Introduction

Solids of Plato

What is the Da Vinci Code?

Platonic Solids

Archimedean solids

Mystery of the Egyptian calendar

Quasicrystals by Dan Shechtman

Penrose tiles

Fullerenes

The artistic world of the Slovenian artist Matiushka Teija Kraszek

Regular polyhedra have attracted the attention of philosophers, builders, architects, artists, and mathematicians since ancient times. They were struck by the beauty, perfection, harmony of these figures.

A regular polyhedron is a volumetric convex geometric figure, all the faces of which are the same regular polygons and all the polyhedral angles at the vertices are equal to each other. There are many regular polygons, but there are only five regular polyhedra. The names of these polyhedra come from Ancient Greece, and they indicate the number ("tetra" - 4, "hexa" - 6, "octa" - 8, "dodeca" - 12, "ikosa" - 20) faces ("hedra").

These regular polyhedra were called Platonic solids after the ancient Greek philosopher Plato, who gave them a mystical meaning, but they were known even before Plato. The tetrahedron personified fire, since its top is directed upwards, like a flaming flame; icosahedron - as the most streamlined - water; the cube - the most stable of the figures - the earth, and the octahedron - the air. The dodecahedron was identified with the entire universe and was considered the most important.

Regular polyhedra are found in nature. For example, the skeleton of a single-celled organism of feodaria resembles an icosahedron in shape. A pyrite crystal (sulphurous pyrite, FeS2) has the shape of a dodecahedron.

Tetrahedron - correct triangular pyramid, and a hexahedron - a cube - figures with which we constantly meet in real life. To better feel the shape of other Platonic solids, you should create them yourself from thick paper or cardboard. It is not difficult to make a flat scan of the figures. The creation of regular polyhedra is extremely entertaining by the very process of shaping.

The complete and bizarre forms of regular polyhedrons are widely used in the decorative arts. Volumetric figures can be made more entertaining if flat regular polygons are represented by other shapes that fit into the polygon. For example: a regular pentagon can be replaced by a star. Such a three-dimensional figure will not have edges. You can collect it by tying the ends of the rays of the stars. And 10 stars are going to be a flat scan. A three-dimensional figure is obtained after fixing the remaining 2 stars.

If your child loves to make crafts with his skillful hands, invite him to assemble a three-dimensional polyhedron dodecahedron figure from flat plastic stars. The result of the work will please your child: he will make an original decorative design with his own hands, which can be used to decorate a children's room. But, the most remarkable thing is that the openwork ball glows in the dark. Plastic stars are made with the addition of a modern harmless substance - a phosphor.

GEOMETRY OF THE PLATONIAN BODIES

rev. dated 06/24/2013 - (updated)

The main five Platonic solids are: octahedron, star tetrahedron, cube, dodecahedron, icosahedron.

Each of the geometric patterns, whether atomic nucleus, microclusters, global grid or distances between planets, stars, galaxies, is one of the five main "Platonic Solids".

Why do these patterns occur so often in nature? One of the first hints: mathematicians knew that these shapes have more "symmetry" than any three-dimensional geometry that we can create.

From Robert Lawlor "Sacred Geometry" we can learn that the Hindus reduced the geometries of the Platonic Solids to the octave structure that we see for sound and light (note and color). The Greek mathematician and philosopher Pythagoras, through the process of successively dividing the frequency by five, first developed the eight "pure" tones of the octave, known as the diatonic scale. He took a single-string "monochord", and measured the exact wavelengths while playing different notes. Pythagoras showed that the frequency (or rate of vibration) of each note could be represented as a ratio between two parts of a string, or two numbers, hence the term "diatonic ratio".

The table below lists the geometry in a particular order, associating it with the helix number fi(). This gives a complete and complete picture of how different vibrations work together. It is based on assigning lengths to the edges of a cube equal to “ 1 ". Then we compare the edges of all other shapes with this value, whether they are larger or smaller. We know that in the Platonic Solids every facet is the same shape, every corner is identical, every node is the same distance from all other nodes, and every line is the same length.

1 Sphere (no faces) 2 Central icosahedron 1/phi 2 3 Octahedron 1/ √2 4 Star tetrahedron √2 5 Cube 1 6 Dodecahedron 1/phi 7 Icosahedron phi 8 Sphere (no faces)

This will help to understand how, with the help of the vibrations of the phi spiral, the Platonic solids gradually flow into one another.

MULTIDIMENSIONALITY OF THE UNIVERSE

The very concept of connecting Platonic geometries to higher planes arises because scientists know: there must be geometry; they found it in the equations. Platonic geometries are required to provide “more room” for invisible additional axes to appear in “hidden” 90° turns. In the data analysis method, each face of the geometric shape represents a different axis or plan in which it could rotate. When we begin to look at the work of Fuller and Jenny, we see that the idea of other planes existing in "hidden" 90° turns is simply an incorrect explanation based on a lack of knowledge of the "sacred" connections between geometry and vibration.

It is very likely that conventional scholars will never understand that ancient cultures could have had a “lost connection” that greatly simplifies and unifies everything. modern theories space physics. While it may seem incredible that a "primitive" culture had access to this kind of information, the evidence is there. Read the classic book of Prasada, for now you can see that Vedic cosmology has an inherent scientific skill.

What do you think you see? is an exploding star with dust ejected from it ... But there is clearly some kind of energy field here, structuring the dust as it expands into a very precise geometric pattern:

The problem is that the typical magnetic fields in conventional physics models simply don't allow for such geometric accuracy. Scientists really don't know how to understand such things!

The image below is the NEW nebula, which is a perfect "square". However, this is still two-dimensional thinking. What is a square in three dimensions?
Of course, the cube!

Seen in infrared, the nebula resembles a giant glowing box in the sky with a bright white inner core. The dying star MWC 922 lies at the center of the system and is spewing out its interior from opposite poles into space. After MWC 922 emits most of its material into space, it will shrink into a dense stellar body known as a white dwarf, hidden in its remnant clouds.

While it's remotely possible that the star's explosion only propagates in one direction, creating a more pyramidal shape, what you're seeing is a perfect cube in space. Because all four sides of the cube are the same length and perfect 90° angles to each other, and again, the cube has the structured “steps” we saw in the previous image, scientists are completely baffled. The cube has even BIGGER SYMMETRY than the "rectangular" nebula!

Such patterns appear not only in the vastness of space. They also occur at the smallest level of atoms and molecules, for example, in the cubic structure of ordinary table salt or sodium chloride. An Pang Tsaya (Japan) photographed quasicrystals of an aluminum-copper-iron alloy in the form of a dodecahedron and an aluminum-nickel-cobalt alloy in the form of a decagonal (ten-sided) prism (see photo). The problem is that you cannot create such crystals using single atoms bonded together.

Another example is the Bose-Einstein condensate. In short, a Bose-Einstein condensate is a large group of atoms behaving like a single “particle” in which each atom composing it simultaneously occupies all space and all time in the entire structure. It is measured that all atoms vibrate at the same frequency, move at the same speed and are located in the same region of space. Paradoxically, but different parts of the system act as a whole, losing all signs of individuality. It is this property that is required for a “superconductor”. Typically, Bose-Einstein condensates can form at extremely low temperatures. However, it is precisely such processes that we observe in microclusters and quasicrystals devoid of individual atomic identity.

