quantifier-logical constructions are widely used both in scientific and everyday speech, their formalization took place only in 1879, in the book of the German logician, mathematician and philosopher Friedrich Ludwig Gottlob Frege "The Calculus of Concepts". Frege's notation looked like cumbersome graphic constructions and was not accepted. Subsequently, many more successful symbols were proposed, but the notation ∃ for the existential quantifier (read "exists", "there is"), proposed by the American philosopher, logician and mathematician Charles Pierce in 1885, and ∀ for the universal quantifier (read "any" , "each", "any"), formed by the German mathematician and logician Gerhard Karl Erich Gentzen in 1935 by analogy with the existential quantifier symbol (the reversed first letters of the English words Existence (existence) and Any (any)). For example, the entry

(∀ε>0) (∃δ>0) (∀x≠x 0 , |x-x 0 |<δ) (|f(x)-A|<ε)

reads as follows: "for any ε>0 there exists δ>0 such that for all x not equal to x 0 and satisfying the inequality |x-x 0 |<δ, выполняется неравенство |f(x)-A|<ε".

Empty set. N. Bourbaki (1939).

A set that does not contain any element. The empty set sign was introduced in the books of Nicolas Bourbaki in 1939. Bourbaki is the collective pseudonym of a group of French mathematicians formed in 1935. One of the members of the Bourbaki group was Andre Weil, the author of the Ø symbol.

Q.E.D. D. Knuth (1978).

In mathematics, a proof is understood as a sequence of reasoning based on certain rules, showing that a certain statement is true. Since the Renaissance, the end of a proof has been denoted by mathematicians as "Q.E.D.", from the Latin expression "Quod Erat Demonstrandum" - "What was required to be proved." When creating the computer layout system ΤΕΧ in 1978, the American computer science professor Donald Edwin Knuth used a symbol: a filled square, the so-called "Halmos symbol", named after the American mathematician of Hungarian origin Paul Richard Halmos. Today, the completion of a proof is usually denoted by the Halmos Symbol. As an alternative, other signs are used: an empty square, a right triangle, // (two slashes), as well as the Russian abbreviation "ch.t.d.".

Geometric symbols. A class of characters that are identical in shape geometric elements widely used in the field of mythological and religious, as well as emblematics and heraldry.

geometric symbols

Swastika straight (left-handed)

The swastika as a solar symbol

A straight (left-handed) swastika is a cross with the ends bent to the left. Rotation is considered to be clockwise (opinions sometimes differ in determining the direction of movement).

A straight swastika is a symbol of blessing, good omen, prosperity, good luck and aversion to misfortune, as well as a symbol of fertility, longevity, health and life. It is also a symbol of the masculine principle, spirituality, which inhibits the flow of lower (physical) forces and allows the energies of a higher, divine nature to manifest.

Reverse swastika (right side)

Swastika on a Nazi military medal

The reverse (right-handed) swastika is a cross with the ends bent to the right. Rotation is considered to be counterclockwise.

The reverse swastika is usually associated with the feminine. Sometimes it is associated with the launch of negative (physical) energies that close the passage to the elevated forces of the spirit.

The Sumerian swastika, formed by four women and their hair, symbolizes the female generative power

Pentagram (pentacle): the general meaning of the symbol

pentagram sign

The pentagram, written in one line, is the most ancient of all the symbols that we own. It had different interpretations in different historical times of mankind. It became the Sumerian and Egyptian sign of the stars.

Later symbolism: five senses; masculine and feminine, expressed by five points; harmony, health and mystical powers. The pentagram is also a symbol of the victory of the spiritual over the material, a symbol of security, protection, a safe return home.

Pentagram as a magical symbol

Pentagrams of the White and Black Magicians

A pentacle with one end up and two down is a sign of white magic, known as the "foot of the druid"; with one end down and two up, it represents the so-called "goat's hoof" and the horns of the devil - a sign change characteristic of symbolism from positive to negative when it is turned over.

The pentagram of the White Magician is a symbol of magical influence and the dominance of a disciplined Will over the phenomena of the world. The will of the Black Magician is directed to destruction, to the refusal to perform a spiritual task, therefore the inverted pentagram is considered as a symbol of evil.

Pentagram as a symbol of a perfect person

Pentagram symbolizing the perfect man

The pentagram, a five-pointed star, is a symbol of a perfect man standing on two legs with outstretched arms. We can say that a person is a living pentagram. This is true both physically and spiritually - a person possesses five virtues and manifests them: love, wisdom, truth, justice and kindness.

Truth belongs to the spirit, love to the soul, wisdom to the intellect, kindness to the heart, justice to the will.

double pentagram

Double pentagram (man and the universe)

There is also a correspondence between the human body and the five elements (earth, water, air, fire and ether): will corresponds to earth, heart to water, intellect to air, soul to fire, spirit to ether. Thus, by his will, intellect, heart, soul, spirit, a person is connected with the five elements working in the cosmos, and he can consciously work in harmony with them. This is the meaning of the symbol of the double pentagram, in which the small one is inscribed in the large one: a person (microcosm) lives and acts inside the Universe (macrocosm).

Hexagram

Hexagram image

Hexagram - a figure made up of two polar triangles, a six-pointed star. It is a complex and solid symmetrical shape in which six small individual triangles are grouped around a large central hexagon. The result is a star, although the original triangles retain their individuality. Since the upward facing triangle is a heavenly symbol, and the downward facing triangle is a symbol of the earth, together they are a symbol of a person who unites these two worlds. It is a symbol of a perfect marriage that binds a man and a woman.

Seal of Solomon

Seal of Solomon, or Star of David

This is the famous magical seal of Solomon, or the Star of David. The top triangle in her image is white and the bottom triangle is black. It symbolizes, first of all, the absolute law of analogy, expressed by the mystical formula: "What is below is similar to what is above."

The Seal of Solomon is also a symbol of human evolution: one must learn not only to take, but also to give, to absorb and radiate at the same time, to radiate for the Earth, to perceive from Heaven. We receive and are filled only when we give to others. This is the perfect union of spirit and matter in man - the union of the solar plexus and the brain.

five pointed star

five pointed star

star of bethlehem

The five-pointed star is interpreted in different ways, including it symbolizes joy and happiness. It is also the emblem of the Semitic goddess Ishtar in her martial incarnation, and in addition, the Star of Bethlehem. For Freemasons, the five-pointed star symbolizes the mystical center.

The Egyptians attached great importance to the five- and six-pointed stars, as is clear from the text preserved on the wall of the funerary temple of Hatshepsut.

seven-pointed star

Seven pointed star of magicians

In the seven-pointed star, the characteristic features of the five-pointed are repeated. The Gnostic star has seven rays.

Seven- and nine-pointed stars drawn in one line are mystical stars in astrology and magic.

The star of the magicians is read in two ways: sequentially along the rays (along the line of the star) and along the circumference. In the course of the rays, there are planets that control the days of the week: Sun - Sunday, Moon - Monday, Mars - Tuesday, Mercury - Wednesday, Jupiter - Thursday, Venus - Friday, Saturn - Saturday.

nine pointed star

Nine-pointed star of magicians

Nine-pointed stars, like seven-pointed ones, if they are drawn in one line, are mystical stars in astrology and magic.

The nine-pointed star, made up of three triangles, symbolizes the Holy Spirit.

Monad

The four constituent parts of a monad

It is a magical symbol called the monad by John Dee (1527–1608), advisor and astrologer to Queen Elizabeth I of England.

Dee presents the nature of magic symbols in terms of geometry and tests the monad in a series of theorems.

Dee explores the monad at such a deep level that he finds links to his theory with Pythagorean harmony, biblical knowledge, and mathematical proportions.

Spiral

Spiral structure of the Milky Way

Spiral shapes are very common in nature, from spiral galaxies to whirlpools and tornadoes, from mollusk shells to human finger prints, and even the DNA molecule has the shape of a double helix.

The spiral is a very complex and ambiguous symbol. But first of all, it is a symbol of the great creative (life) force both at the level of the cosmos and at the level of the microcosm. The spiral is a symbol of time, cyclic rhythms, the change of seasons, birth and death, the phases of "aging" and "growth" of the Moon, as well as the Sun itself.

Tree of Life

Tree of Life in a human being

Tree of Life

The Tree of Life does not belong to any culture - not even to the Egyptians. It is beyond race and religion. This image is an integral part of nature… Man himself is a miniature Tree of Life. He possessed immortality when he was associated with this tree. The Tree of Life can be thought of as the arteries of a large cosmic body. Through these arteries, as through channels, the life-giving forces of the cosmos flow, which nourish all forms of existence, and the cosmic pulse of life beats in them. The Tree of Life is a separate section, part of the scheme of the universal code of life.

Sphere

Armillary sphere (engraving from Tycho Brahe's book)

A symbol of fertility (like a circle), as well as integrity. In ancient Greece, the sign of the sphere was a cross in a circle - the ancient emblem of power. A sphere made up of several metal rings, illustrating the cosmogonic theory of Ptolemy, who believed that the Earth is at the center of the universe, is an ancient emblem of astronomy.

Platonic Solids

Platonic solids inscribed in a sphere

The Platonic solids are five unique shapes. Long before Plato, Pythagoras used them, calling them ideal geometric bodies. Ancient alchemists and such great minds as Pythagoras believed that these bodies are associated with certain elements: cube (A) - earth, tetrahedron (B) - fire, octahedron (C) - air, icosahedron (D) - water, dodecahedron ( E) - ether, and the sphere - emptiness. These six elements are the building blocks of the universe. They create the qualities of the universe.

Planet symbols

Planet symbols

The planets are depicted by a combination of the simplest geometric symbols. This is a circle, a cross, an arc.

Consider, for example, the symbol for Venus. The circle is located above the cross, which personifies a kind of "spiritual attraction" that pulls the cross up into the elevated areas belonging to the circle. The cross, subject to the laws of generation, decay and death, will find its redemption if it is raised within this great circle of spirituality. The symbol as a whole represents the feminine in the world, which is trying to spiritualize and protect the material sphere.

Pyramid

The Great Pyramids of Cheops, Khafre and Menkaure

The pyramid is a symbol of the hierarchy that exists in the universe. In any area, the pyramid symbol can help move from the lower plane of plurality and fragmentation to the higher plane of unity.

It is believed that the initiates chose the shape of the pyramid for their sanctuaries because they wanted the lines converging towards the top, rushing towards the Sun, to teach humanity the lesson of unity.

star tetrahedron

star tetrahedron

A star tetrahedron is a figure consisting of two mutually intersecting tetrahedra. This figure can also be perceived as a three-dimensional star of David.

Tetrahedra manifest as two opposite laws: the law of the spirit (radiation, bestowal, selflessness, selflessness) and the law of matter (drawing inward, cooling, freezing, paralysis). Only a person can consciously combine these two laws, since he is the link between the world of spirit and the world of matter.

The star tetrahedron thus represents the two poles of creation in perfect balance.

The course uses geometric language, made up of notations and symbols adopted in the course of mathematics (in particular, in the new geometry course in high school).

The whole variety of designations and symbols, as well as the connections between them, can be divided into two groups:

group I - designations of geometric figures and relations between them;

group II designations of logical operations, constituting the syntactic basis of the geometric language.

The following is a complete list of math symbols used in this course. Particular attention is paid to the symbols that are used to designate the projections of geometric shapes.

Group I

SYMBOLS DESIGNATED GEOMETRIC FIGURES AND RELATIONSHIPS BETWEEN THEM

A. Designation of geometric shapes

1. The geometric figure is denoted - F.

2. Points are indicated by capital letters of the Latin alphabet or Arabic numerals:

A, B, C, D, ... , L, M, N, ...

1,2,3,4,...,12,13,14,...

3. Lines arbitrarily located in relation to the projection planes are indicated by lowercase letters of the Latin alphabet:

a, b, c, d, ... , l, m, n, ...

Level lines are indicated: h - horizontal; f- frontal.

The following notation is also used for straight lines:

(AB) - a straight line passing through the points A and B;

[AB) - a ray with the beginning at point A;

[AB] - a straight line segment bounded by points A and B.

4. Surfaces are denoted by lowercase letters of the Greek alphabet:

α, β, γ, δ,...,ζ,η,ν,...

To emphasize the way the surface is defined, you should specify the geometric elements by which it is defined, for example:

α(a || b) - plane α is determined by parallel lines a and b;

β(d 1 d 2 gα) - the surface β is determined by the guides d 1 and d 2 , the generatrix g and the plane of parallelism α.

5. Angles are indicated:

∠ABC - angle with apex at point B, as well as ∠α°, ∠β°, ... , ∠φ°, ...

6. Angular: the value (degree measure) is indicated by the sign, which is placed above the angle:

The value of the angle ABC;

The value of the angle φ.

A right angle is marked with a square with a dot inside

7. Distances between geometric figures are indicated by two vertical segments - ||.

For example:

|AB| - distance between points A and B (length of segment AB);

|Aa| - distance from point A to line a;

|Aα| - distances from point A to surface α;

|ab| - distance between lines a and b;

|αβ| distance between surfaces α and β.

8. For projection planes, the following designations are accepted: π 1 and π 2, where π 1 is the horizontal projection plane;

π 2 -fryuntal plane of projections.

When replacing projection planes or introducing new planes, the latter denote π 3, π 4, etc.

9. Projection axes are denoted: x, y, z, where x is the x-axis; y is the y-axis; z - applicate axis.

The constant line of the Monge diagram is denoted by k.

10. Projections of points, lines, surfaces, any geometric figure are indicated by the same letters (or numbers) as the original, with the addition of a superscript corresponding to the projection plane on which they were obtained:

A", B", C", D", ... , L", M", N", horizontal projections of points; A", B", C", D", ... , L", M" , N", ... frontal projections of points; a" , b" , c" , d" , ... , l", m" , n" , - horizontal projections of lines; a" ,b" , c" , d" , ... , l" , m " , n" , ... frontal projections of lines; α", β", γ", δ",...,ζ",η",ν",... horizontal projections of surfaces; α", β", γ", δ",...,ζ" ,η",ν",... frontal projections of surfaces.

11. Traces of planes (surfaces) are indicated by the same letters as the horizontal or frontal, with the addition of a subscript 0α, emphasizing that these lines lie in the projection plane and belong to the plane (surface) α.

