Introduction

This course work is designed to:

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

2) improve independent work skills;

3) develop the ability to formulate judgments and conclusions, present them logically, consistently and demonstrably.

Plato's solids

Platonic solids are convex polyhedra, all of whose faces are regular polygons. All polyhedral angles of a regular polyhedron are congruent. As this already follows from calculating the sum of flat angles at the vertex, convex regular polyhedra no more than five. Using the method indicated below, one can prove that there are exactly five regular polyhedra (this was proven by Euclid). They are regular tetrahedron, hexahedron(cube), octahedron, dodecahedron and icosahedron. The names of these regular polyhedra come from Greece. Literally translated from Greek, “tetrahedron”, “octahedron”, “hexahedron”, “dodecahedron”, “icosahedron” mean: “tetrahedron”, “octahedron”, “hexahedron”. "dodecahedron", "twenty-hedron".

Table No. 1

Table No. 2

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

Tetrahedron- a tetrahedron, all of whose faces 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- twenty-sided, a body bounded by twenty polygons; The regular icosahedron is limited by twenty equilateral triangles. (Fig. 1).

The cube and the octahedron are dual, i.e. are obtained from each other if the centers of gravity of the faces of one are taken as the vertices of the other and vice versa. The dodecahedron and icosahedron are similarly dual. The tetrahedron is dual to itself. A regular dodecahedron is obtained from a cube by constructing “roofs” on its faces (Euclidean method); the vertices of the 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 truly regular polyhedra is surprising - after all, there are infinitely many regular polygons on the plane!

All regular polyhedra were known back in Ancient Greece, and the 13th book of Euclid’s Elements is dedicated to them. They are also called Platonic solids, because. they occupied an important place in Plato’s philosophical concept of the structure of the universe. Four polyhedrons personified four essences or “elements” in it. The tetrahedron symbolized fire, because. its top is directed upward; Icosahedron? water, because it is the most “streamlined”; cube - earth, as the most “stable”; 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 their four elements were connected by the following 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.

Regular polyhedra occupied an important place in I. Kepler’s system of harmonious structure of the world. The same belief 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 entire model will have a single center in which the Sun will be located.

Having done a tremendous amount of computational work, in 1596 I. Kepler published the results of his discovery in the book “The Mystery of the Universe.” He inscribes a cube into the sphere of Saturn's orbit, into a cube? the sphere of Jupiter, the tetrahedron in the sphere of Jupiter, and so on, do the sphere of Mars fit into each other sequentially? dodecahedron, sphere of the Earth? Icosahedron, sphere of Venus? octahedron, sphere of Mercury. The mystery of the universe seems to be open.

Today we can say with confidence that the distances between planets are not related to any polyhedra. However, it is possible that without the “Mystery 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 movement of 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,” you can read the following lines: “Nature nurtures in its bosom an inexhaustible number of amazing creatures, which in beauty and diversity far surpass all forms created by human art.” The creatures of nature shown in this book are beautiful and symmetrical. This is an inseparable property of natural harmony. But are there also unicellular organisms visible here? feodaria, the shape of which accurately reflects the icosahedron. What causes this natural geometrization? Perhaps because of all the polyhedra with the same number of faces, it is the icosahedron that has the largest volume and the smallest surface area. This geometric property helps the marine microorganism overcome the pressure of the water column.

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

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

Stakhov A.P.

“The Da Vinci Code”, Platonic and Archimedean solids, quasicrystals, fullerenes, Penrose lattices and the artistic world of Mother Teia Krashek

annotation

The work of the Slovenian artist Matyushka Teja Krašek 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 semiregular polygons (Platonic and Archimedean solids), the Golden Ratio and Fibonacci numbers.

What is the Da Vinci Code?