Another similar process is the action of laser light, known as "coherent" light. All in space and time the laser beam behaves like a single "photon", that is, it is impossible to separate individual photons in a laser beam.

Moreover, in the late 1960s English physicist Herbert Fröhlich suggested that living systems often behave like Bose-Einstein condensates, only on a large scale.

Photographs of the nebula offer stunning visible evidence that the geometry is at play. about more of a role in the forces of the universe than most people would believe. Our scientists can only fight to understand this phenomenon within existing traditional models.

Stakhov A.P.

The Da Vinci Code, Platonic and Archimedean solids, quasicrystals, fullerenes, Penrose lattices and art world Mother Teija Kraszek

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The work of the Slovenian artist Matyushka Teija Krashek is little known to the Russian-speaking reader. At the same time, in the West it is called the "Eastern European Escher" and the "Slovenian gift" to the world cultural community. Her artistic compositions are inspired by the latest scientific discoveries (fullerenes, Dan Shechtman quasicrystals, Penrose tiles), which in turn are based on regular and semi-regular polygons (Plato and Archimedes solids), the Golden Section and Fibonacci numbers.

Surely every person has thought more than once about the question why Nature is able to create such amazing harmonious structures that delight and delight the eye. Why artists, poets, composers, architects create amazing works of art from century to century. What is the secret of their Harmony and what laws underlie these harmonious creatures?

The search for these laws, the "Laws of Harmony of the Universe", began in ancient science. It is during this period of human history that scientists come to a series of amazing discoveries that permeate the entire history of science. The first of them is considered to be a wonderful mathematical proportion expressing Harmony. It is called differently: "golden ratio", "golden number", "golden mean", "golden ratio" and even "divine proportion". The Golden Section is also called PHI number in honor of the great ancient Greek sculptor Phidias (Phidius), who used this number in his sculptures.

The thriller The Da Vinci Code, written by popular English writer Dan Brown, has become a 21st century bestseller. But what does the Da Vinci Code mean? There are various answers to this question. It is known that the famous "Golden Section" was the subject of close attention and enthusiasm for Leonardo da Vinci. Moreover, the very name "Golden Section" was introduced into European culture by Leonardo da Vinci. At the initiative of Leonardo, the famous Italian mathematician and learned monk Luca Pacioli, Leonardo da Vinci's friend and scientific adviser, published the book "Divina Proportione", the first mathematical work in world literature on the Golden Section, which the author called "Divine Proportion". It is also known that Leonardo himself illustrated this famous book, drawing 60 wonderful drawings for it. It is these facts, which are not very well known to the general scientific community, that give the right to put forward a hypothesis that the Da Vinci Code is nothing but the Golden Section. And confirmation of this hypothesis can be found in a lecture for students Harvard University which he remembers main character book "The Da Vinci Code" by Prof. Langdon:

“Despite its almost mystical origin, the PHI number has played a unique role in its own way. The role of the brick in the foundation of building all life on earth. All plants, animals and even human beings are endowed with physical proportions, approximately equal to the root of the ratio of the number of PHI to 1. This omnipresence of PHI in nature ... indicates the connection of all living beings. It used to be believed that the PHI number was predetermined by the Creator of the universe. Scientists of antiquity called one point six hundred and eighteen thousandths "divine proportion."

Thus, the famous irrational number PHI = 1.618, which Leonardo da Vinci called the Golden Mean, is the Da Vinci Code!

Another mathematical discovery of ancient science is regular polyhedra, which were named "Platonic Solids" and "semi-regular polyhedra", named "Archimedean solids". It is these amazingly beautiful spatial geometric shapes that underlie the two largest scientific discoveries 20th century - quasicrystals(the author of the discovery is the Israeli physicist Dan Shechtman) and fullerenes(Nobel Prize 1996). These two discoveries are the most significant confirmation of the fact that it is the Golden Proportion that is the Universal Code of Nature (“The Da Vinci Code”), which underlies the Universe.

The discovery of quasicrystals and fullerenes has inspired many contemporary artists to create works that reflect the most important physical discoveries of the 20th century in artistic form. One of these artists is the Slovenian artist Mother Theia Kraszek. This article introduces the artistic world of Matyushka Teija Krashek through the prism of the latest scientific discoveries.

A person shows interest in regular polygons and polyhedra throughout his conscious activity - from a two-year-old child playing with wooden cubes to a mature mathematician. Some of the regular and semi-regular bodies occur naturally as crystals, others as viruses that can be seen with an electron microscope.

What is a regular polyhedron? A polyhedron is called regular if all its faces are equal (or congruent) to each other and at the same time are regular polygons. How many regular polyhedra are there? At first glance, the answer to this question is very simple - as many as there are regular polygons. However, it is not. In Euclid's Elements we find a rigorous proof that there are only five convex regular polyhedra, and that only three types of regular polygons can be their faces: triangles, squares and pentagons (regular pentagons).

Many books have been devoted to the theory of polyhedra. One of the most famous is the book of the English mathematician M. Wenniger "Models of polyhedra". In Russian translation, this book was published by the Mir publishing house in 1974. The epigraph to the book is the statement of Bertrand Russell: "Mathematics possesses not only truth, but also high beauty - beauty honed and strict, sublimely pure and striving for genuine perfection, which is characteristic only of the greatest examples of art."

The book begins with a description of the so-called regular polyhedra, that is, polyhedra formed by the simplest regular polygons of the same type. These polyhedra are called Platonic solids(Fig. 1) , named after the ancient Greek philosopher Plato, who used regular polyhedra in his cosmology.

Picture 1. Platonic solids: (a) octahedron ("Fire"), (b) hexahedron or cube ("Earth"),

We will begin our consideration with regular polyhedra, whose faces are equilateral triangles. The first of these is tetrahedron(Fig.1-a). In a tetrahedron, three equilateral triangles meet at one vertex; while their bases form a new equilateral triangle. The tetrahedron has smallest number faces among the Platonic solids and is a three-dimensional analog of a flat right triangle, which has the smallest number of sides among regular polygons.

The next body, which is formed by equilateral triangles, is called octahedron(Fig.1-b). In an octahedron, four triangles meet at one vertex; the result is a pyramid with a quadrangular base. If you connect two such pyramids with bases, you get a symmetrical body with eight triangular faces - octahedron.

Now you can try to connect five equilateral triangles at one point. The result is a figure with 20 triangular faces - icosahedron(Fig.1-d).

The next regular polygon shape is − square. If we connect three squares at one point and then add three more, we get perfect shape with six sides, called hexahedron or cube(Fig. 1-c).

Finally, there is another possibility of constructing a regular polyhedron based on using the following regular polygon − Pentagon. If we collect 12 pentagons in such a way that three pentagons meet at each point, we get another Platonic solid, called dodecahedron(Fig.1-e).

The next regular polygon is hexagon. However, if we connect three hexagons at one point, then we get a surface, that is, it is impossible to build a three-dimensional figure from hexagons. Any other regular polygons above a hexagon cannot form solids at all. From these considerations it follows that there are only five regular polyhedra whose faces can only be equilateral triangles, squares and pentagons.