So: h 0α - horizontal trace of the plane (surface) α;

f 0α - frontal trace of the plane (surface) α.

12. Traces of straight lines (lines) are indicated by capital letters, which begin words that define the name (in Latin transcription) of the projection plane that the line crosses, with a subscript indicating belonging to the line.

For example: H a - horizontal trace of a straight line (line) a;

F a - frontal trace of a straight line (line) a.

13. The sequence of points, lines (of any figure) is marked with subscripts 1,2,3,..., n:

A 1, A 2, A 3,..., A n;

a 1 , a 2 , a 3 ,...,a n ;

α 1 , α 2 , α 3 ,...,α n ;

F 1 , F 2 , F 3 ,..., F n etc.

The auxiliary projection of the point, obtained as a result of the transformation to obtain the actual value of the geometric figure, is denoted by the same letter with the subscript 0:

A 0 , B 0 , C 0 , D 0 , ...

Axonometric projections

14. Axonometric projections of points, lines, surfaces are indicated by the same letters as nature with the addition of the superscript 0:

A 0, B 0, C 0, D 0, ...

1 0 , 2 0 , 3 0 , 4 0 , ...

a 0 , b 0 , c 0 , d 0 , ...

α 0 , β 0 , γ 0 , δ 0 , ...

15. Secondary projections are indicated by adding a superscript 1:

A 1 0 , B 1 0 , C 1 0 , D 1 0 , ...

1 1 0 , 2 1 0 , 3 1 0 , 4 1 0 , ...

a 1 0 , b 1 0 , c 1 0 , d 1 0 , ...

α 1 0 , β 1 0 , γ 1 0 , δ 1 0 , ...

To facilitate the reading of the drawings in the textbook, several colors were used in the design of the illustrative material, each of which has a certain semantic meaning: black lines (dots) indicate the initial data; green color is used for lines of auxiliary graphic constructions; red lines (dots) show the results of constructions or those geometric elements to which special attention should be paid.

B. Symbols Denoting Relations Between Geometric Figures

no.	Designation	Content	Symbolic notation example
1	≡	Match	(AB) ≡ (CD) - a straight line passing through points A and B, coincides with the line passing through points C and D
2	≅	Congruent	∠ABC≅∠MNK - angle ABC is congruent to angle MNK
3	∼	Similar	ΔABS∼ΔMNK - triangles ABC and MNK are similar
4	||	Parallel	α||β - plane α is parallel to plane β
5	⊥	Perpendicular	a⊥b - lines a and b are perpendicular
6		interbreed	with d - lines c and d intersect
7		Tangents	t l - line t is tangent to line l. βα - plane β tangent to surface α
8	→	Are displayed	F 1 → F 2 - the figure F 1 is mapped onto the figure F 2
9	S	projection center. If the projection center is not a proper point, its position is indicated by an arrow, indicating the direction of projection	-
10	s	Projection direction	-
11	P	Parallel projection	p s α Parallel projection - parallel projection to the plane α in the direction s

B. Set-theoretic notation

no.	Designation	Content	Symbolic notation example	An example of symbolic notation in geometry
1	M,N	Sets	-	-
2	A,B,C,...	Set elements	-	-
3	{ ... }	Consists of...	F(A, B, C,... )	Ф(A, B, C,...) - figure Ф consists of points A, B, C, ...
4	∅	Empty set	L - ∅ - the set L is empty (contains no elements)	-
5	∈	Belongs to, is an element	2∈N (where N is the set of natural numbers) - the number 2 belongs to the set N	A ∈ a - point A belongs to the line a (point A lies on line a)
6	⊂	Includes, contains	N⊂M - the set N is a part (subset) of the set M of all rational numbers	a⊂α - line a belongs to the plane α (understood in the sense: the set of points of the line a is a subset of the points of the plane α)
7	∪	Union	C \u003d A U B - set C is a union of sets A and B; (1, 2. 3, 4.5) = (1.2.3)∪(4.5)	ABCD = ∪ [BC] ∪ - broken line, ABCD is union of segments [AB], [BC],
8	∩	Intersection of many	М=К∩L - the set М is the intersection of the sets К and L (contains elements belonging to both the set K and the set L). M ∩ N = ∅- intersection of sets M and N is the empty set (sets M and N do not have common elements)	a = α ∩ β - line a is the intersection planes α and β and ∩ b = ∅ - lines a and b do not intersect (have no common points)

Group II SYMBOLS DESIGNATING LOGICAL OPERATIONS

no.	Designation	Content	Symbolic notation example
1	∧	conjunction of sentences; corresponds to the union "and". Sentence (p∧q) is true if and only if p and q are both true	α∩β = ( K:K∈α∧K∈β) The intersection of surfaces α and β is a set of points (line), consisting of all those and only those points K that belong to both the surface α and the surface β
2	∨	Disjunction of sentences; corresponds to the union "or". Sentence (p∨q) true when at least one of the sentences p or q is true (i.e. either p or q or both).	-
3	⇒	Implication is a logical consequence. The sentence p⇒q means: "if p, then q"	(a||c∧b||c)⇒a||b. If two lines are parallel to a third, then they are parallel to each other.
4	⇔	The sentence (p⇔q) is understood in the sense: "if p, then q; if q, then p"	А∈α⇔А∈l⊂α. A point belongs to a plane if it belongs to some line belonging to that plane. The converse is also true: if a point belongs to some line, belonging to the plane, then it also belongs to the plane itself.
5	∀	The general quantifier reads: for everyone, for everyone, for anyone. The expression ∀(x)P(x) means: "for any x: property P(x)"	∀(ΔABC)( = 180°) For any (for any) triangle, the sum of the values ​​of its angles at the vertices is 180°
6	∃	The existential quantifier reads: exists. The expression ∃(x)P(x) means: "there is x that has the property P(x)"	(∀α)(∃a). For any plane α, there exists a line a not belonging to the plane α and parallel to the plane α
7	∃1	The uniqueness of existence quantifier, reads: there is a unique (-th, -th)... The expression ∃1(x)(Px) means: "there is a unique (only one) x, having the property Rx"	(∀ A, B)(A≠B)(∃1a)(a∋A, B) For any two different points A and B, there is a unique line a, passing through these points.
8	(px)	Negation of the statement P(x)	ab(∃α )(α⊃а, b). If lines a and b intersect, then there is no plane a that contains them
9	\	Negative sign	≠ - the segment [AB] is not equal to the segment .a? b - the line a is not parallel to the line b

Geometric symbols are all kinds of lines - straight, curved, broken and combined. These are geometric shapes - a circle, a cross, a triangle, etc.. And also these are bodies, such as a ball, a cube, a pyramid, etc. In two-dimensional space, these unusual symbols take the form of figures.

The geometric ones represented the structure of outer space, as well as the structure of ritual space (temple, tomb) and the forms of sacred objects. With the help of geometric symbols, the structure and structure of social society, as well as the spiritual (ethical) space (love, faith, hope, perseverance, etc.) were depicted. Let's analyze in more detail the most popular geometric symbols used both in magic and in science.

MOST COMMON GEOMETRIC SYMBOLS:

lines

Most often, straight lines, broken (zigzag), spirals and volts are used in magic, which correlated with thunder, water, earth, snake, etc. Also, as a magic symbol, they can use a continuous line broken at a right angle, otherwise called a meander. This line symbolized the absence of beginning and end - eternity. In ancient Greece, the meander was compared with a labyrinth, and in ancient China - with reincarnation.

Spiral

The spiral is a rather ambiguous symbol. The spiral as a magical symbol was used in ancient Egypt, Mesopotamia, India, China, Europe, Japan, Oceania, pre-Columbian America, the Scandinavian countries and Crete. The spiral is a symbol of solar and lunar energy, thunder, lightning, whirlwind and creative forces.

Triangle

The shape of this geometric figure determines its symbolism. The triangle symbolizes the number 3, as well as the trinity in all its combinations: birth-life-death, body-mind-soul, father-mother-children, sky-earth-underworld.

Among other things, the triangle is a symbol of the fruitfulness of the earth, marriage, flame, mountains, pyramids, physical stability, the head of God.

If you connect three triangles, you get the Pythagorean symbol of health. Also, this symbol is the emblem of the Masons.

The swastika inside the triangle is a symbol of cosmic harmony.

A triangle placed within the borders of a square is a symbol of the combination of everything divine and human, heavenly and earthly, spiritual and bodily.

The triangle inside the circle is a symbol of trinity in a single whole, and two intersecting triangles are divinity, the combination of fire and water, the victory of spirit over matter.

Star of David

The six-pointed star of David, or otherwise the hexagram, according to legend, was the coat of arms of the Israeli king David in the tenth century BC. It is this unusual fact that served as the basis for the name of this symbol. Also, this symbol was depicted on the amulet of the Babylonian king Kurigalsu, a contemporary of the biblical Moses, and on the seal of King Solomon.

Pentagram

The pentagram (five-pointed star) is a symbol of the microcosm, as well as the human figure. Designates the five mysterious centers of power, the five senses of man, the five elements in nature, the five limbs of the human body. With the help of the pentagram, a person can control low creatures and demand help from high creatures.

Square

The square is a symbol of stability and constancy, as well as the perfect form of a closed and mystical union of the four elements.

Pentagon

The Pentagon is a regular pentagon in the form of a star. It is a symbol of eternity, perfection and the universe. Also, the pentagon can serve as an amulet of health. If this symbol is drawn on the doors, then it will drive away witches and evil entities. The Pentagon is used in various magical conspiracies and rituals.

Hexagon

Hexagon - a regular hexagon - is a symbol of beauty and harmony. It is also the image of a person - two arms, two legs, a head and a torso. Due to the fact that on the one hand the hexagon has corners, and on the other hand it is close to the shape of a circle, in mystical rites it is related to the idea of ​​energy and peace, as well as to the Sun.

A circle

The circle is a universal symbol of integrity, harmony and perfection. The rounded shape has been considered sacred since ancient times, as it was the most natural shape in nature. The circle symbolized what is called in the modern world - the space-time continuum, as well as what lies outside of time and space. The circle has no beginning, no end, no top, no bottom.

A circle with a dot in the center is a symbol of a complete time cycle. In astrology the circle is the symbol of the Sun, and in alchemy it is the symbol of the Sun and the Moon.

The circle inside which is placed - denotes Paradise and its four rivers flowing from the center, as well as the Tree of Life.

Cross

The emergence of the symbol of the cross comes from the Neolithic era. The cross is one of the most common religious symbols of the highest sacred values. Unlike the circle and the square, whose main symbolic idea is to distinguish between the inside and the outside, the cross emphasizes the idea of ​​the center and the main directions leading from it. In fact, the cross is the center of the world and the connection point between heaven and earth is the cosmic axis.

The cross often acted as a model of a person or an anthropomorphic deity. At the same time, the cross also moderates the spiritual aspect, the ability for infinite and harmonious stretching in the vertical and horizontal directions.

In the vertical direction - this is the ascent of the spirit, the aspiration to God, eternity: stellar, intellectual, positive, active, male power.

In the horizontal direction, it is an earthly, rational, passive, negative, female force. In general, the cross forms an androgyne (an individual of one sex that has signs of the other sex), and also reflects dualism in nature and the union of opposites. The cross represents the spiritual union and integrity of the human spirit in vertical-horizontal aspects, which is necessary for the fullness of life. In other words, the cross is the figure of a man with outstretched arms, as well as a symbol of the descent of the spirit into matter.

Various forms of the cross are known. The cross with a loop in the upper part was understood as a key that opens the gates to divine knowledge. The T-shaped part of the symbol referred to wisdom - a drop-shaped circle - to the eternal, beginning. Kest with a loop

T-shaped cross - tau-cross. Among the ancient Egyptians, this symbol denoted the location of the horns of a bull or a ram - the vertical part is the muzzle of the animal. Among the ancient Jews, it is a symbol of the expected messiah. In ancient Rome - criminals were crucified on such a cross - it was used as an instrument of execution.

Later, in various religious movements and political unions, they invented their own, certain forms: Burgundy, Maltese, Andreevsky, etc.

Swastika

The swastika is a cross with equal-sized loops, the ends of which are bent in the form of the Greek letter gamma - a religious Hindu symbol. In Asia and Europe, the swastika was considered a secret magical sign. This is the sun, the source of life and fertility, and at the same time - a symbol of thunder and heavenly fire.

Introduction.

There are a huge number of emblems and symbols in the world. It is human nature to symbolize, otherwise he cannot. Any word is a symbol of something. The history of symbolism is the history of the species Homo Sapiens. Naturally, any symbol is an expression of essence. People stick to symbolism in everything. The symbol is important because it is a short expression of the essence.

The theme of symbolism was, is and will be one of the most intriguing, interesting and relevant topics. Probably, there is not a single science that does not use symbolism.

The first chapter is devoted to a detailed consideration of symbolism, the study of culture as a symbolic system, as well as a variety of sign systems.

The second chapter is devoted directly to the geometric symbols themselves and reveals the basis of the topic of the essay. This chapter discusses five geometric symbols: dot (center), circle, cross, swastika, and spiral. Each of the symbols will be given many different meanings, correlations with cultures and eras.

The third chapter shows that each symbol, element and sign is inextricably linked with another, the hypothesis is proved that there is an inextricable link between symbols that can connect all symbols into something unified and contain a special incomprehensible knowledge.

There is also an appendix to the abstract containing drawings and photographs depicting certain symbols of various eras and cultures.

The appendix to the abstract contains a glossary of terms, a dictionary of names and illustrations for the abstract. The glossary of terms contains concepts and names that are of great importance in this work. The dictionary of names includes the most prominent representatives of cultural, philosophical, mathematical, anthropological and other teachings.

The meaning and reflection of symbols on the culture and consciousness of people.

Anyone who touches on the topic of "symbolology" will most often encounter two fundamentally different positions. On the one hand, there is an opinion that symbolism is something antediluvian, obsolete, which in our time no serious person will do; but there is another extreme: symbolism is the key to understanding the spiritual world. A person needs symbols in order to be able to enter the inexpressible into the area of ​​the felt, tangible, and then meaningfully understand it. It is easy to prove that the symbolic penetrates even into the realm of ordinary spoken language. But it is also present in the slogans and signs of politics, in the allegoricalness of the religious spiritual world, in the icons and ciphers of foreign and prehistoric cultures, in legal laws and objects of art, in poetry and historical images - wherever the "bearer of meaning" conveys something, beyond its banal external form. A wedding ring, a cross, a national flag, traffic lights, a red rose, black mourning clothes, candles on a festive table - countless objects, gestures, mental images and turns of speech connect thoughts with meaning carriers. The ever-increasing abstraction and rationalization of the world of ideas seems to dry up the once almost boundless stream of images.