Surely every person has thought more than once about the question of 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 was during this period of human history that scientists came to a number of amazing discoveries, which permeate the entire history of science. The first of them is rightfully considered to be a wonderful mathematical proportion expressing Harmony. It is called differently: “golden proportion”, “golden number”, “golden average”, “golden ratio” and even "divine proportion" The Golden Ratio is also called number of PHI in honor of the great ancient Greek sculptor Phidias, who used this number in his sculptures.

The thriller "The Da Vinci Code", written by the popular English writer Dan Brown, has become a bestseller of the 21st century. But what does the Da Vinci Code mean? There are different answers to this question. It is known that the famous “Golden Section” was the subject close attention and hobbies of Leonardo da Vinci. Moreover, the very name “Golden Section” was introduced into European culture by Leonardo da Vinci. At Leonardo’s initiative, the famous Italian mathematician and scientific monk Luca Pacioli, a friend and scientific adviser to Leonardo da Vinci, 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 us the right to put forward the hypothesis that the “Da Vinci Code” is nothing more than the “Golden Ratio”. And confirmation of this hypothesis can be found in a lecture for students at Harvard University, which recalls main character books "The Da Vinci Code" by prof. Langdon:

“Despite its almost mystical origins, the PHI number played a unique role in its own way. The role of a 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 PHI number to 1. This ubiquity of PHI in nature... indicates the connection of all living things. It was previously believed that the PHI number was predetermined by the Creator of the universe. Scientists of antiquity called one point six hundred and eighteen thousandths the “divine proportion.”

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

Another mathematical discovery of ancient science is regular polyhedra which were named "Platonic solids" And "semiregular polyhedra", called "Archimedean solids". It is these amazingly beautiful spatial geometric figures that underlie two of the largest scientific discoveries of the 20th century - quasicrystals(the author of the discovery is Israeli physicist Dan Shekhtman) 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 (“Da Vinci Code”), which underlies the Universe.

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

Platonic solids

A person shows interest in regular polygons and polyhedra throughout his entire conscious activity - from a two-year-old child playing with wooden blocks to a mature mathematician. Some of the correct and semi the right bodies occur in nature in the form of crystals, others - in the form of viruses, which can be viewed using electron microscope.

What is a regular polyhedron? A regular polyhedron is such a polyhedron, all of whose 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 - there are as many regular polygons as there are. However, it is not. In Euclid's Elements we find a rigorous proof that there are only five convex regular polyhedra, and their faces can only be three types of regular polygons: triangles, squares And pentagons (regular pentagons).

Many books are devoted to the theory of polyhedra. One of the most famous is the book of the English mathematician M. Wenniger “Models of Polyhedra”. This book was published in Russian translation by the Mir publishing house in 1974. The epigraph to the book is a statement by Bertrand Russell: “Mathematics possesses not only truth, but also high beauty - beauty that is sharpened and strict, sublimely pure and striving for true 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 usually 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”),

(c) octahedron (“Air”), (d) icosahedron (“Water”), (e) dodecahedron (“Universal Mind”)

We will begin our consideration with regular polyhedra, the faces of which are equilateral triangles. The first one is tetrahedron(Fig.1-a). In a tetrahedron, three equilateral triangles meet at one vertex; at the same time, their bases form a new equilateral triangle. The tetrahedron has smallest number faces among the Platonic solids and is a three-dimensional analogue of a flat regular 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 their 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 will be a figure with 20 triangular faces - icosahedron(Fig.1-d).

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

Finally, there is another possibility of constructing a regular polyhedron, based on the use of 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, 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, the faces of which 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 centers of gravity 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-e) . Tetrahedron(Fig. 1-a) is dual to itself. A dodecahedron is obtained from a cube by constructing “roofs” on its faces (Euclidean method); the vertices of a tetrahedron are any four vertices of the cube that are not pairwise adjacent along an edge, that is, all other regular polyhedra can be obtained from the cube. The very fact of the existence of only five truly regular polyhedra is surprising - after all, there are infinitely many regular polygons on the plane!