There are amazing geometric connections between all regular polyhedra. For example, cube(Fig.1-b) and octahedron(Fig.1-c) are dual, i.e. are obtained from each other if the centroids of the faces of one are taken as the vertices of the other and vice versa. Similarly dual icosahedron(Fig.1-d) and dodecahedron(Fig.1-d) . Tetrahedron(Fig.1-a) is dual to itself. The dodecahedron is obtained from the cube by constructing "roofs" on its faces (Euclid's method), the vertices of the tetrahedron are any four vertices of the cube that are not pairwise adjacent along the edge, that is, all other regular polyhedra can be obtained from the cube. The very fact that there are only five really regular polyhedra is amazing - after all, there are infinitely many regular polygons on the plane!

Numerical characteristics of the Platonic solids

The main numerical characteristics Platonic solids is the number of sides of the face m, the number of faces converging at each vertex, m, number of faces G, number of vertices AT, number of ribs R and the number of flat corners At on the surface of a polyhedron, Euler discovered and proved the famous formula

V - P + G = 2,

linking the number of vertices, edges, and faces of any convex polyhedron. The above numerical characteristics are given in Table. one.

Table 1

Numerical characteristics of the Platonic solids

Golden ratio in dodecahedron and icosahedron

The dodecahedron and its dual icosahedron (Fig. 1-d, e) occupy a special place among Platonic solids. First of all, it must be emphasized that the geometry dodecahedron and icosahedron directly related to the golden ratio. Indeed, the edges dodecahedron(Fig.1-e) are pentagons, i.e. regular pentagons based on the golden ratio. If you look closely at icosahedron(Fig. 1-d), then you can see that five triangles converge at each of its vertices, the outer sides of which form pentagon. Already these facts are enough to make sure that the golden ratio plays an essential role in the construction of these two Platonic solids.

But there is deeper mathematical evidence for the fundamental role played by the golden ratio in icosahedron and dodecahedron. It is known that these bodies have three specific spheres. The first (inner) sphere is inscribed in the body and touches its faces. Let us denote the radius of this inner sphere as R i. The second or middle sphere touches her edges. Let us denote the radius of this sphere by R m . Finally, the third (outer) sphere is circumscribed around the body and passes through its vertices. Let's denote its radius by Rc. In geometry, it is proved that the values of the radii of the indicated spheres for dodecahedron and icosahedron, which has an edge of unit length, is expressed in terms of the golden ratio t (Table 2).

table 2

The golden ratio in the spheres of the dodecahedron and icosahedron


icosahedron
Dodecahedron

Note that the ratio of radii = is the same as for icosahedron, and for dodecahedron. Thus, if dodecahedron and icosahedron have the same inscribed spheres, then their circumscribed spheres are also equal to each other. The proof of this mathematical result is given in Beginnings Euclid.

In geometry, other relations are also known for dodecahedron and icosahedron confirming their connection with the golden ratio. For example, if we take icosahedron and dodecahedron with an edge length equal to one, and calculate their external area and volume, then they are expressed through the golden ratio (Table 3).

Table 3

Golden ratio in the outer area and volume of the dodecahedron and icosahedron

	icosahedron	Dodecahedron
outer area

Thus, there is a huge number of relationships obtained by ancient mathematicians, confirming the remarkable fact that it is the golden ratio is the main proportion of the dodecahedron and icosahedron, and this fact is especially interesting from the point of view of the so-called "dodecahedral-icosahedral doctrine", which we will consider below.

Plato's cosmology

The regular polyhedra considered above are called Platonic solids, since they occupied an important place in Plato's philosophical concept of the structure of the universe.

Plato (427-347 BC)

Four polyhedrons personified in it four essences or "elements". Tetrahedron symbolized Fire, since its top is directed upwards; icosahedron water, since it is the most "streamlined" polyhedron; Cube earth, as the most "stable" polyhedron; Octahedron Air, as the most “airy” polyhedron. Fifth polyhedron, Dodecahedron, embodied "everything that exists", "Universal mind", symbolized the entire universe and was considered the main geometric figure of the universe.

The ancient Greeks considered harmonious relationships to be the basis of the universe, so the four elements were connected by such a proportion: earth / water = air / fire. The atoms of the "elements" were tuned by Plato in perfect consonances, like the four strings of a lyre. Recall that consonance is a pleasant consonance. In connection with these bodies, it would be appropriate to say that such a system of elements, which included four elements - earth, water, air and fire - was canonized by Aristotle. These elements remained the four cornerstones of the universe for many centuries. It is quite possible to identify them with the four states of matter known to us - solid, liquid, gaseous and plasma.

Thus, the ancient Greeks associated the idea of the "through" harmony of being with its embodiment in the Platonic solids. The influence of the famous Greek thinker Plato also affected Beginnings Euclid. In this book, which for centuries was the only textbook of geometry, a description of "ideal" lines and "ideal" figures is given. The most "ideal" line - straight, and the most "ideal" polygon - regular polygon, having equal sides and equal angles. The simplest regular polygon can be considered equilateral triangle, since it has the smallest number of sides that can delimit a part of the plane. It's interesting that Beginnings Euclid begin with a description of the construction right triangle and end with five Platonic solids. notice, that Platonic solids devoted to the final, that is, the 13th book Began Euclid. By the way, this fact, that is, the placement of the theory of regular polyhedra in the final (that is, as it were, the most important) book Began Euclid, gave rise to the ancient Greek mathematician Proclus, who was a commentator on Euclid, to put forward an interesting hypothesis about the true goals pursued by Euclid, creating his Beginnings. According to Proclus, Euclid created Beginnings not for the purpose of presenting geometry as such, but to give a complete systematized theory of the construction of "ideal" figures, in particular five Platonic solids, along the way highlighting some of the latest achievements in mathematics!

It is no coincidence that one of the authors of the discovery of fullerenes, Nobel laureate Harold Kroto, in his Nobel lecture, begins his story about symmetry as “the basis of our perception of the physical world” and its “role in attempts to explain it comprehensively” precisely with Platonic solids and "the elements of all things": “The concept of structural symmetry dates back to antiquity...” The most famous examples can, of course, be found in Plato’s Timaeus, where in section 53, referring to the “Elements”, he writes: “Firstly, to each (!) , of course, it is clear that fire and earth, water and air are bodies, and every body is solid ”(!!) Plato discusses the problems of chemistry in the language of these four elements and connects them with four Platonic solids (at that time only four, while Hipparchus did not discover the fifth - the dodecahedron). Although at first glance such a philosophy may seem somewhat naive, it indicates a deep understanding of how Nature actually functions.

Semiregular polyhedra

Many more perfect bodies are known, called semi-regular polyhedra or Archimedean bodies. They also have all polyhedral angles equal and all faces are regular polygons, but several different types. There are 13 semi-regular polyhedra whose discovery is attributed to Archimedes.

Archimedes (287 BC - 212 BC)

Lots of Archimedean solids can be divided into several groups. The first of these consists of five polyhedra, which are obtained from Platonic solids as a result of their truncation. A truncated body is a body with a cut off top. For Platonic solids truncation can be done in such a way that both the resulting new faces and the remaining parts of the old ones are regular polygons. For example, tetrahedron(Fig. 1-a) can be truncated so that its four triangular faces turn into four hexagonal ones, and four regular triangular faces are added to them. In this way, five Archimedean solids: truncated tetrahedron, truncated hexahedron (cube), truncated octahedron, truncated dodecahedron and truncated icosahedron(Fig. 2).