Each person has his own mythology and elevates certain personalities (real or mythical) to the level of a symbol, therefore, a different, multilateral understanding of symbols by people gives a huge range of various explanations for certain images. The huge wealth of symbols of foreign cultures attracts people's attention. In order to prove the prevalence of universal human images and to explain their meanings, one has to continually turn to the foundations of various figurative worlds. one

The nature of the studied material on the topic of “symbolism” with its various nuances is such that the categories “symbol”, “allegory”, “metaphor”, “sign”, “emblem” and “sign” distinguished in theory are not so easy to separate from each other in practice. friend. It is important that many symbols do not have an unambiguous explanation, but due to their traditional nature, they have a dual meaning. So, for example, not always and not everywhere fire, as a symbolically contradictory element, is an element that warms and illuminates - sometimes it symbolizes a sign that can cause pain and death; and the heart is not always intended for love - after all, real symbols, even at different levels of knowledge, “report information” that is different, but always significant. Sometimes you can justify the reasons why a certain symbol is interpreted in this way, and not otherwise. But more often a person interprets subjectively, that is, in the image and likeness of the divine world order he understands. He imagines himself surrounded by signals that enable him to consciously submit to the great holy order. Anyone who evaluates the images of previous eras from today's point of view and only registers signs of imperfect logic and insufficiency in the knowledge of nature will pass by the various attitudes of symbolic thinking.

If the exact sciences can be designated as a monological form of knowledge (the intellect contemplates a thing and speaks about it), then the interpretation of a symbol is essentially a dialogic form of knowledge: the meaning of a symbol really exists only within human communication, within a situation of dialogue, outside of which only the empty form of a symbol can be observed. . Studying a symbol, we not only disassemble and consider it as an object, but at the same time allow its creator to appeal to us, to be a partner in our mental work. If the thing only allows it to be examined, then the symbol itself “looks” at us (see the words of R.M. Rilke in the verse “The Archaic Torso of Apollo”: “There is not a single place here that would not see you. You must change his life”; moreover, the fact that we are talking about a headless and therefore eyeless torso deepens the metaphor, depriving it of superficial visibility!).
That the study of symbols can cause polemical controversy is shown by an excerpt from one of the anti-Masonic books (Friedrich Michtl. "World Freemasonry"). It talks about how much conscious work on symbols slows down thinking. Whoever communicates with the world in this way, the book says, is not able to “give free and natural space to the wealth of thoughts, again and again thinking is interrupted by custom, which has become second nature” 2 .

It will be necessary to bring to this work the words of Manfred Lurker about the very concept of “symbol”, which quite clearly express what will be discussed in this work: “The meaning of a symbol lies not in itself, but points to something more ... A symbol is mystery and revelation at the same time.”

It must be admitted that some symbols can also play a negative role in the life of both individuals and entire societies. Not only in the Aztec state, such ritual symbols as “sacrificial blood, heart, sun”, led to the terrible destruction of people, but also other symbols in the era of the twentieth century, which is closer to us (the height of wars and contradictions all over the world), for example, “banner, leader, blood and earth, swastika, fire.”

1 Bidderman G. Encyclopedia of symbols: Per. with him. / Common Ed. And foreword. Svenitskaya I. S., Republic Publishing House, 1996.

2 This is also mentioned in the book of the neuropathologist M. Ludendorff, dedicated to induced insanity due to occult teachings, section "Artificial dementia due to symbolism."

But still, it is undeniable that countless ancient symbolic ideas belong to the most valuable treasures of mankind and brought to life great creations in the history of culture - pyramids, cathedrals, temples, symphonies, poems, paintings, religious rites, holidays, dances. We must come to terms with the fact that the symbols fixed in the deepest layers of human consciousness have a certain independent power and, thanks to a kind of feedback, influence their creators. The responsibility of the man who is aware of this fact is that he has the opportunity to choose from the richness of the symbols of all history that which is truly valuable.

The semantic structure of the symbol is multi-layered and is designed for the active internal work of the perceiver.

In a general sense, the treatment of symbols is dual: it can open access to the spiritual wealth of past eras and revive it to life, but with an immoral treatment of this “world of ciphers”, it can bind a person, make him chained, dependent, simply turn him into a functioning robot. .

ChapterI. SYMBOLISM AS A PROPERTY OF CULTURE

1.2 Studies of culture as a symbolic system.

From the point of view of culture, any object or process can be considered in which we are interested not only in its applied significance, but also in the way of interpretation and value coloring of the world hidden in it, which implies a non-utilitarian choice. There are over 200 definitions of culture. In this paper, the concept of culture will be closely considered with the concept of symbolism.

A large place among the definitions of culture is occupied by definitions that interpret culture in terms of symbolic behavior, considering it as "the ability to create symbols" and, "the ability to teach and learn", the ability to create a special, symbolic language. The interest of the 20th century in linguistic problems was also reflected in cultural anthropology, including evolutionism. L. White, a prominent representative of American cultural anthropology, formulated the concept of culture as follows: “culture is a set of phenomena and actions (modes of behavior), objects (tools and things made with their help), ideas (beliefs, knowledge), feelings (relationships values) that depend on the use of symbols. Culture is a symbolic, continuous, cumulative and progressive process." Here we have the broadest definition, where the facts of culture are organized, ordered by indicating their dependence on symbolic activity, which ultimately limits both the content of culture and the field of research interest.

The symbolic definitions of culture can be divided into two groups. The first group includes those that do not go beyond the objectivist-oriented ethnology. In the concept of L. White, a symbol is interpreted as "an object that has a value or meaning given by those who use it." The symbolic world is an objective world that has significance for a person. Here we see a kind of attempt to introduce the human dimension into ethnology: the world created by man is not indifferent to him; designating it, a person thereby turns out to be able to preserve it and pass it on by inheritance. In those cases where the sign and symbol are associated primarily with the verbal, verbal activity of a person, they somehow refer to the objective world, the world of things. This is given by the positivist attitude and ethnological tradition. No matter how a sign and a symbol are defined (in the concept of L. White they are one and the same), the world of signs and symbols, as well as the world of meanings behind them, is a stable world that guarantees a person the possibility of orientation in it.

The symbol as an element and instrument of culture becomes a special subject of attention and scientific research in connection with the formation of a new humanitarian discipline - cultural studies. In some cases, culture as a whole is interpreted as a symbolic reality (as in Cassirer’s “philosophy of symbolic forms”), in others, a methodology is developed to “decipher” the meaning that was unconsciously given to an object of culture, in the third, a symbol is studied as a consciously created message of culture, and in this case, both the poetics of its creation and the mechanisms of its perception are of interest. The most problematic is the understanding of cultural symbols that are devoid of direct emblematicity: such can be an artistic image, a myth, a religious or political act, a ritual, a custom, etc.

1.2 The language of culture as a language of symbols. Variety of sign systems.

The language of culture in the broad sense of this concept refers to those means, signs, symbols, texts that allow people to enter into communicative relations with each other, to navigate in the space of culture. The language of culture is a universal form of comprehension of reality, in which all newly emerging or already existing representations, perceptions, concepts, images and other similar semantic constructions (meaning carriers) are organized.

The main structural unit of the language of culture, from the point of view of semiotics, are sign systems. A sign is a materialized carrier of the image of an object, limited by its functional purpose. A sign is a material, sensually perceived object (phenomenon, action), which acts as a representative of another object, property or relationship. There are linguistic and non-linguistic signs; the latter are divided into signs-copies, signs-signs and signs-symbols; understanding signs is impossible without clarifying their meaning.

Language is formed where the sign is consciously separated from the representation and begins to function as a representative (representative) of this representation, its spokesman.
The signs that make up each of the languages ​​of culture and are intended to express ideas and experiences differ both in their origin and in the degree of similarity of what they represent. Cultural researchers distinguish 5 main sign systems: natural, functional, conventional, verbal, notation systems.
For consideration, let's take the most basic sign systems:

I. Natural. Natural signs are understood as things and natural phenomena in the case when they are pointed to some other objects or phenomena and are considered as a carrier of information about them. In other words, these are signs-signs, for example, smoke is a sign of fire.
II.Functional signs are also signs. But, as a rule, these are things and phenomena that have a direct pragmatic purpose, but are included in human activity in addition to their immediate functions, they still receive a symbolic function, that is, they provide some information about things and phenomena.

III.Convention signs. If for natural and functional signs the sign function is a side function and is performed by them, as it were, “in combination”, then for conventional signs this function is the main one.

3 Baudouin Descharnet, Luc Nefontaine, Symbol, Le Symbole; "University Library"

Publishers: AST, Astrel, 2007

Conventional signs are signs in the full sense of the word. Their meanings are set not by the objects and processes they inform about, but by agreements between people.

There are 4 types of conventional signs: signals, indices, images and, directly, the symbols themselves.

A symbol is like a sign that is associated with the objectivity it denotes in such a way that the meaning of the sign and its object are represented only by the sign itself and are revealed only through its interpretation.

Along with individual conventional signs introduced for one reason or another, in the course of the development of culture, various systems of conventional signs arise. For example, heraldry, a system of traffic signs, ceremonial systems associated with the performance of various kinds of rituals (wedding, funeral, festive, religious and religious, taking office - coronation, inauguration, etc.). We can say that each area of ​​socio-cultural life has its own symbolic system. 4

1.3 Symbol as an ideological and figurative construction. Symbolism as one of the forms of culture (and philosophy), changing over the centuries.

Symbol(Greek σύμβολον - a sign, an identifying sign) - a universal category of aesthetics, best of all amenable to disclosure through comparison with adjacent categories of the image, on the one hand, and the sign, on the other. Taking the words broadly, we can say that a symbol is an image taken in the aspect of its symbolism, and that it is a sign endowed with all the organicity of myth and the inexhaustible polysemy of the image. Every symbol is an image (and every image is, at least to some extent, a symbol); but if the category of an image presupposes an objective identity to itself, then the category of a symbol focuses on the other side of the same essence - on the image going beyond its own limits, on the presence of some meaning intimately merged with the image, but not identical to it. The objective image and the deep meaning appear in the structure of the symbol as two poles, one inconceivable without the other (because the meaning loses its appearance outside the image, and the image breaks up into its components outside the meaning), but also divorced from each other and generating tension between themselves, in which and is the essence of the symbol. Passing into a symbol, the image becomes "transparent"; the meaning “shines through” through it, being given precisely as a semantic depth, a semantic perspective that requires a difficult “entry” into oneself. 5

I. Forms of material culture (clothes, customs, architecture, etc.) may change over time, but the symbols continue to be very carefully reproduced from generation to generation. The interpretation of symbols may undergo changes, the meaning originally laid down in them may be lost, but their form does not change or almost does not change. Here you can see the fact that information transmitted by people in verbal form can be very significantly distorted due to the subjectivity of its understanding by a particular person, but the form (material sign) is simpler, visual, and therefore more stable for human perception than content.

4 Bely A. Symbolism as a worldview / Comp., entry. Art. and approx. L.A. Sugay. - M.: Respublika, 1994.

5 Averintsev S.S. Sophia-Logos. Vocabulary. 2nd, rev. ed. - K .: Spirit and Literature, 2001, p. 155-161.

Researchers quite well explain similar drawings or symbols that appear among different peoples, the image of which was associated with visually perceived objects of the real world (plants, animals, etc.). For example, the “curl” (“spiral”) was originally associated with rare plant sprouts, later transformed into an animal endowed with a similar shape - a ram (and other animals with spiral horns), which became, because of this, a sacred animal.

Therefore, symbols often lose their primary meaning and can be interpreted differently, often acquiring the opposite meaning.

So the question of the nature of the symbol arises. Many scientists try to explain it on the basis of individual human psychology. So, there is a version according to which similar living conditions give rise to identical symbolic forms, but in this case, the interpretation of these forms should be the same for different peoples. However, this does not happen. In addition, there is a whole group of graphic symbols, the origin of which is unclear and cannot be explained on the basis of human psychology alone. Such is the form of abstract symbols having geometric outlines, which is not motivated by objects of the real world and therefore cannot be explained by the principle of association.

The symbol as an ideological and figurative construction has a huge semantic richness. The abstractness of its form allows, in a collapsed form, at the same time, to present a special meaning that creates a perspective for the endless development of society; knowledge that can be perceived and applied in expanded form.

In art, symbols are often placed in traditional, canonical systems, but their esoteric (secret) meaning may be lost. In this case, there is an inversion of meanings - an incorrect "reading" of the meaning and, as a result of this, its distortion. However, most often in art there are symbols endowed with an exoteric (generally accessible) meaning, although the exit to a more complex level of comprehension of their content is not excluded.

The main feature of any symbol, or type of symbol, is to indicate the "unknown" by presenting it in a clear and understandable construction, but the manifestation of this feature will always be different, as it depends on the area of ​​\u200b\u200bfunctioning of the symbol. Thus, two large groups of symbols can be distinguished: exoteric, containing a secret meaning inaccessible to the uninitiated, and exoteric, having generally accessible meanings.

So, according to the "theory of the symbol" A.F. Losev identified eight exoteric symbols:

1) Scientific symbols are symbols that are fundamentally devoid of imagery. A scientific symbol only then becomes a symbol (and not just a meaning) when it has a high degree of generality (generalization).

2) Philosophical symbols - differ from scientific symbols only in the degree of their ultimate generalization. Philosophical concepts are not only abstract images, but it is also a way of understanding reality, associated with its analysis and identification of patterns.

3) Artistic symbols - any art, even the most realistic, cannot do without the construction of "symbolic imagery". The artistic image here is also a generalization (for example, the “image” of motherhood in Renaissance portraits goes back to the archetype of the Mother of God).

4) Mythological symbols - convey a certain content (or knowledge) in a mythopoetic or allegorical form.

5) Religious symbols - reflect transcendental images (one of the most important terms of Kant's philosophy) and esoteric knowledge. Therefore, the religious myth is always magical and mysterious.

6) Humanly expressive symbols - are associated with the moral aspects of society, the rules of conduct.