Numerical characteristics of Platonic solids

Main numerical characteristics Platonic solids is the number of sides of the face m, the number of faces meeting at each vertex, m, number of faces G, number of vertices IN, number of ribs R and number of flat angles U on the surface of a polyhedron, Euler discovered and proved the famous formula

B P + G = 2,

connecting number of vertices, edges and faces of any convex polyhedron. The above numerical characteristics are given in Table. 1.

Table 1

Numerical characteristics of Platonic solids

Polyhedron	Number of edge sides m	The number of faces meeting at a vertex n	Number of faces	Number of vertices	Number of ribs	Number of flat angles on the surface
Tetrahedron
Hexahedron (cube)
Icosahedron
Dodecahedron

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, edges dodecahedron(Fig.1-d) 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, external sides which form pentagon. These facts alone are enough to convince us that the golden ratio plays a significant role in the design 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 by R i. The second or middle sphere touches its ribs. Let us denote the radius of this sphere by Rm. Finally, the third (outer) sphere is described around the body and passes through its vertices. Let's denote its radius by Rc. In geometry it has been proven that the values ​​of the radii of the indicated spheres for dodecahedron And icosahedron, having an edge of unit length, is expressed through the golden proportion t (Table 2).

table 2

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 identical 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 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 proportion (Table 3).

Table 3

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

	Icosahedron	Dodecahedron
External area

Thus, there is a huge number of relationships obtained by ancient mathematicians, confirming the remarkable fact that exactly 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 look at below.

Plato's cosmology

The regular polyhedra discussed 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 four essences or “elements” in it. Tetrahedron symbolized Fire, since its top is directed upward; Icosahedron — Water, since it is the most “streamlined” polyhedron; Cube — Earth, as the most “stable” polyhedron; Octahedron — Air, as the most “airy” polyhedron. The fifth polyhedron Dodecahedron, embodied “all things”, “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 their four elements were connected by the following proportion: earth/water = air/fire. The atoms of the “elements” were tuned by Plato in perfect consonances, like the four strings of a lyre. Let us remember 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 “end-to-end” harmony of existence with its embodiment in the Platonic solids. The influence of the famous Greek thinker Plato also affected Beginnings Euclid. This book, which for centuries was the only textbook on geometry, describes “ideal” lines and “ideal” figures. The most “ideal” line is straight, and the most “ideal” polygon is 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 limit part of the plane. I wonder what Beginnings Euclid begins with a description of the construction regular triangle and end with a study of five Platonic solids. notice, that Platonic solids the final, that is, the 13th book is dedicated to Began Euclid. By the way, this fact, that is, the placement of the theory of regular polyhedra in the final (that is, as if 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 that Euclid pursued when 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 the five Platonic solids, simultaneously 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 “elements of all things”: “The concept of structural symmetry dates back to ancient antiquity...” The most famous examples can, of course, be found in Plato’s Timaeus, where in section 53, relating to the Elements, he writes: “First, 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 the four Platonic solids (at that time only four, until Hipparchus did not discover the fifth one - the dodecahedron). Although at first glance such a philosophy may seem somewhat naive, it indicates a deep understanding of how Nature actually works."

Archimedean solids

Semiregular polyhedra

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

Archimedes (287 BC – 212 BC)

A bunch of Archimedean solids can be divided into several groups. The first of them consists of five polyhedra, which are obtained from Platonic solids as a result of their truncation. A truncated body is a body with the top cut off. 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 will be regular polygons. Eg, 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 can be obtained Archimedean solids: truncated tetrahedron, truncated hexahedron (cube), truncated octahedron, truncated dodecahedron And truncated icosahedron(Fig. 2).