(a)	(b)	(in)


(G)	(e)

Figure 2. Archimedean solids: (a) truncated tetrahedron, (b) truncated cube, (c) truncated octahedron, (d) truncated dodecahedron, (e) truncated icosahedron

In his Nobel lecture, the American scientist Smalley, one of the authors of the experimental discovery of fullerenes, speaks of Archimedes (287-212 BC) as the first researcher of truncated polyhedra, in particular, truncated icosahedron, however, with the stipulation that perhaps Archimedes appropriates this merit and, perhaps, icosahedrons were truncated long before him. Suffice it to mention those found in Scotland and dated around 2000 BC. hundreds of stone objects (apparently for ritual purposes) in the form of spheres and various polyhedra(bodies bounded on all sides by flat faces), including icosahedrons and dodecahedrons. The original work of Archimedes, unfortunately, has not been preserved, and its results have come down to us, as they say, “second hand”. During the Renaissance all Archimedean solids one after another were "discovered" anew. In the end, Kepler in 1619 in his book "World Harmony" ("Harmonice Mundi") gave an exhaustive description of the entire set of Archimedean solids - polyhedra, each face of which is regular polygon, and all peaks are in an equivalent position (like carbon atoms in the C 60 molecule). Archimedean solids consist of at least two various types polygons, as opposed to 5 Platonic solids, all faces of which are the same (as in the C 20 molecule, for example).

Figure 3. Construction of the Archimedean truncated icosahedron
from Platonic icosahedron

So how do you construct Archimedean truncated icosahedron from Platonic icosahedron? The answer is illustrated with the help of Fig. 3. Indeed, as can be seen from Table. 1, 5 faces converge at any of the 12 vertices of the icosahedron. If at each vertex 12 parts of the icosahedron are cut off (cut off) by a plane, then 12 new pentagonal faces are formed. Together with the already existing 20 faces, which have turned from triangular to hexagonal after such a cut, they will make up 32 faces of a truncated icosahedron. In this case, there will be 90 edges, and 60 vertices.

another group Archimedean solids make up two bodies called quasi-correct polyhedra. The "quasi" particle emphasizes that the faces of these polyhedra are regular polygons of only two types, with each face of one type surrounded by polygons of another type. These two bodies are called rhombicuboctahedron and icosidodecahedron(Fig. 4).

Figure 5. Archimedean solids: (a) rhombicuboctahedron, (b) rhombicosidodecahedron

Finally, there are two so-called "snub" modifications - one for the cube ( snub cube), the other is for the dodecahedron ( snub dodecahedron) (Fig. 6).


(a)	(b)

Figure 6 Archimedean solids: (a) snub cube, (b) snub dodecahedron

In the mentioned book by Wenniger "Models of Polyhedra" (1974) the reader can find 75 different models of regular polyhedra. "The theory of polyhedra, in particular convex polyhedra, is one of the most fascinating chapters of geometry"- this is the opinion of the Russian mathematician L.A. Lyusternak, who did a lot in this area of mathematics. The development of this theory is associated with the names of prominent scientists. A great contribution to the development of the theory of polyhedra was made by Johannes Kepler (1571-1630). At one time he wrote the sketch "About a snowflake", in which he made the following remark: “Among the regular bodies, the very first, the beginning and progenitor of the rest is the cube, and its consort, if I may say so, is the octahedron, for the octahedron has as many angles as the cube has faces.” Kepler was the first to publish full list thirteen Archimedean solids and gave them the names by which they are known to this day.

Kepler was the first to study the so-called star polyhedra, which, unlike the Platonic and Archimedean solids, are regular convex polyhedra. At the beginning of the last century, the French mathematician and mechanic L. Poinsot (1777-1859), whose geometric works relate to star-shaped polyhedra, developed the work of Kepler and discovered the existence of two more types of regular non-convex polyhedra. So, thanks to the work of Kepler and Poinsot, four types of such figures became known (Fig. 7). In 1812, O. Cauchy proved that there are no other regular star-shaped polyhedra.

Figure 7 Regular star polyhedra (Poinsot solids)

Many readers may have a question: “Why study regular polyhedra at all? What is the use of them?" This question can be answered: “And what is the use of music or poetry? Is everything beautiful useful? The polyhedra models shown in Figs. 1-7, first of all, make an aesthetic impression on us and can be used as decorative ornaments. But in fact, the wide manifestation of regular polyhedra in natural structures caused great interest in this branch of geometry in modern science.

What is a calendar?

A Russian proverb says: "Time is the eye of history." Everything that exists in the Universe: the Sun, Earth, stars, planets, known and unknown worlds, and everything that exists in nature, living and inanimate, everything has a space-time dimension. Time is measured by observing periodically repeating processes of a certain duration.

Even in ancient times, people noticed that the day always gives way to night, and the seasons pass in a strict sequence: spring follows winter, summer follows spring, autumn follows summer. In search of a clue to these phenomena, man drew attention to the heavenly bodies - the Sun, the Moon, the stars - and to the rigorous periodicity of their movement across the sky. These were the first observations that preceded the birth of one of the most ancient sciences - astronomy.

Astronomy based the measurement of time on the movement of celestial bodies, which reflects three factors: the rotation of the Earth around its axis, the revolution of the Moon around the Earth, and the movement of the Earth around the Sun. On which of these phenomena the measurement of time is based, different concepts of time also depend. Astronomy knows stellar time, sunny time, local time, waist time, maternity leave time, atomic time, etc.

The sun, like all other luminaries, is involved in movement across the sky. In addition to the daily movement, the Sun has the so-called annual movement, and the entire path of the annual movement of the Sun across the sky is called ecliptic. If, for example, we notice the location of the constellations at a certain evening hour, and then repeat this observation every month, then a different picture of the sky will appear before us. The view of the starry sky changes continuously: each season has its own picture of the evening constellations, and each such picture is repeated every year. Consequently, after the expiration of the year, the Sun in relation to the stars returns to its original place.

For the convenience of orientation in the stellar world, astronomers divided the entire sky into 88 constellations. Each of them has its own name. Of the 88 constellations, a special place in astronomy is occupied by those through which the ecliptic passes. These constellations, in addition to their own names, also have a generalized name - zodiacal(from Greek word"zoop" - an animal), as well as symbols (signs) widely known throughout the world and various allegorical images included in the calendar systems.

It is known that in the process of moving along the ecliptic, the Sun crosses 13 constellations. However, astronomers found it necessary to divide the path of the Sun not into 13, but into 12 parts, uniting the constellations Scorpio and Ophiuchus into one - under the general name Scorpio (why?).

The problems of measuring time are dealt with by a special science called chronology. It underlies all calendar systems created by mankind. The creation of calendars in antiquity was one of the most important tasks of astronomy.

What is a "calendar" and what are calendar systems? Word calendar comes from the Latin word calendarium, which literally means "debt book"; in such books the first days of each month were indicated - calends, in which in Ancient Rome debtors pay interest.