7) Ideological and motivating symbols are semantic structures expressed either in visual form (for example, graphic and “heraldic” signs – a star, a swastika, a cross, etc.). These symbols define the principle of social action and the method of its implementation.

8) External-technical symbols - are the principle of the implementation of an infinite series of actions, in their content they are symbols-signs that can be divided into two large subgroups: imitative and neutral.

The more and deeper nature and society are perceived and studied by man, the more the reality surrounding us is filled with various symbols. Only in the case when a symbol, as an ideological and figurative construction, allows, due to its abstractness, to represent special knowledge available to society, a symbol or an artistic symbolic image acquires the meaning of universality and has a special stability and power of influence.

II. There is a well-known judgment in history and philosophy that a person lives not only in the real world, but also in the symbolic one. Symbolic culture anticipates the experience of everyone, and even disagreement with its requirements does not cancel this dependence.

Already at the origins of philosophical thinking, we find the art of constructing symbols, in cases where the concept collides with the transcendent - that is, beyond in relation to any particular area, to the world as a whole.

The specific differences of a symbol from all other sign tropes, such as concepts, myths, signs, are the following functions: 1) the ability of a symbol to endlessly reveal its content in the process of correlation with its objectivity while maintaining and “irrevocable” this symbolic form; 2) the ability of a symbol to establish communication, which, in turn, creates (actually or potentially) a community of “initiates”, i.e. subjects who are in the field of action and relative intelligibility of symbols (for example, a church, a direction in art, an esoteric circle, cultural ritual) 3) the steady attraction of the symbol to the ascent from the given "parts" to the actual and supposed "whole". The symbol in this case is the meeting point of that which is in itself incompatible. 6

As for the concept of "symbol" itself, for example, in Ancient Greece in its primary meaning it was extremely specific: an identifying sign, evidence of the unity of two disparate parts, by combining which one could obtain the original "whole", and thus, through concrete materiality evidence, confirm internal involvement.

The European Middle Ages makes the symbol one of the general cultural principles, however, the emblematic possibilities of the symbol become the subject of reflection and cultivation in the first place, while its own specificity is revealed only in the creative practice of the cultural rise of the 13th - early 14th centuries. The situation did not change significantly until the last quarter of the 18th century: the Renaissance, Mannerism, Baroque, Enlightenment are rich in their symbolic artistic and religious worlds, but at the same time they see nothing in the symbol but a means of allegory and "heraldic" detection.

6 Surina MO, Color and Symbol in Art, Design and Architecture. – Ed. 2nd, with changes. And extra. - M .: ICC "Mart", Rostov n / D.

A new turn of the theme arises in connection with the Kantian doctrine of the imagination (18th century). Here the symbol for the first time acquires the status of a special way of spiritual exploration of reality. At the same time, Goethe comes to the intuition of the "proto-phenomenon", that is, a kind of objective symbol, born of organic nature. In the philosophy of German romanticism (Novalis, F. Schlegel, Schelling, Kreutzer, etc.), a whole philosophy of the symbol unfolds, revealing its specificity in connection with the main themes of romantic aesthetics (creativity, genius, irony). A version close to romanticism is given by Schopenhauer, who depicts the world as a symbol of an empty will in ideas and ideas. Kierkegaard's concept of "indirect messages" can be considered as a variant of the romantic theme of the symbol.

In the second half of the 19th century comprehension of the problem of the symbol takes on philosophizing art: myth comes to music and literature, interpreted not as a formal shell of meaning, but as a meaning-generating element (R. Wagner, a practitioner and theorist, is most indicative). Since the 1880s symbolism as an artistic movement and theoretical self-justification, absorbing both the romantic heritage and the ideas of the philosophy of life, creates a new philosophy of the symbol, which claims to be a total mythologization of not only creativity, but also the life of the creative subject.

Russian branch of symbolism of the late 19th - early. 20th century gives abundant philosophical fruits: in the constructions of V. S. Solovyov, Andrei Bely, Vyach. I. Ivanov, P. A. Florensky, A. F. Losev, symbolism receives a systematic multi-variant philosophical justification.

Currents of Western thought in the 20th century. represent several models of symbol understanding. Cassirer's "Philosophy of Symbolic Forms" makes the symbol a universal way of explaining spiritual reality. The philosophy of language reveals the symbolic potential that allows natural language to play the role of a world-creating force. The structuralism of Levi-Strauss explores the mechanisms of the functioning of the symbol in the primitive unconscious, without avoiding projections on modern culture.

The latest philosophy of the West preserves the problematic of the symbol in transformed forms to the extent that the task of delimiting and evaluating various types of sign activity of a person and culture remains relevant. 7

Symbols are directly reflected in the images of literature, music, theater. Their primary mythological content is fixed by a wide range of humanitarian disciplines: mythology, ethnography, literary criticism.

7 From the article by A. L. Dobrokhotov, Great Encyclopedia of Cyril and Methodius, Moscow, 2003.

ChapterII. Geometric signs as symbols of different cultures

2.1 The simplest geometric symbols

When the Initiates of the past painted

vertical or horizontal line

circle or dot, and then combined from

them cross, triangle, square, swastika

pentagram, hexagram, or snake,

biting its own tail... they invested

in each figure eternal knowledge.

O.M. Ivanhov

Almost all geometric symbols consist of combinations of several geometric elements - simple components, each of which at the same time has its own special meaning, contributing to the overall composition. The simplest of these magical "particles"-symbols are a point, varieties of arcs, a circle, as well as a square, a rectangle and a triangle.

In fact, the meanings of these seemingly simple figures are quite complex.

2.1.1 Point

In mystical representations, the dot is a symbol of the center, the source of life, a symbol of primary creative energy, which is sometimes presented as so concentrated that only something intangible, such as a hole, can reflect it. The ancient symbolism of the point as an extremely compressed energy, widespread in mystical literature, is extremely close to modern physical and astronomical theories about the origin of the Universe.

In order for the energy to come out of the primary state and manifest itself, it needs a point of separation. The point is dimensionless and has not yet gone out of unity, but it is necessary for manifestation. Since the point consists of a single factor, it carries the number of unity - 1.

The point is the quintessence (basis) of all signs. eight

2.1.2. A circle

Circle 9 is a symbol that has an ancient mythological basis. Besides the point (center), the circle is the only geometric figure, not eternity. This is a symbol of completeness, completeness, which can contain the idea of ​​both constancy and dynamism.

Since the circle (and sphere) is a figure that has no beginning and no end, it is the most important and universal of all geometric forms in mystical teachings.

8 Encyclopedia of symbols / comp. V.M. Roshal. – M.: AST; "Publishing house" Owl ", 2005

9 See Appendix 3, illustrations 1, 2, comments to illustrations 1, 2.

And since it can stand for other important symbols (wheel, disk, ring, dial, sun, moon, zodiac), its symbolism is rather difficult to define.

So, in many cases, the mythological tradition represents the cosmos as a ball (graphically it is a circle) in certain symbolic variants: a turtle, a disk, etc.

In magical symbolism, the circle means spiritual forces. Being without beginning or end. The sleeping eye of God, the sphere of soul life, where souls are in an intermediate state. In the teachings of magic, the circle also has the function of protecting from evil spirits, which, during the spell ceremonies, arises around the magician and cannot be stepped over. ten

According to the views of Platonists and Neoplatonists, the circle is the most perfect form, the embodiment of God and the unlimited center of the cosmos. For the ancients, the boundless system was seen as circular: in their opinion, all the planets looked like this, including the alleged earth disk surrounded by water, they were also convinced of this by the cyclical processes and the change of seasons. Symbolic meanings and functions when using the circle to measure time (sundial) and space (the main astrological and astronomical starting points) were an inseparable unity.

Celestial symbolism and belief in heavenly power underlay primitive rituals and early architecture around the world: circular dances and ritual round dances around a fire, an altar, or an idol; the pipe of peace passed around the circle among the Indians of North America, the round shapes of yurts, tents and camps of nomadic peoples; whirling shamans, circular structure of megalithic signs and structures (Reconstruction of a megalithic sanctuary. Stonehenge 11, Southern England, c. 1800 BC) in the Neolithic period.

The circle had both protective and divine meanings, as, for example, among the Celts, and this meaning is still preserved in folklore, but in new, modern forms: remember the mysterious rings on farmers' fields and flying saucers. The circle was also seen as an object of harmony, like the Round Table in the legend of King Arthur, or the expression "witchy circle of acquaintances" widely used in modern English idioms. In many images, the circle was given dynamism with the help of rays, wings, flames, which is especially noticeable in Sumerian, ancient Egyptian and Mexican iconography. In these cases, the circle symbolizes the power of the sun or the creative, fruitful cosmic forces. The 12 concentric circles can represent celestial hierarchies (such as the angelic choirs symbolizing Heaven in Renaissance art), the circles of hell, or, in Zen Buddhism, levels of spiritual development.

In the Christian tradition, three circles depict the Divine Trinity, the boundaries of time, the elements, the periods of the sun and the phases of the moon. The circle can be a masculine sign (such as the sun) or feminine (the mother's womb). The circle (feminine) around the cross (masculine) is a symbol of the unity of opposites in Egypt, also found in Northern Europe, China and the Middle East. The Chinese yin-yang symbol, 13 representing the interdependence of male and female, uses a circle divided by an S-shaped line into two colors, each with a small circle of the opposite color in the center.

The dot in the circle is the astrological symbol of the sun and the alchemical symbol of gold. A circle with a dot in the center is the open eye of God, a symbol of the Universe, a projection of the image of the world in plan. The point in the center is like a peak that collects everything, it is from the peak that one can see the unity of life in all its manifestations.

10 Encyclopedia of symbols / comp. V.M. Roshal. – M.: AST; "Publishing house" Owl ", 2005

11 See Appendix 3, illustration 4, commentary to ill. 4.

12 See Appendix 3, illustration 5, commentary to ill. 5.

13 See Appendix 3, illustration 6, commentary to ill. 6

Between the point-center and the periphery of the circle there is a continuous exchange, and this exchange creates life on the entire area of ​​the circle.

This figure can be found everywhere in nature: the solar system; a cell consisting of a nucleus and a peripheral membrane; atom…

The symbolic contrast to the circle is the square, which, in contrast to it, denotes the earthly world and the material.

According to K. Jung, a circle combined with a square is a symbol of the connection between the soul or "I" (circle) and the body or reality (square). Remarkable is the coincidence of this interpretation with the Buddhist tradition, where the mandala, on which the circle is inscribed in a square, symbolizes the transition from the material world to the spiritual one. In the western and eastern traditions, a square inscribed in a circle represents the sky enclosing the earth. In architectural structures based on a square, cross or rectangle - for example, Romanesque churches or some pagan temples - round vaults and domes carry heavenly symbolism. The proverbial task of “squaring the circle”, the transformation of a square (by purely geometrical means) into a circle of equal area, means the effort of a person to make it impossible for his own essence to pass into the essence of a deity, i.e. morally rise to the divine. This problem, usually unsolvable by geometric means, often emerging during the Renaissance as an allegory of the human desire for "deification", also plays a large role in alchemical symbolism.

In contrast to this, in the Kabbalistic tradition, a circle inscribed in a square is a symbol of the "spark of God" in a mortal body, a symbol of divine "flicker" inside the material shell.

Naturally, the circle as a symbol is not limited to highly developed cultures; among various Indian groups, it “symbolizes, for example, the cosmic gesture of the “Great Spirit”, since the “path of the Moon” and (from the point of view of an earthly observer) the “path of the Sun” and the “movement of the stars”, as well as natural development create round shapes ” (Nicksdorff from under Sterk, 1987).

In Zen Buddhism, according to the fundamental principle, the circle means enlightenment, the perfection of man. The Chinese yin-yang symbol in a circle (t'ai-chi, originally one) contains duality. In Europe, the perception of cosmic spheres in a circular projection, entering into each other in the form of shells, takes over the worldview of the Middle Ages and is poetically presented in Dante's "Divine Comedy" in the form of circles of hell; the hierarchy of angels as guardians of the spheres takes possession of this vast world order. fourteen

All of the above allows us to conclude that the circle, as a symbol that has an ancient mythological basis, is also the beginning of the entire universe.

2.2 Basic symbols of geometry

Geometric shapes are similar

framework of reality, while

images still contain, so to speak,

some flesh, skin and muscle.

O. M. Aivankhov

Sacred geometry is the study of the forms that underlie our being and testify to the divine order in our reality.

14 from A Dictionary of Symbols by Jack Tresidder.

We can trace this order from the invisible atom to the gigantic, infinitely distant stars. Sacred symbols have a universal cosmic nature, they are stable and are passed on from generation to generation without changes.

sacred geometry

"The main goal of all exploration of the external world should be the discovery of rational order and harmony, which God sent down to the world and revealed to us in the language of mathematics." 15

In contrast to the modern isolation of various branches of knowledge, ancient societies recognized the universal unity of all sciences, the unity of harmony and beauty, which is expressed in the inseparability of science, religion, art, mythology, mathematics, linguistics, architecture, trade and politics. All these are different ways of considering the all-encompassing unity and course of the world process, as well as an attempt to establish a state of equilibrium with it. This unity is best understood in terms of sacred geometry.

Sacred geometry is the way of knowing the Universe and man. Pythagoras referred to sacred geometry as "the most secret science of God." It explores not only the proportions and relationships of forms, which are the matrices of the laws and structures of the universe, but also the dynamic processes of life, reflecting the interaction of energies and different planes of consciousness. She embodied the discoveries of many initiatory schools and metaphysical traditions. Harmoniously combining various types of art and science, the insights of mystics and the principles of quantum physics, sacred geometry proves that form is a concentration of psychic energy, a generator of power, a gate to other spaces.

The true geometer does not study pure geometry because it is useful: he studies it because he admires its beauty. For some people, sacred geometry is the study of ancient temples, places and the copying of cosmic forms made with the help of special equipment from satellites. For others, it is a means of going beyond the physical body in order to travel to other dimensions. But in fact, the science and art of sacred geometry is a means of becoming an individual, a way of knowing the Divine and a method of understanding earthly experience.

Using the language of sacred geometry, the great sages left important messages for us, embodied in architectural, musical and pictorial works, as well as forming the basis of mystery performances. "Truly the visible is the image of the invisible." Having learned to decipher these messages, one can find many keys to understanding being, since geometric images are interconnected with all elements of existence.