(A)	(b)	(V)

(G)	(d)

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 caveat that perhaps Archimedes takes credit for this 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 edges), including icosahedrons and dodecahedrons. The original work of Archimedes, unfortunately, has not survived, and its results have come to us, as they say, “second-hand.” During the Renaissance everything Archimedean solids one after another were “discovered” again. After all, Kepler in 1619 in his book “World Harmony” (“Harmonice Mundi”) gave a comprehensive description of the entire set of Archimedean solids - polyhedra, each face of which represents 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 different types of polygons, as opposed to 5 Platonic solids, all faces of which are identical (as in the C 20 molecule, for example).

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

So how to design Archimedes truncated icosahedron from Platonic icosahedron? The answer is illustrated using 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 with a plane, then 12 new pentagonal faces are formed. Together with the existing 20 faces, which after such cutting turned from triangular to hexagonal, they will make up 32 faces of the truncated icosahedron. In this case, there will be 90 edges and 60 vertices.

Another group Archimedean solids consist of two bodies called quasi-regular 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) rhombocuboctahedron, (b) rhombicosidodecahedron

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

(A)	(b)

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

In the aforementioned 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 outstanding scientists. Johannes Kepler (1571-1630) made a great contribution to the development of the theory of polyhedra. At one time he wrote a 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, if I may say so, spouse is the octahedron, for the octahedron has as many angles as the cube has faces.” Kepler was the first to publish a complete list of the thirteen Archimedean solids and gave them the names by which they are known today.

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 related to stellate 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 stellated polyhedra.

Figure 7. Regular stellated polyhedra (Poinsot solids)

Many readers may ask: “Why study regular polyhedra at all? What is the use of them? This question can be answered: “What is the benefit of music or poetry? Is everything beautiful useful? Models of polyhedra shown in Fig. 1-7, above all, make an aesthetic impression on us and can be used as decorative decorations. But in fact, the widespread appearance of regular polyhedra in natural structures has caused enormous interest in this branch of geometry in modern science.

The Mystery of the Egyptian Calendar

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 the nature of living and nonliving things, 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 day always gives way to night, and the seasons pass in a strict sequence: after winter comes spring, after spring comes summer, after summer comes autumn. In search of a solution to these phenomena, man paid attention to the celestial bodies - the Sun, the Moon, the stars - and the strict periodicity of their movements across the sky. These were the first observations that preceded the birth of one of the most ancient sciences - astronomy.

Astronomy bases 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. The different concepts of time depend on which of these phenomena the measurement of time is based on. Astronomy knows stellar time, sunny time, local time, waist time, maternity leave time, atomic time, etc.

The sun, like all other luminaries, participates in movement across the sky. In addition to the daily movement, the Sun has a 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 appearance of the starry sky changes continuously: each season has its own pattern of evening constellations, and each such pattern is repeated every year. Consequently, after a year, the Sun returns to its original place relative to the stars.

For ease of orientation in the starry 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 general name - zodiac(from the Greek word “zoop” = animal), as well as symbols (signs) widely known throughout the world and various allegorical images included in 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, combining the constellations Scorpio and Ophiuchus into a single 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 ancient times was one of the most important tasks of astronomy.

What is a “calendar” and what types exist? 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 - Kalends, in which in ancient Rome debtors paid interest.

Since ancient times in the countries of Eastern and South-East Asia When compiling calendars, great importance was attached to the periodicity of the movements of the Sun, Moon, and also Jupiter And Saturn, two giant planets of the solar system. There is reason to believe that the idea of ​​creating Jovian 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 is Saturn makes a complete revolution around the Sun in about 30 years (29.458 years). Wanting to harmonize 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 full revolutions around the Sun, and Jupiter 5 revolutions.

When creating annual calendars, astronomical phenomena are used: the change of day and night, change 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, sunny, 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, it was later discovered that this length 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 days of the month. These were 5 holidays connecting neighboring calendar years. Thus, the Egyptian calendar year had the following structure: 365 = 12ґ 30 + 5. Note that the Egyptian calendar 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 with 20 days per month. The next question regarding the Egyptian calendar: why did each month have exactly 30 days (more precisely, days)? Some questions can also be raised about the Egyptian system of time measurement, in particular regarding 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 into a day, that is, why 1 day = 24 (2½ 12) hours? Next: 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 find it expedient to believe that there are 12 zodiac signs, although in fact, during 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?