Since ancient times in the countries of Eastern and South-East Asia when making calendars great importance gave the periodicity of the movement of the Sun, Moon, as well as Jupiter and Saturn, the two giant planets of the solar system. There is reason to believe that the idea of creating jupiterian calendar with celestial symbolism of the 12-year animal cycle associated with rotation Jupiter around the Sun, which makes a complete revolution around the Sun in about 12 years (11.862 years). On the other hand, the second giant planet of the solar system - Saturn makes a complete revolution around the Sun in about 30 years (29.458 years). Wanting to coordinate the cycles of motion of the giant planets, the ancient Chinese came up with the idea of introducing a 60-year cycle of the solar system. During this cycle, Saturn makes 2 complete revolutions around the Sun, and Jupiter - 5 revolutions.

When creating annual calendars, astronomical phenomena are used: the change of day and night, the change in lunar phases and the change of seasons. The use of various astronomical phenomena led to the creation of three types of calendars among various peoples: lunar, based on the movement of the moon, solar, based on the movement of the sun, and lunisolar.

Structure of the Egyptian calendar

One of the first solar calendars was Egyptian, created in the 4th millennium BC. The original Egyptian calendar year consisted of 360 days. The year was divided into 12 months of exactly 30 days each. However, later it was found that such a duration of the calendar year does not correspond to the astronomical one. And then the Egyptians added 5 more days to the calendar year, which, however, were not the days of the months. It was 5 public holidays connecting adjacent calendar years. Thus, the Egyptian calendar year had the following structure: 365 = 12ґ 30 + 5. Note that it is the Egyptian calendar that is the prototype of the modern calendar.

The question arises: why did the Egyptians divide the calendar year into 12 months? After all, there were calendars with a different number of months in the year. For example, in the Mayan calendar, the year consisted of 18 months of 20 days per month. The next question regarding the Egyptian calendar is why each month had exactly 30 days ( more precisely days)? Some questions can be raised about the Egyptian system of measuring time, in particular about the choice of such units of time as hour, minute, second. In particular, the question arises: why was the hour unit chosen in such a way that it fits exactly 24 times a day, that is, why 1 day = 24 (2ґ 12) hours? Further: why 1 hour = 60 minutes and 1 minute = 60 seconds? The same questions apply to the choice of units of angular quantities, in particular: why is the circle divided into 360°, that is, why 2p = 360° = 12ґ 30°? To these questions are added others, in particular: why did astronomers consider it expedient to consider that there are 12 zodiacal signs, although in fact, in the process of its movement along the ecliptic, the Sun crosses 13 constellations? And one more "strange" question: why did the Babylonian number system have a very unusual base - the number 60?

Relationship of the Egyptian calendar with the numerical characteristics of the dodecahedron

Analyzing the Egyptian calendar, as well as the Egyptian systems for measuring time and angular values, we find that four numbers are repeated with amazing constancy: 12, 30, 60 and the number 360 \u003d 12ґ 30 derived from them. The question arises: is there any then a fundamental scientific idea that could give a simple and logical explanation for the use of these numbers in the Egyptian systems?

To answer this question, we turn again to dodecahedron shown in Fig. 1-d. Recall that all geometric ratios of the dodecahedron are based on the golden ratio.

Did the Egyptians know the dodecahedron? Historians of mathematics admit that the ancient Egyptians had knowledge of regular polyhedra. But did they know all five regular polyhedra, in particular dodecahedron and icosahedron how the most difficult ones? The ancient Greek mathematician Proclus attributes the construction of regular polyhedra to Pythagoras. But many mathematical theorems and results (in particular, Pythagorean theorem) Pythagoras borrowed from the ancient Egyptians during his very long "business trip" to Egypt (according to some reports, Pythagoras lived in Egypt for 22 years!). Therefore, we can assume that Pythagoras also borrowed knowledge about regular polyhedra from the ancient Egyptians (and possibly from the ancient Babylonians, because according to legend, Pythagoras lived in ancient Babylon for 12 years). But there is other, more solid evidence that the Egyptians had information about all five regular polyhedra. In particular, in the British Museum there is a dice from the Ptolemaic era, which has the shape icosahedron, that is, the "Platonic solid", the dual dodecahedron. All these facts give us the right to put forward the hypothesis that The Egyptians knew the dodecahedron. And if this is so, then a very harmonious system follows from this hypothesis, which makes it possible to explain the origin of the Egyptian calendar, and at the same time the origin of the Egyptian system for measuring time intervals and geometric angles.

Earlier we established that the dodecahedron has 12 faces, 30 edges and 60 flat corners on its surface (Table 1). Based on the hypothesis that the Egyptians knew dodecahedron and its numerical characteristics are 12, 30. 60, then what was their surprise when they discovered that the cycles of the solar system are expressed by the same numbers, namely, the 12-year cycle of Jupiter, the 30-year cycle of Saturn and, finally, the 60- summer cycle of the solar system. Thus, between such a perfect spatial figure as dodecahedron, and solar system, there is a deep mathematical connection! This conclusion was made by ancient scientists. This led to the fact that dodecahedron was adopted as the "main figure", which symbolized Harmony of the Universe. And then the Egyptians decided that all their main systems (calendar system, time measurement system, angle measurement system) should correspond to numerical parameters. dodecahedron! Since, according to the ideas of the ancients, the movement of the Sun along the ecliptic had a strictly circular character, then, having chosen 12 signs of the Zodiac, the arc distance between which was exactly 30 °, the Egyptians surprisingly beautifully agreed annual movement Suns along the ecliptic with the structure of their calendar year: one month corresponded to the movement of the Sun along the ecliptic between two neighboring signs of the Zodiac! Moreover, the movement of the Sun by one degree corresponded to one day in the Egyptian calendar year! In this case, the ecliptic was automatically divided into 360°. Dividing each day into two parts, following the dodecahedron, the Egyptians then divided each half of the day into 12 parts (12 faces dodecahedron) and thus introduced hour is the most important unit of time. Dividing one hour into 60 minutes (60 flat corners on the surface dodecahedron), the Egyptians in this way introduced minute is the next important unit of time. Similarly, they entered give me a sec- the smallest unit of time for that period.

Thus, choosing dodecahedron as the main "harmonic" figure of the universe, and strictly following the numerical characteristics of the dodecahedron 12, 30, 60, the Egyptians managed to build an extremely harmonious calendar, as well as systems for measuring time and angular values. These systems were in full agreement with their "Theory of Harmony", based on the golden ratio, since it is this proportion that underlies dodecahedron.

These surprising conclusions follow from the comparison dodecahedron with the solar system. And if our hypothesis is correct (let someone try to refute it), then it follows that for many millennia, humanity has been living under the sign of the golden ratio! And every time we look at our watch face, which is also built on the use of numerical characteristics dodecahedron 12, 30 and 60, we touch the main "Mystery of the Universe" - the golden section, without knowing it!

On November 12, 1984, in a short article published in the authoritative journal Physical Review Letters by the Israeli physicist Dan Shechtman, experimental evidence was presented for the existence of a metal alloy with exceptional properties. When studied by electron diffraction methods, this alloy showed all the signs of a crystal. Its diffraction pattern is composed of bright and regularly spaced dots, just like a crystal. However, this picture is characterized by the presence of "icosahedral" or "pentangonal" symmetry, which is strictly forbidden in a crystal due to geometric considerations. Such unusual alloys were called quasicrystals. In less than a year, many other alloys of this type were discovered. There were so many of them that the quasi-crystalline state turned out to be much more common than one might imagine.