Geometry is an amazing science. She does not submit to private views, hardly recognizes new authorities, offers amazingly accurate answers to many things and is pure beauty. Nature itself enjoys its achievements; examples of this are everywhere, from the spirals of shells and small daisy flowers to the symmetry of hexagonal honeycombs and the golden proportions of natural stone formations. "Nature shows that it is equally rich, equally inexhaustible in the product of both the most outstanding and the most insignificant creations" (I. Kant). Sacred geometry predetermines the forms of molecules and crystals that make up our bodies and the Cosmos. In fact, it is the key to the creation and understanding of the universe.

In ancient initiatory practices, geometry was referred to as "the first and noblest of the sciences."

15 I. Kepler.

The term sacred geometry is used by archaeologists, anthropologists, philosophers, culturologists and people whose work is connected with spiritual activity. It is used to cover the system of religious, philosophical and spiritual archetypes that are observed in various cultures throughout human history and are somehow connected with geometric views regarding the structure of the Universe and man. This term covers all Pythagorean and Neoplatonic geometry.

In ancient Greece, the study of the essence of beauty, the mystery of beauty, based on certain geometric patterns, formed into a separate branch of science, aesthetics, which among ancient philosophers was inextricably linked with cosmology. The ancient Greeks had a geometric vision of universal order. They perceived the universe as a vast expanse of diverse interconnected elements.

Many scientists, for example, P. Dirac and M. Kline, noted the inability of modern mathematics to describe the world around us and felt the need to create a new mathematics. Such a new mathematics (although existing for many millennia; new in the sense of methodology) is sacred geometry. Blavatsky also noted: "For the philosophers-Kabbalists and Hermetic philosophers, everything in nature is represented in a triune aspect; everything is plural and trinity in unity, and can be symbolically represented by various geometric figures."

There are several examples of the action of sacred geometry in different eras and cultures.

1) The ancient Greeks attributed various properties to the Platonic solids and certain geometrically derived relationships, endowing them with a special meaning. "God geometrizes," said Plato. For example, the cube symbolized kingship and earthly foundations, while the golden ratio was considered a dynamic principle that embodies the highest wisdom. Thus, a building dedicated to a deified ruler could bear traces of a cube, while a temple dedicated to a heavenly god was built in such a way that the golden ratio lay at its base.

2) When the Hindus (ancient and modern) were going to erect any kind of religious building - from a small roadside chapel to a monumental temple - they first executed a simple geometric drawing on the ground, properly determining the directions to the east and west and building a square on their basis. This is a simple procedure at the level of a school geometry course. After that, the entire building is erected on the resulting diagram. Geometric calculations are accompanied by chants and prayers. All this is done with the aim of activating the radiating properties of the structure and converting energy using the architectural properties of the building. The Christian religion uses the cross as its main symbol; in geometric terms, in the Middle Ages, it appeared in the form of an unfolded cube (cf. with an example from ancient Greece, where the cube was correlated with kingship). Many Gothic cathedrals were built using calculations derived from the exact geometry of the cube and the double cube. This tradition continues in modern Christian churches.

3) The ancient Egyptians discovered that regular polygons could be enlarged with a constant aspect ratio by adding a strictly marked area (which would later be called the gnomon by the Greeks). The Egyptians associated the concept of a continuing relationship of expansion of a rectangular area with the god Osiris, who is therefore often seen in ancient Egyptian frescoes, seated on a square throne (square = kingship). At the base of the throne, a square with an L-shaped gnomon is clearly visible, although usually the construction was carried out in such a way as to hide the gnomon from the eyes of the uninitiated.

4) The spirals on the Ionic pillars of the ancient Greek temples were placed on the principle of a rotating rectangle - this is the method of creating a logarithmic spiral. The use of such spirals in Greek temple architecture indicates that the architects deliberately used the principles of sacred geometry in their creations. The idea of ​​spatial ordering in the form of a spiral also excites modern architects. The technical mobility and flexibility of such a system makes it possible to adequately respond to the dynamics of the development of society.

5) In medieval semantic geometry, the properties of geometric figures were correlated with the virtues of heraldry and etiquette.

These examples can be given ad infinitum. One of the most startling ideas that permeates the sacred teachings of all ancient civilizations is that the universe exists as a harmonious and beautiful whole, whether we feel it or not. The basis of beauty is harmony. The Egyptian goddess Maat was the embodiment of the principle of the natural order of things, proportional measure and balance as the eternal truth of nature. The Greeks, who studied with the Egyptians, associated with civilization the word cosmos, literally translated as "embroidery" and expressing the harmony and beauty inherent in the world.

To comprehend cosmic harmony, one should be based on the following primary sources of awareness of the universal patterns of a harmonious whole:

Observations of nature, its cycles, rhythms and architecture;
-study of mathematical samples of numbers in geometry;
- direct revelation.

So, the practical methods of sacred geometry:

They keep us in a state of awareness of who we are, where we came from and why we are here now;

They teach deep reflection on the mysteries of being and ways of gaining spiritual perfection;

They turn to ancient and modern knowledge about the spiritual worlds, which provide an opportunity to establish balance on all planes of existence;

They endow the soul with responsibility for their actions, compassion and love.

Sacred geometry combines the wisdom of many mystical schools, both existing long before our era, and modern, linking esotericism with the latest achievements of quantum physics. This amazing science recognizes all the typical forms of manifestation of higher knowledge, considering them as cups containing information about the manifested world and about the place of man in it. Everything is energy, vibration, harmony and dissonance of frequency; everything is geometry.

The science of sacred geometry shows the quality of the connection between unique and individual differences and demonstrates how diverse elements can be organized into a whole - while maintaining their individuality. It combines the physical, material aspects of Creation with the spiritual essence. This is the interaction of the visible and the invisible, the manifested and the non-obvious, the finite and the infinite, the mundane and the sublime. Sacred geometry has played and continues to play a major role in the art, architecture and philosophy of numerous cultures for thousands of years.

Sacred geometric forms, these primordial bodies are given to mankind in order to transmit true knowledge about God and the Cosmos with their help. Geometric style as a way of understanding being covers all spheres of a person's worldview. sixteen

16 Geometry, Sacred Geometry - the Key to Harmony, M., Ed. Rosman, 1998

2.2.1. Cross

The cross as a symbol of space

The common symbol of humanity is the cross. The symbol of the cross can be found in the most ancient religions, among the most ancient civilizations: in Mesopotamia, Egypt, China, etc. Who invented it? Nobody - because it exists in nature.

This is an ancient universal symbol and, above all, a symbol of the Cosmos, reduced to its simplest form.

The cross is the center of the world; symbol of fire and light; a symbol of the sacred center of the Earth, where the earthly horizontal intersects with the celestial vertical - and this is the point of communication between Heaven and Earth. Four main axes emanate from the center of the cross, symbolizing the four cardinal points: north, south, west, east. The cosmic axis passes through the zenith and Nadir points, symbolizing the cosmic Tree of Life. The north-south direction is the solstice axis, and the east-west direction is the equinox axis.

The central (fifth) point of the cross unites the four main elements of matter. They correspond to four powerful exalted forms. They stand in world space on the four cardinal points, forming a cosmic cross. They lead the world processes, manage them and are the servants of the One God, who is life - the Sun. These are the archangels. They change during each cosmic day, guided and inspired by the Spirit of the Sun. They are the original force, which manifests itself in the Cosmos and in the human soul in the form of three forces: thought, feeling (emotions) and will.

In the north stands the archangel Uriel (Uriel, Sandalion), whose light has a bluish radiance, and it is weaker than that of others. He has a lofty stern appearance.

In the south direction stands the archangel, who is the ruler of solar development - Raphael (Raphael). He has an exalted, full of power appearance.

In the west is the one whose being shines in the silvery light. He has an exalted, loving appearance; His name is Gabriel (Gabriel).

The fourth angel radiates his pink and golden light from the east. His name is Michael; he has an exalted victorious appearance, bearing in itself the properties of the other three.

Each of the archangels is associated with one of the members of the human being. During the spiritual transformation, the power of Michael unites with the three already existing, and thanks to him they are illuminated by a higher power.

In addition, the cross is a symbol of four mighty air currents - winds. The northern end of the cross symbolizes the north wind, the most powerful, all-conquering; as well as head and intellect. The southern end is the south wind; fire and feeling, as well as melting and burning. The western end symbolizes the soft west wind from the land of the spirits; the breath of death and the journey into the unknown that awaits everyone. The east end, respectively, the east wind, the heart is the source of love and life.

The cross also acts as a symbol of the four elements - air, earth, water and fire. The arrangement of any elements in the form of a cross helps to balance the natural elements, normalizes their work.

The cross is the main symbol of the connection between micro- and macrocosm. It symbolizes the combination of spirit and mother, the involvement of the spirit (expressed by the vertical line) in time (expressed by the horizontal line).

A person standing with his hands outstretched to the side is also a cross, it is an image of the microcosm, a reflection of the vast Universe in each individual. The cross personifies the universal archetypal man, capable of infinite and harmonious expansion both horizontally and vertically. The vertical line is heavenly, spiritual and intellectual, positive, active, masculine. Horizontal line - Earthy, rational, passive, negative, feminine.

The cross balances the physical energy of a person with emotional, mental and spiritual energies. Man also contains the cross at the physiological level. The movement of his hands is cross-shaped: the movement of the right hand is associated with the left hemisphere of the brain, the left hand with the right.

The cross contains two principles: male and female, meeting to work together in the universe. One must learn to make the masculine and feminine principles work together in oneself, active and passive, separating and absorbing, spirit and matter, man and woman, intellect and heart, wisdom and love.

Able to expand in any direction, the cross symbolizes eternal life.

Thus, the cross is a cosmic symbol that should be studied and treated with the greatest respect. Wears a cross - good, but subject to understanding of its meaning. Any thing, any creature that we do not know how to handle and be in harmony with, can be fatal to us. If a person does not have anything good in his mind or in his heart, even the cross will not be able to transform or protect him. 17

Cross shapes

The forms of the cross are various. They differ in the number of crossbars and the number of ends of the cross (from three to six, as in Chaldea and Israel, or even eight), and proportions.

Greek, or square cross 18

Greek, or square cross: the horizontal crossbar is located in the middle of the vertical one; cross of Saint George.

This is a cross of the simplest form with ends of equal length. In early Christianity, the Greek cross symbolized Christ.

On the national flag of Greece, this cross, white on a blue background, first appeared in 1820, symbolizing the struggle against the rule of the Muslim Turks.

It is also a symbol of secular, earthly power, but received from God.

Square crosses based on the Greek cross

These include the following forms of the cross: capitate, club, star, Mighty crosses, the cross of the crusaders, the Teutonic cross, the cross-hammer, the cross of illumination.

crossed cross(Holy, Germanic) Gnostics have a sign of the four sacraments (philosophical beginnings).

crusader cross represents five golden crosses on a silver background. This cross was adopted as a coat of arms by the Norman conqueror Gottfried of Bouillon, who became the Guardian of the Holy Tomb and the first ruler of Jerusalem after its liberation from the Muslims at the end of the First Crusade in 1099.

The cross of the crusaders is often (sometimes called the Jerusalem cross) is often used on the bedspreads in the altar: the large cross symbolizes Christ, the four small ones - the four evangelists, spreading the doctrine on four sides. Five crosses together can also symbolize the wounds of Christ.

17 Losev A.F. Philosophy. Mythology. Culture. – M.: Politizdat, 1991.

18 See Appendix 3, illustration 10, commentary to ill. ten

Teutonic cross. The four small crosses at the ends symbolize the four gospels.

cross hammer one of the main heraldic crosses, so named from the French potenee - "support", because its shape is similar to the supports used in antiquity.

Cross Lighting, symbols of Christ, were called upon to ward off the devil and his demons and were an important attribute of the consecration ceremony.

latin cross 19

Another name for the Latin cross is the long cross. Its horizontal bar is located above the middle of the vertical bar. The Latin cross is the most common Christian religious symbol in the Western world. According to tradition, it is believed that Christ was removed from this cross, hence its other name is the Cross of the Crucifixion; it is also called the cross of the West, the Cross of Life, the cross of Suffering. This form, so similar to a man with outstretched arms, symbolized God in Greece and China long before the advent of Christianity. For the Egyptians, the cross rising from the heart symbolized kindness.

Long crosses based on the Latin cross

This is, first of all, cross of st peter(inverted Latin cross), which since the 4th century is one of the symbols of St. Peter, who was crucified with his head up on an inverted cross in 65 AD. e. during the reign of Emperor Nero in Rome. This cross was also the emblem of the Knights Templar.

club cross in heraldry it is also called a cross with clover leaves.

The clover leaf is a symbol of the Trinity, and the cross expresses the same idea. It is also used to refer to the resurrection of Christ.

Cross dagger originated in the Middle Ages, when priests marked in the book the place where they needed to cross themselves.

St. Andrew's Cross 20

It is also called diagonal or oblique. On such a cross the Apostle Saint Andrew suffered and was martyred. According to the legend, he considered himself unworthy to be crucified on the same cross as Christ, and therefore asked his executioners to turn him over.

The Romans used this symbol to mark the border, the passage beyond which was forbidden.

It also symbolizes perfection, the number 10.

In shape, this cross resembles the letter X, the first letter of the name of Christ in Greek writing.

Saint Andrew is the patron saint of Russia, and when Peter the Great created the Russian navy, he adopted a blue oblique cross on a white background for the flag of the fleet. This flag was used until the 1917 revolution.

Also, the St. Andrew's Cross (white on a blue background) was adopted as the national idea of ​​Scotland around the 12th century. And in 1801, the cross was included in the heraldry of Great Britain. It is also present on the flag of the island of Jersey.

Tau Cross (Saint Anthony's Cross) 21

The tau cross is so named because of its resemblance to the Greek letter "T" (tau). It symbolizes life, the key to supreme power.

In heraldry, this is the Almighty Cross.

The ancient Egyptians used the Tau Cross to represent fertility and life. In biblical times, since this symbol was the last letter of the Hebrew script, the tau cross came to mean the end of the world.

19 See Appendix 3, illustration 11, commentary to ill. eleven

20 See Appendix 3, illustration 12, commentary to ill. 12

21 See Appendix 3, illustration 13, commentary to ill. thirteen

At the beginning of the 13th century, Francis of Assisi made this cross his national emblem.

The Tau Cross also acts as a crucifix. Due to its resemblance to the gallows, as it was made in antiquity, it is also called the "cross of the gallows."

In magic, the T-shaped cross means the descent of the spirit from the higher plane into the earthly region (vertical) through the time region (horizontal).

Egyptian cross Ankh (Ankh) cross 22

The ankh is the most significant symbol among the ancient Egyptians, also known as the "cross with a handle." This cross combines two symbols: a circle (as a symbol of eternity) and a tau-cross suspended from it (as a symbol of life); together they denote immortality, eternal life.

The Egyptian ankh also denotes hidden wisdom, the key to the secrets of life and knowledge.

Christian Ankh (Ankh) cross 23

In times closer to us, this sign was used by sorceresses in rituals, divination, divination, healing and helping women in childbirth. During the hippie movement in the late 1960s, the ankh was a popular symbol of peace and truth.

Maltese cross 24

The Maltese cross is also called the eight-pointed cross. It symbolizes the four great gods of Assyria: Ra, Anna, Belus and Hea. It was the emblem of the Knights of the Order of Malta.

From the very beginning, a white cross of this form on a black background was the emblem of the military and religious order of the Hospitallers, also called the Joannites, who dedicated themselves to liberation from the Muslims of the Holy Land during the Crusades (1095-1272). Driven out in 1291, they moved their "headquarters" to Rhodes (in 1310) and later to Malta (in 1529), hence the name.

Today, the Maltese cross can be seen in Britain as the designation of the Sanitary Brigade of St. John.

Cross with crossbars 25

Ecclesiastical crosses with two crossbars mean archbishops and patriarchs, and with stirrup crossbars - the pope.

Lorraine cross or cross of Laurent, has two transverse lines. Joan of Arc, who was born on January 6, 1412 in Domrem, near Laurent, is said to have had this cross as her emblem. This form of the cross was also approved by Charles de Gaulle in June 1940 as a symbol of the liberation of France from Nazi occupation, as well as a symbol of the Free French organization.

Patriarchal Cross- with two horizontal bars - used by archbishops and cardinals. This is a symbol of the Orthodox Church, it is also called the Catholic cross with two crossbars. It is often found on the coats of arms of archbishops. This cross is widespread in Greece and is sometimes called Angevin or Lorraine.

papal cross with three horizontal bars, also known as the triple cross, is used in processions in which the pope participates. Three cross lines symbolize power and the Tree of Life.

eight pointed cross- This is the cross of the Russian Orthodox Church. It is also called the eastern cross or the cross of St. Lazarus.

Raised crosses 26

The most famous of the raised crosses is the cross of Golgotha. It's a latin cross

22 See Appendix 3, illustration 14, commentary to ill. fourteen

23 See Appendix 3, illustration 15, commentary to ill. fifteen

24 See Appendix 3, illustration 16, commentary to ill. sixteen

25 See Appendix 3, illustration 17, commentary to ill. 17

26 See Appendix 3, illustration 18, commentary to ill. eighteen

also called the cross of ascent or descent. One of the most concise altar crosses.

hollow crosses 27

The simplest and most hollow crosses - gamma cross, or gammadion; so named because it consists of four Greek letters G (gamma). Often such a cross can be seen on the clothes of the priests of the Orthodox Church.

german cross built with four letters F in the following sequence: "Frisch, Fromm, Fruhlich, Frei" (strong, God-fearing, cheerful and free).

The most complex of the hollow - Roman sacred cross.

Cross of Constantine (Sign "Chi-Ro") 28

The Cross of Constantine is a monogram known as "Chi-Rho". Consists of X (the Greek letter "chi") and R ("ro") - these are the first two letters of the name Hista in Greek. The legend says that it was this cross that Emperor Constantine saw in the sky on the way to Rome to his co-ruler and at the same time opponent Maximilian; along with the cross, he saw the inscription "In hoc vinces", translated as "Conquer this". According to another legend, he dreamed of this cross on the night before the battle, while the emperor heard a voice: “In hoc signo vinces” (“With this sign you will win”). Both legends claim that it was this prediction that converted Constantine to Christianity. Later he made this monogram his emblem (instead of the eagle).

Cross swastika 29

The swastika is usually regarded as an independent ancient symbol of cosmic energy, but, in essence, it is also a kind of cross, the “broken” ends of which convey rotational movement. Christians used this symbol (the so-called "hidden cross") during times of persecution for their religion. It was also believed that it consists of four letters of the Greek alphabet G ("gamma").

Cross in a circle (Masonic cross) 30

The Masonic cross is a cross inscribed in a circle, meaning a holy place and a cosmic center. The four dimensions of space in the celestial circle symbolize the totality that includes the Great Spirit. Such a cross was either made in stone or depicted on the walls of Roman Gothic temples, symbolizing their sanctification.

Pacifist Cross (Cross of Peace) 31

This symbol was designed by Gerald Holton in 1958 for the then emerging movement for nuclear disarmament. To develop a new symbol, he used the semaphore alphabet: he made a cross from its symbols for "N" (nuclear, nuclear) and "D" (disarmament, disarmament) and placed them in a circle, which symbolized a global agreement. Soon this cross became one of the most common signs of the 60s of the twentieth century, symbolizing both peace and anarchy.

The symbolism of the cross in various cultures and religions

The cross, representing two intersecting lines, has served as a religious, protective symbol in almost every culture of the world since prehistoric times. The symbol of the cross of different countries and eras is similar in philosophical content and execution. The similarity of sign systems is observed even among peoples historically separated by space and time.

Among the Romans, the cross personifies punishment for atrocities.

27 See Appendix 3, illustration 19, commentary to ill. nineteen

28 See Appendix 3, illustration 20, commentary to ill. 20

29 See Appendix 3, illustration 21, commentary to ill. 21

30 See Appendix 3, illustration 22, commentary to ill. 22

31 See Appendix 3, illustration 23, commentary to ill. 23

In Phoenicia, the cross meant life and health.

In Chaldea, the six days of creation and the six phases of time and the life span of the world were depicted with a cross.

In Jewish Kabbalah, the six-pointed cross means the six days of creation, the six phases of time and the duration of the world.

The cross was most widespread in Christianity. First of all, it is a symbol of Christ, his crucifixion and his glory and, thus, the Christian faith.

In the Roman, Persian, Jewish traditions, crucifixion was a cruel and humiliating way of punishment for such segments of the population as slaves, pirates, rebels, criminals and other “non-citizens”. Thus, in the time of Christ, it hardly looked like a symbol that could attract new believers. Even after the baptism of the Roman emperor, the cross remained a secondary symbol compared to the Christogram. After Christianity had spread sufficiently, it began to dominate, while absorbing pre-Christian meanings into its symbolism and thereby deepening a new tradition: the cross as a symbol of redemption through the self-sacrifice of Christ. 32

The ordinary cross has become a comforting symbol of human suffering.

In medieval symbolism, the cross, according to legend, was made of a tree taken from the Tree of Knowledge, which, being the cause of the fall, became the very instrument of salvation.

In Hinduism and Buddhism, the cross is an image of the unity of the lower and higher spheres of being: the vertical crossbar means ascension to heaven, and the horizontal one means earthly life.

In China, the cross is considered a staircase not the sky, the number 10 (a symbol of universality) in Chinese hieroglyphic writing is also indicated by the cross.

In Africa, the sign of the cross can symbolize patronage, protection, cosmic unity, fate, and the cross inscribed in a circle - supreme power.

The American Indians have a cross - the shape of a person, rain, stars, fire from firewood, virginity, four cardinal points and four winds. The center of the cross is the earth and man, driven by the opposing forces of the gods and winds.

In Maori, the cross is the goddess of the moon, the common good. 33

All of the above allows us to conclude that the cross, being a universal symbol, has always been and remains one of the most precious and significant in various eras and peoples. Its forms are varied, and its power, for the most part, consists in balancing the physical energy of a person with the mental and spiritual. But the main uniqueness and value of this symbol is that at all times, in all cultures, the cross symbolized and to this day symbolizes eternal spiritual life.

32 See Appendix 3, illustrations 8, 9, comments to ill. 8, 9

33 Encyclopedia of symbols / comp. V.M. Roshal. – M.: AST; "Publishing house" Owl ", 2005

2.2.2 Swastika

The swastika is the most ancient of the graphic symbols and one of the most common at all times in all cultures. He is credited with Aryan origin. The swastika is considered a non-iconic image of the ancient supreme Aryan god of the Sun and Sky. It most often refers to solar symbols, since it is depicted along with the solar disk. The exact meaning of this symbol is unknown, because it is ambiguous.

The name itself comes from the Sanskrit words “su” (“good”) and “asti” (“being”), which means “good existence”, “welfare” and

is a symbol of blessing, good omen, prosperity, good luck and aversion to misfortune, as well as a symbol of fertility, longevity, health and life. So in the "Ramayana" (Indian epic) it is said that when King Rama moved with his army across the Ganges River to go to conquer India and the island of Ceylon, symbols of good luck - swastikas - were depicted on the nose of his ships.

The swastika can be considered a hidden image of the cross. In archeology, such a cross is called the crux gammata, or gammadion, since it is a combination of four Greek capital letters G (“gamma”). This interpretation emphasizes that "gamma" is the first letter in the name of the Goddess of the Earth - Gaia, so here the swastika is seen more as a symbol of the fertility of the earth, rather than as a solar symbol.

Another name for the swastika is the hooked cross. In heraldry, the swastika is known as the crampon cross, from crampon - an iron hook.

The swastika is a regular equilateral cross, the ends of which are "broken" at a right angle, which creates the illusion of rotation of this sign.

It is known that this sign has always been considered as an emblem of life and light due to its belonging to those symbols that, imitating the apparent movement of the Sun around the Earth, are considered to rotate around its axis.

The swastika sign probably depicted the sun wheel, in many cultures it was associated with the gods of the sun or sky, especially in the Indo-Iranian tradition.

Often it is a symbol of the solar passage through the heavens, turning day into night, and hence the wider meaning as a symbol of fertility and the rebirth of life.

In addition to the energy of rotation inherent in the swastika, other graphic symbols of this sign were also used: the curved ends form a square surrounding the static center; the cross as the four corners of a square, over which the Sun moves in a circle, turning them into a circle (that is, rounding the square and making a circle a square); the cross as a combination of vertical and horizontal lines, which means spirit and matter, as well as the four levels of existence. In the symbolism of the North American Indians, the swastika was associated with the sacred number four, the four gods of the winds, the four seasons; in China - with four cardinal directions, and was also a symbol of the number 10,000 ("accumulation of lucky symbols of Ten Thousand Forces").

It is also assumed that the swastika is an image of a person with two arms and two legs, or a combination of male and female principles, dynamic and static, mobile and motionless, harmony and balance, inhalation and exhalation, departure from the center and return to it, beginning and end . In addition, it symbolizes a kind of labyrinth, water in motion. Maybe it is an image of a forked lightning (a combination of two Z-shaped lightning symbols) or two burning torches and their circular motion, or a doubled Scandinavian Sun Serpent ...

There is another opinion that the swastika was formed by the intersection of a meander (a geometric ornament from a continuous curve or a line broken at a right angle, forming a series of spirals. Developed in the art of ancient Greece). Sometimes it is considered as a variant of the Tau Cross. It is also believed that this is a symbol of humility and humility, like arms crossed in a sign of humility on the chest.

The intersecting swastikas, sometimes called Solomon's knots, symbolize divine incomprehensibility and infinity.

The swastika is a cross in motion. The movement can be directed to the right (then the ends of the branches of the cross are turned to the left) or to the left (the ends of the branches are turned to the right). However, opinions often differ on the definition of movement.

The cross rotating to the right (clockwise) means that we are screwing, squeezing, preventing energies from manifesting: holding them in order to control them. This is a symbol of spirituality, inhibiting the flow of physical forces. An example of this is yoga, supporting the body in immobility, “screwing up” their lower nature so that the energies of their higher, divine nature manifest themselves.

When rotating in the opposite direction (counterclockwise), it means that we “unscrew”, release the brakes, launching physical and instinctive energies and thereby closing the passage to the elevated forces of the spirit: we give ourselves to the mechanical, earthly side.

There are two forms of the swastika: straight and reverse - depending on which way its ends are bent (sometimes they are called "hands").

Direct swastika: left-handed 34 (ends are bent to the left), rotation is considered to occur clockwise.

Reverse swastika: right-handed 35 (ends are bent to the right), rotation is considered to be counterclockwise.

It is believed that the direct and reverse swastikas symbolize the masculine and feminine principles, solar and lunar, movement clockwise and counterclockwise, as well as, apparently, the two hemispheres of the brain (left and right), heavenly and chthonic (underground) power, ascending spring and the descending autumn sun.

A straight swastika is associated with sunny and fertile symbols such as a lion, a ram, a deer, a horse, birds, a lotus. It can be found on altars, statues, robes, vases, weapons, coins, as well as spindles, where it is believed to signify rotational movement.

The winged disk on the swastika is a symbol of solar energy in Egypt and Babylon.

Among the ancient Greeks, the swastika is an attribute of Zeus as the god of heaven and Helios as the solar god; also found in Hera, Ceres and Artemis.

Among the ancient Romans, the swastika symbolized Jupiter, Tonans, and Pluvius.

Among the ancient Scandinavians and Teutons, this is the battle ax or hammer of Thor as the god of air, thunder and lightning, good luck. Thor's hammer was sometimes depicted as a swastika with two zigzag lightning bolts. In Lithuania, the swastika has the properties of a talisman that brings good luck.

The Celts have luck brought by the gods of thunder.

In the Masonic tradition, the swastika was used as a symbol of misfortune and evil. It was also a secret sign of the Gnostics and was used instead of the cross in the Christian sect of the Manichaeans, meaning humility.

As an emblem of "Aryan" racial purity, the swastika began to be used before the First World War by members of anti-Semitic socialist groups in Germany and Austria.

In the early Christian catacomb churches, it was also the emblem of Christ.

For Asian Muslims, the swastika means the four cardinal points and control over the four seasons by the angels: the West is the Recording Angel, the South is the Angel of Death, the North is the Angel of Life, the East is the Announcer Angel.