Israeli physicist Dan Shechtman

The concept of a quasicrystal is of fundamental interest because it generalizes and completes the definition of a crystal. A theory based on this concept replaces the age-old idea of "a structural unit repeated in space in a strictly periodic manner" with the key concept far order. As emphasized in the article "Quasicrystals" famous physicist D Gratia, “This concept has led to the expansion of crystallography, the rediscovered riches of which we are just beginning to explore. Its significance in the world of minerals can be put on a par with the addition of the concept of irrational numbers to rational ones in mathematics.

What is a quasicrystal? What are its properties and how can it be described? As mentioned above, according to fundamental law of crystallography strict restrictions are imposed on the crystal structure. According to classical ideas, a crystal is composed ad infinitum from a single cell, which should densely (face to face) “cover” the entire plane without any restrictions.

As is known, dense filling of the plane can be carried out using triangles(Fig.7-a), squares(Fig.7-b) and hexagons(Fig. 7-d). By using pentagons (pentagons) such filling is impossible (Fig. 7-c).


a)	b)	in)	G)

Figure 7 Dense filling of the plane can be done using triangles (a), squares (b) and hexagons (d)

These were the canons of traditional crystallography that existed before the discovery of an unusual alloy of aluminum and manganese, called a quasicrystal. Such an alloy is formed by ultrafast cooling of the melt at a rate of 10 6 K per second. At the same time, during a diffraction study of such an alloy, an ordered pattern is displayed on the screen, which is characteristic of the symmetry of the icosahedron, which has the famous forbidden symmetry axes of the 5th order.

Several scientific groups around the world over the next few years studied this unusual alloy through electron microscopy. high resolution. All of them confirmed the ideal homogeneity of matter, in which the 5th order symmetry was preserved in macroscopic regions with dimensions close to those of atoms (several tens of nanometers).

According to modern views, the following model has been developed for obtaining the crystal structure of a quasicrystal. This model is based on the concept of "basic element". According to this model, the inner icosahedron of aluminum atoms is surrounded by the outer icosahedron of manganese atoms. Icosahedrons are connected by octahedra of manganese atoms. The "base element" has 42 aluminum atoms and 12 manganese atoms. In the process of solidification, there is a rapid formation of "basic elements", which are quickly connected to each other by rigid octahedral "bridges". Recall that the faces of the icosahedron are equilateral triangles. To form an octahedral bridge of manganese, it is necessary that two such triangles (one in each cell) come close enough to each other and line up in parallel. As a result of such a physical process, a quasi-crystalline structure with "icosahedral" symmetry is formed.

In recent decades, many types of quasi-crystalline alloys have been discovered. In addition to having "icosahedral" symmetry (5th order), there are also alloys with decagonal symmetry (10th order) and dodecagonal symmetry (12th order). The physical properties of quasicrystals began to be investigated only recently.

What is the practical significance of the discovery of quasicrystals? As noted in Gratia's article cited above, “the mechanical strength of quasi-crystalline alloys increases dramatically; the absence of periodicity leads to a slowdown in the propagation of dislocations compared to conventional metals ... This property is of great practical importance: the use of the icosahedral phase will make it possible to obtain light and very strong alloys by introducing small particles of quasicrystals into an aluminum matrix.

What is the methodological significance of the discovery of quasicrystals? First of all, the discovery of quasicrystals is a moment of great triumph of the "dodecahedral-icosahedral doctrine", which permeates the entire history of natural science and is a source of deep and useful scientific ideas. Secondly, quasicrystals destroyed the traditional notion of an insurmountable divide between the world of minerals, in which "pentagonal" symmetry was forbidden, and the world of wildlife, where "pentagonal" symmetry is one of the most common. And we should not forget that the main proportion of the icosahedron is the "golden ratio". And the discovery of quasicrystals is another scientific confirmation that, perhaps, it is the “golden proportion”, which manifests itself both in the world of wildlife and in the world of minerals, is the main proportion of the Universe.

When Dan Shechtman gave experimental proof of the existence of quasicrystals with icosahedral symmetry, physicists in search of a theoretical explanation for the phenomenon of quasicrystals, drew attention to a mathematical discovery made 10 years earlier by the English mathematician Roger Penrose. As a "flat analogue" of quasicrystals, we chose penrose tiles, which are aperiodic regular structures formed by "thick" and "thin" rhombuses, obeying the proportions of the "golden section". Exactly penrose tiles were adopted by crystallographers to explain the phenomenon quasicrystals. At the same time, the role Penrose diamonds in the space of three dimensions began to play icosahedra, with the help of which dense filling of three-dimensional space is carried out.

Consider again carefully the pentagon in Fig. eight.

Figure 8 Pentagon

After drawing diagonals in it, the original pentagon can be represented as a set of three types geometric shapes. In the center is a new pentagon formed by the intersection points of the diagonals. In addition, the pentagon in Fig. 8 includes five isosceles triangles colored in yellow, and five isosceles triangles colored red. The yellow triangles are "gold" because the ratio of the hip to the base is equal to the golden ratio; they have acute angles of 36° at the apex and acute angles of 72° at the base. The red triangles are also "golden", since the ratio of the hip to the base is equal to the golden ratio; they have an obtuse angle of 108° at the apex and acute angles of 36° at the base.

And now let's connect two yellow triangles and two red triangles with their bases. As a result, we get two "golden" rhombus. The first one (yellow) has sharp corner at 36° and an obtuse angle at 144° (Fig. 9).


(a)	(b)

Figure 9. " Golden" rhombuses: a) "thin" rhombus; (b) "thick" rhombus

Rhombus in Fig. 9-a we will call thin rhombus, and the rhombus in Fig. 9-b - thick rhombus.

The English mathematician and physicist Rogers Penrose used "golden" rhombuses in Fig. 9 for the construction of the "golden" parquet, which was named Penrose tiles. Penrose tiles are a combination of thick and thin diamonds, shown in Fig. ten.

Figure 10. Penrose tiles

It is important to emphasize that penrose tiles have "pentagonal" symmetry or symmetry of the 5th order, and the ratio of the number of thick rhombuses to thin ones tends to the golden ratio!

And now let's talk about another outstanding modern discovery in the field of chemistry. This discovery was made in 1985, that is, a few years later than quasicrystals. We are talking about the so-called "fullerenes". The term "fullerenes" refers to closed molecules such as C 60 , C 70 , C 76 , C 84 , in which all carbon atoms are located on a spherical or spheroidal surface. In these molecules, carbon atoms are located at the vertices of regular hexagons or pentagons that cover the surface of a sphere or spheroid. The central place among fullerenes is occupied by the C 60 molecule, which is characterized by the highest symmetry and, as a result, the highest stability. In this molecule, resembling a soccer ball tire and having the structure of a regular truncated icosahedron (Fig. 2e and Fig. 3), carbon atoms are located on a spherical surface at the vertices of 20 regular hexagons and 12 regular pentagons, so that each hexagon borders on three hexagons and three pentagons, and each pentagon is bordered by hexagons.