34 See appendix 3, illustration 24, commentary to ill. 24

35 See appendix 3, illustration 25, commentary to ill. 25

The swastika has been adopted as an amulet by many Eastern cultures. In addition, it is one of the most common magical symbols. The symbolism of the swastika as a sign of vitality, solar energy and cyclical rebirth often coincides with the symbolism of the Creator, especially in the traditions of Buddhism 36 and Jainism 37 .

For the followers of Jainism, this is the divine power, the creator of Heaven and Earth. The four hands symbolize the four levels of existence: protoplasmic life, plants and animals, man, celestial beings.

The swastika, “twisted” with the ends to the left (straight) is a Buddhist symbol of infinite existence, which was depicted on the foot or chest of the Buddha (the fixed core of the Wheel of Becoming). In Buddhism, the swastika is often found at the beginning and end of inscriptions, and is carved on ancient Buddhist medals.

The blue swastika means the infinite perfection of Heaven, the red symbolizes the infinite perfection of the virtue of the heart of the Buddha, the yellow symbolizes infinite prosperity, the green symbolizes the infinite perfection contained in agriculture.

In Hinduism, the swastika is sometimes used to seal jars of holy water from the Ganges.

In Japan, the swastika is a symbol of long life and prosperity. In Japanese, it means the heart of the Buddha, good luck, good wishes.

However, the symbol of the right-handed (reverse) swastika can cause negative associations. In India, for example, it symbolizes the night and black magic.

The most famous type of reverse swastika in the twentieth century was the German Hakenkreuz - "hooked cross". Hitler, who mastered the art of mass consciousness manipulation, used the dynamism inherent in this symbol for the needs of his party and in August 1920 placed the swastika on Nazi banners, thereby appropriating its invocative power. “The effect was like a bombshell,” Hitler later wrote. From 1935 to 1945, the swastika under the imperial eagle (on a white or black background) was the symbol of the "Third Reich" 38 . In this regard, in European culture, this symbol is steadily associated with the Nazi regime and ideology.

An interesting fact is that there is a very common misconception that the Nazis chose the right-handed swastika as their emblem, thereby perverting the precepts of the ancient sages and defiling the sign itself, which is more than five thousand years old. In reality, this is not so. In the cultures of different peoples, both left-handed and right-handed swastikas are found.

The swastika remained a purely symbolic figure for a long time, but gradually its symbolism was forgotten and the swastika turned into a motif of decoration. It was applied both to sacred objects and to household items.

The image of both direct (left-handed) and reversed (right-handed) swastikas can be found in many cultures: on the tablecloth of the Navajo tribe, on Greek ceramics, Cretan coins, Roman mosaics, on objects excavated during the excavations of Troy and in many other cultures.

In Russia at the end of the 19th century, the swastika was often found on the patterns of towels in the Novgorod and Oryol provinces. At the beginning of the 20th century, on the 1000 and 250 ruble banknotes issued after the February Revolution, the swastika occupied a central position, serving as a background for the outstretched eagle, and on the 250 ruble tickets, in addition, there were two more images of the swastika around the edges on the reverse side. The first banknotes of Soviet Russia, issued in 1918, also contained the swastika (for example, a banknote in denominations of 10,000 rubles).

36 See Appendix 3, illustration 26, commentary to ill. 26

37 See Appendix 3, illustration 27, commentary to ill. 27

38 See Appendix 3, illustration 29, commentary on ill. 29

The shape of the swastika can sometimes take on completely unexpected shapes. For example, figures such as triskelion and triquetra can also be considered a type of swastika.

Triskelion 39 is probably built from three triangles standing on their vertices, each of which has one side removed. The figure gives the impression that it moves along an imaginary line of the earth, rotating around its axis, from which the image of rapidly rotating triangles appeared at a later time. Triquetra 40 expresses the idea of ​​movement in the form of a rolling wheel.

All of the above allows us to conclude that the swastika is by no means a symbol belonging to any one nation, but on the contrary, it connects many lands and peoples, which gives certain grounds for searching for a common ancestral home of mankind, and also makes it possible to identify common laws and principles of symbolic thinking of people. 41

39 See Appendix 3, illustration 28, commentary to ill. 28

40 See Appendix 3, illustration 30, commentary to ill. thirty

41 Bagdasarov R.V., Swastika: a sacred symbol. Ethno-religious essays. Ed. second fix. - M.: White Alvy, 2002.

2.2.3 Spiral

Spiral as a sacred form

Another sacred form in our lives is the spiral. We use spirals all the time without even noticing it. We live in a galaxy with spiral arms. The organ of hearing in our ears is shaped like a spiral...

A spiral is a common form in nature. Sacred geometry explores two types of spirals: the golden ratio (golden section) spiral and the Fibonacci spiral. Comparison of these spirals allows us to draw the following conclusion. The spiral of the golden section is ideal: it is similar to God, the Primary Source. When considering the pattern of the golden spiral, it can be seen that the four upper squares on both spirals are of the same size. The difference is where they start. The lower part of the Fibonacci spiral occupies a zone equal to half the upper zone: the golden ratio spiral below occupies a zone 0.618 from the upper zone. The Fibonacci 42 spiral is built using six equal squares (it's "over"), while the golden ratio spiral starts much deeper (it never actually starts - it goes on indefinitely like God). And although their points of origin are different, they approach each other very quickly.

When studying Egypt, scientists discovered that the three pyramids at Giza were built in a spiral. They thought it was a golden spiral and not a Fibonacci spiral. But later (in the 1980s) it was found that both spirals are present there, superimposed on each other.

Another example: many books claim that the Pharaoh's Chamber in the Great Pyramid is a rectangle of the golden ratio, but this is not so. It is also related to the Fibonacci series.

Spirals in nature

Spiral shapes are very common in nature, ranging from spiral galaxies 43 to whirlpools and tornadoes, from shellfish shells to patterns on human fingers, and even, as science has discovered, the DNA molecule contained in every cell of a living organism has the shape of a double helix.

“We fly through space with Sirius A in a spiral shaped like a DNA 44 molecule. We have a common fate with this star. Such a movement indicates that DNA molecules and chromosomes carry information about certain parts of the cosmos. There are key periods when certain events occur. They are related to the genetic alignment between Sirius, Earth and the rest of the cosmos. A very special attunement is currently taking place” (B. Frissel).

Considering the swirling galaxy, Drunvalo Melchizedek writes:

“The spiral has two arms, one opposite the other, exactly 180 degrees apart. Notice how the light is very dark between the reflective sleeves. The dark colored spirals rotate at 180 degrees to each other and 90 degrees to the white light spirals. If you look directly at the center, you will see that the two opposite arms of the galaxy are at exactly 180 degrees to each other...

Here, a spiral of white light exits in one direction, and at 180 degrees from it, another spiral of white light exits in the opposite direction. Dark sleeves - women's - go between the light ones.

42 See Appendix 3, illustration 33, commentary to ill. 33

43 See Appendix 3, illustrations 31, 32, comments on ill. 31, 32

44 See Appendix 3, illustration 34, commentary to ill. 34

This explains why the dark light between the light arms of the spiral is different from the darkness in the rest of space (this was discovered by scientists). This is because the black light in the spiral is feminine energy, and the blackness of outer space is the Great Void, which is not the same thing. Scientists don’t quite understand why they are different.”

The symbolism of the spiral is contained in everything that has a helical structure - this is the ear, mollusc shells, octopus tentacles, coiled coils of a snake, animals such as a cat and a dog that can arch its back, various plants ... Spirals can be found in pine and spruce cones, sunflowers and many other plants, in the horns of some animals, including a deer ... If you put your open palm vertically in front of you, pointing your thumb to your face, and, starting from the little finger, successively clench your fingers into a fist, you get a movement that is a Fibonacci spiral.

A double helix is ​​clearly visible on a pine cone: the first helix goes in one direction, the second in the other. If we count the number of scales in a spiral rotating in one direction and the number of scales in another spiral, we can see that these are always two consecutive numbers from the Fibonacci series: for example, if there are 8 scales in one direction, then there will be 13 in the other; if one has 13, then the other has 21. Numerous examples of double helixes found throughout nature always conform to this rule. In particular, sunflower spirals always correlate with the Fibonacci series.

Another example of the manifestation of sacred geometry in nature is the nautilus shell: “There is an unwritten rule that any good book on sacred geometry should show a nautilus shell. Many books say that this is a golden ratio spiral, but this is not true - it is a Fibonacci spiral.

You can see the perfection of the arms of the spiral, but if you look at the center or the beginning, it doesn't look so perfect. The two innermost shell bends are actually equal, and the ratio of their lengths is 1, which is far from the coefficient fi(1.618). The second and third bends are slightly closer to fi. Then, finally, this elegant smooth spiral is obtained. You might think that at the very beginning this little clam made a mistake; he didn't seem to know what he was doing. No, he works beautifully, this is not a mistake. It just follows the math of the Fibonacci series exactly.” (Drunvalo Melchizedek)

The proportion expressed by "φ" - the ratio, was widely known in antiquity. It was used in the construction of the Pyramids and the Parthenon; it was studied by the great masters of the Renaissance - Leonardo da Vinci and Raphael, today it is widely used in architecture by Le Corbusier. For its aesthetic properties, it was given the name of the "golden section" or "divine proportion". 45

Symbolism of the spiral

The spiral is a highly complex symbol that has been used since Paleolithic times. It is found in pre-dynastic Egypt, Crete, Mycenae, Mesopotamia, India, China, Japan, pre-Columbian America, Europe, Scandinavia and Britain; it is also found in Oceania (but not Hawaii).

And everywhere the spiral is, first of all, a symbol of the great creative (vital) force both at the level of the cosmos and at the level of the microcosm.

The spiral, which combines the shape of a circle and the impulse of movement, is also a symbol of time, cyclical rhythms, the change of seasons, birth and death, the phases of "aging" and the growth of the moon, as well as the sun itself. It also symbolizes the emanations of the Sun and Moon, air and water currents, thunder and lightning. As divergent and convergent,

45 Fadeeva T.M., Sacred Space. M., 2002. Chapters from the book. Page 126-137.

a spiral can mean growth and spread, expansion and contraction, twisting and unwinding. It can also symbolize continuity. It can be an image of the rotating sky, the movement of the Sun, the rotation of the earth. In the form of an air whirlwind during a thunderstorm or in the form of a whirlpool, it symbolizes fertility and the dynamic aspect of being. As a tornado, it is associated with the Chinese descending dragon. The spiral and the tornado have the same symbolism, especially when they act as symbols of energy in nature.

Spirals-whirlwinds are associated with spinning and weaving the web of life and the veil of the Mother Goddess, the steward of destinies and the weaver of the veil of illusions.

In addition, the spiral has the same symbolism as the labyrinth.

In metaphysical terms, it symbolizes the realities of existence, the various modalities of being, the wanderings of the soul and its final return to the center.

The spiral, as part of a smooth and endless line, also symbolizes development, continuation, continuity, centripetal and centrifugal movement, the rhythm of breathing and life itself.

The double helix symbolizes the increase and decrease in the power of the Sun and

The moons, as well as the changing rhythms of evolution and involution, life and death, etc. It can mean two hemispheres, two poles, day and night, all the rhythms of nature, shakta-shakti, manifestation and non-manifestation, as well as a sequence of cycles. It is a typical androgyne symbol and is associated with bidirectional symbolism.

The spiral shape of the snakes on the caduceus, like other double spirals, symbolizes the balance of opposites. The same meaning is contained in the Taoist sign "yin-yang", which is a kind of double helix.

The opposing forces visible in the whirlpools, whirlwinds and flames are reminiscent of the ascending, descending or rotating energy that governs the Cosmos.

The ascending spiral is a masculine, phallic sign, the descending is feminine, which makes the double helix a symbol of fertility and childbearing.

The compressed coil spring is a symbol of latent power, as is the serpentine ball of energy at the base of the back, which is considered an important element of the teachings of yoga.

The spiral is also connected with the navel as the center of power and life.

She is a magical symbol reflecting the journey to the center where enlightenment, wisdom and intuition will be found. If the purpose of such a spiral journey towards the center is to acquire wisdom, then sometimes the spiral is depicted as a coiled snake.

Representing the "path", the spiral also serves as an expression of a powerful charge of energy.

In popular magical tradition, there are many "serpent amulets", sometimes called "Saint Hilda's amulets", which are actually ammonites (petrified shells) with a snake's head attached to the open end. Such amulets are still sold in Whitby (England) and nearby villages and are believed to have protective powers for those who wear them. The ammonites are said to be the fossilized bodies of snakes that Saint Hilda sent over the edge of the cliff above Whitby to drive them out of her abbey.

Among the Celts, the spiral can also be a symbol of the flame.

In Crete and Mycenae, the coiled tentacles of the octopus were associated with the spiral, thunder, rain and water.

In Taoism and Buddhism, the "precious pearl" or "dragon sword" was sometimes depicted in the form of a spiral.

She is also associated with the souls of gods and kings, with the rain-causing reptiles, and with the coiled and sleeping Kundalini serpent.

The gods of tornadoes and natural elements and movements, such as Rudra or Pushan, have hairstyles in the form of spirals or shells.

In art, a spiral is one of the most common decorative patterns - from Europe (double spirals in the Celtic tradition or spirals on Roman capitals) to the Pacific Ocean (Maori spiral carving in New Zealand, tattoos of the islanders of Polynesia). Maori carving is based on the arrangement of fern leaves, which demonstrates the connection between spiral patterns and natural phenomena. It is this connection that often determines the symbolism of the spiral, although its ambiguity is so great that special keys are sometimes required to decipher its meaning. It is also worth noting that the symbolism in spiral patterns is present rather involuntarily, its conscious use is much less common.

Spirals carved on megalithic monuments depict a journey through the labyrinths of the underworld and give hope for a possible return from there.

The sign of the double helix is ​​interesting, in which both elements - self-unfolding and self-concentration ("evolution and involution") are connected in an inseparable unity. In this one can see the image of "becoming and disappearing" as a process of eternal circulation. 46

All of the above allows us to conclude that the spiral is a fairly complex symbol for interpretation and understanding. Having appeared since the Paleolithic times, this symbol has found its origins in different cultures and nationalities. But, despite various applications and interpretations, the spiral, first of all, was and remains a symbol of the great creative (life) force both at the cosmic level and at the microcosmic level.

46 Fadeeva T.M., "GOLDEN SECTION", Publishing house "Enlightenment", M., 2002

ChapterIII. The interaction of geometric signs and symbols when they are combined.

Geometric shapes are not just works of art. They must be perceived in connection with those innermost phenomena that they help to express and decorate. All sacred structures are based on the original geometric Logos: Egyptian and Mexican pyramids, temples in India, pagodas of China and Japan, Indian tents of North America, churches and cathedrals of Christendom.

Each element is inextricably linked with the other, testifying to the unity of the apparent differences between fire and water, air and earth. No element dominates, they all balance each other and form a very stable structure. All sacred geometry and architecture is based on this simple fact.

Taken together, almost all geometric symbols can be considered as some kind of special inimitable element that has a special meaning and interpretation.

Five symbols were used in this essay: dot, circle, cross, swastika and spiral.

For example, symbols such as the swastika and the spiral have a lot of common features and meanings. First of all, both symbols have the illusion of rotation (in the Swastika, due to the shape of a regular equilateral cross, the ends of which are “broken” at a right angle). Two symbols express an imitation of the visible movement of the Sun around the earth, rotation around its axis, the constant continuous movement of vortices. These symbols have a certain creative force in action, the generation of natural and temporary symbols, the feminine and masculine principles. It is also very important to note that both symbols are interpreted as some kind of labyrinths in which the movement goes either from the center or towards the center.

As for all symbols, interacting with each other, they represent a kind of "living body", and the body is logically structured so that each symbol in it is the beginning or continuation of another.

The author managed to draw up a drawing consisting of all the symbols used in the abstract 47 . As a result, the drawing is a “living element” that can be interpreted in different ways, but based on all the symbols that make up the drawing, the following chain is obtained: the point, being the beginning of the entire universe, concentrates all energy in itself. Absolutely any symbol, element or sign can be formed from a dot. Following the symbols from this work, we can say that the point is the center, the beginning of the cross. The swastika has the meaning of a bent cross, the ends of which, due to the “brokenness”, create rotational energy. The rotational energy of the swastika is similar to the rotational, twisting movement of a spiral. Rotating, the spiral forms the shape of a circle. All five symbols have the meaning of vitality, solar energy, cyclical rebirth.

As a result, the drawing resulting from these five symbols is nothing more than an opportunity to show that life never stands still, continuous restoration, self-reproduction, cyclical rebirths will always be present in nature, in people's lives, in everything that lives .

47 See Appendix 3, illustration 35, commentary on ill. 35

Conclusion.

The purpose of this essay was to study geometric symbols and analyze their meaning in nature, science and in the life of various civilizations.

The following tasks were implemented:

Conduct a selection of literature on this topic.

Consider symbolism as a property of culture.

Study the geometric symbols chosen for work and combine them into a single one, showing that each symbol, element and sign is inextricably linked with another.

The subject of the study was the process of identifying different meanings of one symbol in different cultures and eras, as well as the proof that all symbols are polysemantic.

The object of the study was geometric symbols.

As a result of the material, the hypothesis was proved that there is an inextricable link between the symbols, which can connect all the symbols into something unified and contain a special incomprehensible knowledge.

In conclusion, it should be noted that the symbols are identical everywhere, although they bear the imprints of the culture and traditions of different peoples. Time has not fundamentally changed the structure of symbolism. It only gradually revealed new semantic layers without destroying the previous system of concepts. And our own human vision, imagination and understanding must grow and rise to new heights in order to fully appreciate the amazing legacy left to us on the surface of the earth, in museums, libraries and spheres of the spiritual culture of mankind.

Appendix 1

Glossary of terms

Abstraction- (German Abstraktion, French abstraction< лат. abstrāctio удаление, отвлечение).

1. Mental distraction from certain aspects, properties or connections of objects and phenomena in order to highlight their essential features. (Without abstraction, it is impossible to form concepts ) . 2. Abstract concept, theoretical generalization of experience (as a result of abstraction).

Allegory- (German Allegorie< греч. allēgoria < allos другой, иной + agoreyō говорю). В литературе и изобразительном искусстве: выражение чего-н. отвлеченного в конкретном художественном образе; иносказание. | Примеры аллегорий: весы - правосудие, крест - страдание, якорь - надежда и т. п.

Androgyne- (Greek androgynos< anēr (andros) мужчина + gynē женщина). Мифологическое обоеполое человеческое существо, которое боги разделили на две особи - мужчину и женщину.

Anthropology– (Greek anthrōpos man + logos teaching). The science of the biological nature of man.

Artemis of Ephesus- in Greek mythology, the daughter of Zeus, the goddess of the hunt, the patroness of women in childbirth. Depicted with a bow and arrows, sometimes with a crescent on her head.

Aztecs- (Spanish Aztecas), Indian people. Other names are tenochki and meshika. Modern Aztecs live in Mexico. 1.2 million people (1992). Aztec language. Believers are Catholics.

Buddha- awakened, awakened, Enlightened, Knowing the transcendental light. 1) in Buddhism, the highest state of spiritual perfection; 2) the name of the ancient Indian sage Shakyamuni after he acquired a special spiritual experience (gift).

Hera- in Greek mythology, the queen of the gods, sister and wife of Zeus; patroness of marriage. Daughter of Kronos and Rhea. Differs in imperiousness, cruelty and jealous disposition. She pursues with her hatred the beloved and children of Zeus. In retaliation, Zeus, who gave birth to Athena, gives birth to Hephaestus without his participation.

Heraldry- (from the Middle Ages. Lat. heraldus - herald), heraldry. From the 2nd half of the 19th century - an auxiliary historical discipline that studies coats of arms; earlier, in the 13th-1st half of the 19th centuries, the drawing up of noble, guild and land coats of arms. In the most common sense, heraldry deals with coats of arms that appeared in the European Middle Ages.

Zeus The supreme god in Greek mythology. Having thrown his father, the titan Kronos, into Tartarus, he became the lord of the gods and people. The attributes of Zeus were the aegis (shield), the scepter, sometimes the eagle; Olympus (Olympian Zeus) was considered the seat.

"Golden Ratio"- (golden proportion, division in the extreme and average ratio, harmonic division), segment division AC into two parts so that most of it AB belongs to the smaller Sun like the whole segment AC refers to AB(i.e. AB : Sun = AC : AB). Approximately this ratio is equal to 5 / 3, more precisely 8 / 5, 13 / 8, etc. The principles of the golden section are used in architecture and in the visual arts. The term "golden ratio" was introduced by Leonardo da Vinci.

cabalistics, cabalistics- (German Kabbalistik< др.-евр. - см. bondage2). 1. cabal and related magical performances or rites. 2. transfer. Something incomprehensible, confusing or mysterious

Quintessence- (German Quintessenz, French quintessence< лат. quinta essentia пятая сущность). Самое главное, наиболее существенное, важное; то же, что essence.| In ancient philosophy - ether, the fifth element - the main element of celestial bodies, opposed to the four earthly ones: water, earth, fire and air.

Cumulative process- (French cumulatif, German kumulativ< лат. cumulātio увеличение; скопление). Основанный на принципе накопления, концентрации чего-н.

Freemasonry - A religious and ethical trend with mystical rites, usually combining the tasks of moral self-improvement with the goals of the peaceful unification of mankind in a religious fraternal union.

Metaphor- (Greek metaphora transfer). Turn of speech, consisting in the use of words and expressions in a figurative sense based on similarity, comparison.

Mythology- (German Mythologie< греч. Mythologia). 1. Совокупность myths some people.

2. Scientific discipline that studies myths.

Neoplatonism– (neo… + Platonism). A philosophical trend that arose in the Roman Empire in the 3rd century, combining the idealism of Plato with Eastern mysticism. Neoplatonist is a follower of Neoplatonism.

Parthenon- Temple of Athena Parthenos on the Acropolis in Athens, a monument of ancient Greek high classics. Remarkable majestic beauty of shapes and proportions. Destroyed in 1687; partially restored.

Platonism- (French platonisme< греч. Platōn Платон). 1. Учение древнегреческого философа Платона (427-347 гг. до н.э.) и его последователей, противопоставлявшее реальному миру вещей мир сверхчувственных идей; разновидность idealism

controversy- (German: Polemik< фр. polйmique < греч. polemikos воинственный, враждебный). Спор при обсуждении научных, литературных, политических вопросов.

Semiotics- (Greek Sēmiōtikē). The science of signs and sign systems

Sirius- star - 1.5 magnitude, the brightest in the sky. Sirius is a double star, its component Sirius Major is the first discovered white dwarf.

transcendental- In idealistic philosophy: being outside the world.

Triskelion- (from Greekτρισκελης - three-legged) is an ancient sign, three running legs coming out of one point. The triskelion is also a symbol Sicily.

Triquetra- (lat. triquetrum - triangular) - the oldest symbolic sign common among the Nordic peoples of Europe - Irish, Frisians, Scandinavians.

Christogram- krism, the monogram of Jesus Christ, the most common of the sacred monograms, formed from the Greek letters "X" (chi) and "R" (ro).

Ceres- in Roman mythology, the goddess of agriculture and fertility.

Shaktism(from Sanskrit sakti - strength, energy), a trend in Hinduism, based on the veneration of shakti, the deified female energy, which is understood as a hypostasis of the energy of God

Esoteric symbols- (Greek esōterikos internal). Secret, hidden, intended exclusively for the initiated.

Exoteric Symbols- (Greek exōterikos external). specialist. Not constituting a secret, intended for the uninitiated.

Emanation- (from the late Latin emanatio - outflow, origin), the central concept of Neoplatonism, meaning the transition from the highest ontological level of the universe (the One) to the lower, less perfect ones. Following the Neoplatonists, Eriugena and the late Schelling adhered to the theory of emanation. Emanation as a decrease of being is opposite to ascending development, perfection.

Emblem- an auxiliary historical discipline that studies symbols and signs of belonging, property, etc. (except for coats of arms and seals)

Ethnology- (German Ethnologie, French ethnologie< греч. ethnos народ + logos наука, учение). Наука, изучающая материальную и духовную культуру народов.

Annex 2

Name Dictionary

Ivankhov O. M. (1900-1986)- French philosopher and educator, born in Bulgaria in 1900, initiate, one of the greatest contemporary Teachers of mankind, founder of the Universal White Brotherhood. A man of inexhaustible strength of the Spirit, full of the deepest Love and sympathy for people, for all life on this Earth. Ivankhov bases the esoteric teaching entirely on Christianity, using this only possible path to the minds and hearts of his students.

Dirac Paul (Adrien Maurice) (1902-1984) one of the greatest theoretical physicists of the 20th century. Proposed a second quantization method. He laid the foundations of quantum electrodynamics and the quantum theory of gravity. Nobel Prize (1933, jointly with E. Schrödinger).

Drunvalo Melchizedek- world famous scientist, ecologist, inventor, esoteric, healer and teacher, author of the books "The Ancient Secret of the Flower of Life" and "Live in the Heart". Drunvalo's teaching openly presents esoteric information coming from the depths of centuries on how to consciously use the principles of Sacred Geometry for spiritual growth - the geometry of forms that underlies life and all other manifestations in the Universe.

Ivanov I. (1862-1939)- Mathematician, Corresponding Member of the Academy of Sciences of the USSR (1925; Corresponding Member of the Russian Academy of Sciences since 1924). Works on the theory of numbers.

Kant Immanuel (1724-1804)- German philosopher, founder of German classical philosophy; professor at the University of Koenigsberg, foreign honorary member of the St. Petersburg Academy of Sciences (1794). In 1747-55 he developed a cosmogonic hypothesis of the origin of the solar system from the original nebula ("General Natural History and Theory of the Sky", 1755).

Kierkegaard (Kirkegaard) Soren (1813-55)- Danish theologian, philosopher, writer. He contrasted the "objectivism" of Hegel's dialectic with the subjective ("existential") dialectics of personality, which, according to Kierkegaard, goes through three stages on the path to God: aesthetic, ethical and religious.

Leonardo da Vinci (1452-1519)- one of the greatest representatives of Italian Renaissance art, painter, sculptor, musician. poet, architect and scientist.

Losev A. F. (1893-1988)- Russian philosopher, historian of philosophy and aesthetics, philologist.

Plato (428 or 427 BC - 348 or 347)- Ancient Greek philosopher. A student of Socrates, ca. 387 founded a school in Athens (Platonic Academy). Ideas (the highest among them is the idea of ​​good) are eternal and unchanging intelligible prototypes of things, of all transient and changeable being; things are likeness and reflection of ideas.

Raphael Santi (1483-1520)- Italian painter and architect. Representative of the High Renaissance. With classical clarity and sublime spirituality, he embodied the life-affirming ideals of the Renaissance. At the end of 1508, at the invitation of Pope Julius II, he moved to Rome, where, along with Michelangelo, he took a leading position among the artists who worked at the court of Julius II and his successor Leo X.

White Leslie (1900-1975) is an American culturologist. He revived the evolutionary approach to the study of cultures, graduated from the sociology department of the University of Chicago. In 1964 he was elected president of the American Anthropological Association.

Fibonacci (Leonardo of Pisa) (1180-1240)- Italian mathematician In his main work, The Book of the Abacus (1202), he was the first to systematically outline the achievements of Arabic mathematics, which contributed to their acquaintance in Western Europe.

Florensky P. A. (1882-1937)- Russian scientist, religious philosopher, theologian. In the essay “The Pillar and Ground of the Truth. The experience of Orthodox theodicy" developed the doctrine of Sophia (God's Wisdom) as the basis of the meaningfulness and integrity of the universe. In the works of the 20s. sought to build a "concrete metaphysics" (research in the field of linguistics and semiotics, art history, philosophy of worship and icons, mathematics, experimental and theoretical physics, etc.). Repressed; rehabilitated posthumously.

Holton Gerald(b. 1922) American historian and philosopher of science. He taught at Harvard University. Known as the founder of a new direction in the study of the history of science - thematic analysis, designed to complement the standard analysis of the logical structure of scientific knowledge. In neopositivism, analysis was limited mainly to two types of statements, which were qualified as empirical and analytical (logo-mathematical).

Jung Carl Gustav (1875-1961)- Swiss psychologist and philosopher, founder of "analytical psychology". He developed the doctrine of the collective unconscious, in the images of which (the so-called archetypes) he saw the source of universal human symbolism, including myths and dreams (“Metamorphoses and symbols of libido”).

Appendix 3

Illustrations for the abstract

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