The term "fullerene" originates from the name of the American architect Buckminster Fuller, who, it turns out, used such structures when constructing domes of buildings (another use of a truncated icosahedron!).

"Fullerenes" are essentially "man-made" structures derived from fundamental physics research. For the first time they were synthesized by the scientists G. Kroto and R. Smalley (who received in 1996 Nobel Prize for this discovery). But they were unexpectedly found in the rocks of the Precambrian period, that is, fullerenes turned out to be not only "man-made", but natural formations. Now fullerenes are being intensively studied in laboratories. different countries, trying to establish the conditions for their formation, structure, properties and possible areas of application. The most fully studied representative of the fullerene family is fullerene-60 (C 60) (it is sometimes called buckminster-fullerene. Fullerenes C 70 and C 84 are also known. Fullerene C 60 is obtained by evaporation of graphite in a helium atmosphere. This forms a fine, soot-like powder containing 10% carbon, when dissolved in benzene, the powder gives a red solution, from which C 60 crystals are grown. physical properties. So, at high pressure, C 60 becomes hard, like a diamond. Its molecules form a crystalline structure, as if consisting of perfectly smooth balls, freely rotating in a face-centered cubic lattice. Due to this property, C 60 can be used as a solid lubricant. Fullerenes also have magnetic and superconducting properties.

Russian scientists A.V. Yeletsky and B.M. Smirnov in his article "Fullerenes", published in the journal "Uspekhi fizicheskikh nauk" (1993, volume 163, no. 2), notes that "fullerenes, the existence of which was established in the mid 80s and efficient technology the isolation of which was developed in 1990, has now become the subject of intensive research by dozens of scientific groups. The results of these studies are closely monitored by application firms. Since this modification of carbon has presented scientists with a number of surprises, it would be unwise to discuss predictions and possible consequences study of fullerenes in the next decade, but one should be prepared for new surprises.”

Matjuska Teja Krasek received her bachelor's degree in painting from the College visual arts(Ljubljana, Slovenia) and is a freelance artist. Lives and works in Ljubljana. Her theoretical and practical work focuses on symmetry as a connecting concept between art and science. Her artwork has been presented at many international exhibitions and published in international journals (Leonardo Journal, Leonardo on-line).

M.T. Kraszek at his exhibition ‘Kaleidoscopic Fragrances’, Ljubljana, 2005

The artistic work of Matyushka Teija Kraszek is associated with various types of symmetry, Penrose tiles and rhombuses, quasicrystals, the golden section as the main element of symmetry, Fibonacci numbers, etc. With the help of reflection, imagination and intuition, she tries to find new relationships, new levels of structure, new and different kinds of order in these elements and structures. In her works, she makes extensive use of computer graphics as a very useful medium for creating artworks, which is the link between science, mathematics and art.

On Fig. 11 shows the composition of T.M. Crashek associated with Fibonacci numbers. If we choose one of the Fibonacci numbers (for example, 21 cm) for the length of the side of the Penrose diamond in this perceptibly unstable composition, we can observe how the lengths of some segments in the composition form the Fibonacci sequence.

Figure 11. Matushka Teija Kraszek "Fibonacci numbers", canvas, 1998.

A large number of artistic compositions of the artist are devoted to Shechtman's quasi-crystals and Penrose lattices (Fig. 12).


(a)	(b)


(in)	(G)

Figure 12. The world of Theia Kraszek: (a) The world of quasicrystals. Computer Graphics, 1996.
(b) Stars. Computer Graphics, 1998 (c) 10/5. Holst, 1998 (d) Quasicube. Canvas, 1999

In the composition of Matyushka Teija Kraszek and Clifford Pickover "Biogenesis", 2005 (Fig. 13), a decagon is presented, consisting of Penrose rhombuses. One can observe the relationship between the Petrouse diamonds; every two adjacent Penrose diamonds form a pentagonal star.

Figure 13. Matushka Theia Kraszek and Clifford Pickover. Biogenesis, 2005.

in the picture Double Star GA(Figure 14) we see how the Penrose tiles fit together to form a two-dimensional representation of a potentially hyperdimensional object with a decagonal base. When depicting the painting, the artist used the hard edges method proposed by Leonardo da Vinci. It is this method of representation that allows us to see in the projection of the picture onto a plane a large number of pentagons and pentacles, which are formed by the projections of individual edges of Penrose rhombuses. In addition, in the projection of the picture onto a plane, we see a decagon formed by the edges of 10 adjacent Penrose rhombuses. In essence, in this picture, Matyushka Teija Kraszek found a new regular polyhedron, which quite possibly really exists in nature.

Figure 14. Matushka Teia Kraszek. Double Star GA

In the composition of Crashek "Stars for Donald" (Fig. 15), we can observe the endless interaction of Penrose rhombuses, pentagrams, pentagons, decreasing towards the central point of the composition. Golden ratio ratios are represented in many different ways on different scales.

Figure 15. Matyushka Teija Kraszek "Stars for Donald", computer graphics, 2005.

The artistic compositions of Matyushka Teija Kraszek attracted great attention from representatives of science and art. Her art is equated with the art of Maurits Escher and the Slovenian artist is called the "Eastern European Escher" and the "Slovenian gift" to world art.

Stakhov A.P. "The Da Vinci Code", Platonic and Archimedean solids, quasicrystals, fullerenes, Penrose lattices and the artistic world of Matyushka Teija Kraszek // "Academy of Trinitarianism", M., El No. 77-6567, publ. 12561, 07.11.2005

This coursework is designed to:

1) consolidate, deepen and expand theoretical knowledge in the field of methods for modeling surfaces and objects, practical skills and skills of software implementation of methods;

2) improve the skills of independent work;

3) to develop the ability to formulate judgments and conclusions, to state them logically and conclusively.

Polyhedron

The number of sides of the face, m

The number of faces converging at the vertex, n

Number of faces

Number of vertices

Number of ribs

Number of flat corners on a surface

Tetrahedron

Hexahedron (cube)

icosahedron

Dodecahedron

Plato's solids are convex polyhedra, all of whose faces are regular polygons. All polyhedral angles of a regular polyhedron are congruent. As follows already from the calculation of the sum of flat angles at the vertex, there are no more than five convex regular polyhedra. In the way indicated below, it can be proved that there are precisely five regular polyhedra (this was proved by Euclid). They are the regular tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron. The names of these regular polyhedra come from Greece. AT literal translation from the Greek "tetrahedron", "octahedron", "hexahedron", "dodecahedron", "icosahedron" mean: "tetrahedron", "octahedron", "hexahedron". dodecahedron, dodecahedron.

Table No. 1

Table number 2

Name:	Radius of the circumscribed sphere	Radius of the inscribed sphere
Tetrahedron
Hexahedron

Dodecahedron
icosahedron

Tetrahedron- a tetrahedron, all faces of which are triangles, i.e. triangular pyramid; a regular tetrahedron is bounded by four equilateral triangles. (Fig. 1).

Cube or regular hexahedron- correct quadrangular prism with equal edges, bounded by six squares. (Fig. 1).

Octahedron- octahedron; a body bounded by eight triangles; a regular octahedron is bounded by eight equilateral triangles; one of the five regular polyhedra. (Fig. 1).

Dodecahedron- dodecahedron, a body bounded by twelve polygons; regular pentagon. (Fig. 1).

icosahedron- a twenty-sided body, a body bounded by twenty polygons; a regular icosahedron is bounded by twenty equilateral triangles. (Fig. 1).

The cube and the octahedron are dual, i.e. are obtained from each other if the centroids of the faces of one are taken as the vertices of the other and vice versa. The dodecahedron and the icosahedron are similarly dual. The tetrahedron is dual to itself. A regular dodecahedron is obtained from a cube by constructing “roofs” on its faces (Euclid's method), the vertices of a tetrahedron are any four vertices of the cube that are not pairwise adjacent along an edge. This is how all other regular polyhedra are obtained from the cube. The very fact of the existence of only five really regular polyhedra is amazing - after all, there are infinitely many regular polygons on the plane!

All regular polyhedra were known in ancient Greece, and the 13th book of Euclid's "Beginnings" is dedicated to them. They are also called the bodies of Plato, because. they occupied an important place in Plato's philosophical concept of the structure of the universe. Four polyhedrons personified in it four essences or "elements". The tetrahedron symbolized fire, because. its top is directed upwards; icosahedron? water, because he is the most "streamlined"; cube - earth, as the most "steady"; octahedron? air, as the most "airy". The fifth polyhedron, the dodecahedron, embodied "everything that exists", symbolized the entire universe, and was considered the main one.

The ancient Greeks considered harmonious relationships to be the basis of the universe, so the four elements were connected by such a proportion: earth / water = air / fire.

In connection with these bodies, it would be appropriate to say that the first system of elements, which included four elements? earth, water, air and fire - was canonized by Aristotle. These elements remained the four cornerstones of the universe for many centuries. It is quite possible to identify them with the four states of matter known to us - solid, liquid, gaseous and plasma.

An important place was occupied by regular polyhedra in the system of the harmonious structure of the world by I. Kepler. All the same faith in harmony, beauty and the mathematically regular structure of the universe led I. Kepler to the idea that since there are five regular polyhedra, only six planets correspond to them. In his opinion, the spheres of the planets are interconnected by the Platonic solids inscribed in them. Since for each regular polyhedron the centers of the inscribed and circumscribed spheres coincide, the whole model will have a single center, in which the Sun will be located.

Having done a huge computational work, in 1596 I. Kepler published the results of his discovery in the book "The Secret of the Universe". He inscribes a cube in the sphere of Saturn's orbit, in a cube? the sphere of Jupiter, the sphere of Jupiter - a tetrahedron, and so on successively fit into each other the sphere of Mars? dodecahedron, the sphere of the earth? icosahedron, sphere of Venus? octahedron, the sphere of Mercury. The secret of the universe seems open.

Today it is safe to say that the distances between the planets are not related to any polyhedra. However, it is possible that without the "Secrets of the Universe", "Harmony of the World" by I. Kepler, regular polyhedra there would not have been three famous laws of I. Kepler, which play an important role in describing the motion of the planets.

Where else can you see these amazing bodies? In the book of the German biologist of the beginning of the last century, E. Haeckel, "The Beauty of Forms in Nature," one can read the following lines: "Nature nourishes in its bosom an inexhaustible number of amazing creatures that far surpass all forms created by human art in beauty and diversity." The creations of nature in this book are beautiful and symmetrical. This is an inseparable property of natural harmony. But here you can also see unicellular organisms? feodarii, the shape of which accurately conveys the icosahedron. What caused such a natural geometrization? Maybe because of all the polyhedra with the same number of faces, it is the icosahedron that has the largest volume and smallest area surfaces. This geometric property helps the marine microorganism overcome the pressure of the water column.

It is also interesting that it was the icosahedron that turned out to be the focus of attention of biologists in their disputes regarding the shape of viruses. The virus cannot be perfectly round, as previously thought. To establish its shape, they took various polyhedrons, directed light at them at the same angles as the flow of atoms to the virus. It turned out that only one polyhedron gives exactly the same shadow? icosahedron. His geometric properties, which were mentioned above, allow saving genetic information. Regular polyhedra? the most profitable figures. And nature takes advantage of this. The crystals of some substances familiar to us are in the form of regular polyhedra. So, the cube conveys the shape of sodium chloride crystals NaCl, the single crystal of aluminum-potassium alum (KAlSO4) 2 12H2O has the shape of an octahedron, the crystal of pyrite sulfide FeS has the shape of a dodecahedron, antimony sodium sulfate is a tetrahedron, boron is an icosahedron. Regular polyhedra define shape crystal lattices some chemicals.

So, the regular polyhedra revealed to us the attempts of scientists to approach the secret of world harmony and showed the irresistible attractiveness and beauty of these geometric figures.

Even in ancient times, people noticed that some three-dimensional figures have special properties. These are the so-called regular polyhedra- all their faces are the same, all the angles at the vertices are equal. Each of these figures is stable and can be inscribed in a sphere. With all the variety of different shapes, there are only 5 types of regular polyhedra (Fig. 1).

Tetrahedron- a regular tetrahedron, the faces are equilateral triangles (Fig. 1a).

Cube- correct hexagon, the faces are squares (Fig. 1b).

Octahedron- a regular octahedron, the faces are equilateral triangles (Fig. 1c).

Dodecahedron- a regular dodecahedron, the faces are regular pentagons (Fig. 1d).

icosahedron- a regular twenty-hedron, the faces are equilateral triangles (Fig. 1e).

The ancient Greek philosopher Plato believed that each of the regular polyhedra corresponds to one of the 5 primary elements. According to Plato, the cube corresponds to earth, the tetrahedron to fire, the octahedron to air, the icosahedron to water, and the dodecahedron to ether. In addition, Greek philosophers singled out another primary element - emptiness. It matches geometric shape a sphere into which all the Platonic solids can be inscribed.

All six elements are the building blocks of the universe. Some of them are common - earth, water, fire and air. Today it is known for certain that regular polyhedra, or Platonic solids, form the basis of the structure of crystals, molecules of various chemicals.

The human energy shell is also a spatial configuration. The outer boundary of the human energy field is a sphere, the figure closest to it is a dodecahedron. Then the figures of the energy field replace each other in a certain order, repeating in different cycles. For example, in a DNA molecule, icosahedrons and dodecahedrons alternate.

It has been found that the Platonic solids are able to have a beneficial effect on a person. These forms have the ability to modify, organize energy in the chakras of the human body. Moreover, each crystalline form has a beneficial effect on the chakra, the primary element of which it corresponds to.

The imbalance of energies in Muladhara disappears when using the cube (earth element), Svadhisthana reacts to the impact of the icosahedron (water element), the tetrahedron (fire element) has a beneficial effect on Manipura, Anahata functions are restored with the help of the octahedron (air element). The same figure contributes to the normal functioning of Vishuddha. Both upper chakras - Ajna and Sahasrara - can be corrected with a dodecahedron.

In order to use the properties of the Platonic solids, it is necessary to make these figures from copper wire (size from 10 to 30 cm in diameter). You can draw them on paper or glue them out of cardboard, but copper wire frames are more effective. Models of the Platonic solids need to be attached to the projections of the corresponding chakras and lie down for a bit in deep relaxation.