The connection between the Egyptian calendar and 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 derived from them 360 = 12ґ 30. The question arises: is there any then a fundamental scientific idea that could provide a simple and logical explanation for the use of these numbers in Egyptian systems?

To answer this question, let us turn once again to dodecahedron, shown in Fig. 1-d. Let us 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 information about regular polyhedra. But did they know all five regular polyhedra, in particular dodecahedron And icosahedron What are 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 information, Pythagoras lived in Egypt for 22 years!). Therefore, we can assume that Pythagoras may also have borrowed knowledge about regular polyhedra from the ancient Egyptians (and possibly from the ancient Babylonians, because according to legend, Pythagoras lived in ancient Babylon 12 years old). But there is other, more compelling evidence that the Egyptians had information about all five regular polyhedra. In particular, the British Museum houses a die from the Ptolemaic era, which has the shape icosahedron, that is, the “Platonic solid”, dual dodecahedron. All these facts give us the right to put forward the hypothesis that The dodecahedron was known to the Egyptians. And if this is so, then a very harmonious system follows from this hypothesis, which allows us to explain the origin of the Egyptian calendar, and at the same time the origin of the Egyptian system of measuring time intervals and geometric angles.

Previously, we established that the dodecahedron has 12 faces, 30 edges and 60 flat angles 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 same numbers express the cycles of the solar system, namely, the 12-year cycle of Jupiter, the 30-year cycle of Saturn and, finally, the 60-year 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" who 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 ancients, the movement of the Sun along the ecliptic was strictly circular, then, by choosing 12 signs of the Zodiac, the arc distance between which was exactly 30°, the Egyptians surprisingly beautifully coordinated the annual movement of the Sun 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°. Having divided each day into two parts, following the dodecahedron, the Egyptians then divided each half of the day into 12 parts (12 faces dodecahedron) and thereby introduced hour- the most important unit of time. Dividing one hour into 60 minutes (60 plane angles on the surface dodecahedron), the Egyptians introduced in this way minute– the next important unit of time. In the same way they introduced 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 fully consistent with their “Theory of Harmony”, based on the golden proportion, since it is this proportion that underlies dodecahedron.

These are the surprising conclusions that 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 the dial of our watch, 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 ratio, without even knowing it!

Quasicrystals by Dan Shekhtman

On November 12, 1984, a short paper published in the prestigious journal Physical Review Letters by Israeli physicist Dan Shechtman provided experimental evidence 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 prohibited in the crystal for geometric reasons. 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 quasicrystalline 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. The 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 long-range order. As emphasized in the article “Quasicrystals” by the famous physicist D. Gratia, “This concept led to the expansion of crystallography, the newly discovered riches of which we are only 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 numbers in mathematics."

What is a quasicrystal? What are its properties and how can it be described? As mentioned above, according to basic law of crystallography Strict restrictions are imposed on the crystal structure. According to classical concepts, a crystal is composed ad infinitum from a single cell, which should tightly (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)	V)	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, which existed before the discovery of an unusual alloy of aluminum and manganese, called a quasicrystal. Such an alloy is formed by ultra-fast cooling of the melt at a rate of 10 6 K per second. Moreover, during a diffraction study of such an alloy, an ordered pattern appears on the screen, characteristic of the symmetry of an icosahedron, which has the famous forbidden 5th-order symmetry axes.

Over the next few years, several scientific groups around the world studied this unusual alloy using high-resolution electron microscopy. All of them confirmed the ideal homogeneity of the substance, in which 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 for obtaining the crystal structure of a quasicrystal has been developed. This model is based on the concept of a “basic element”. According to this model, an inner icosahedron of aluminum atoms is surrounded by an outer icosahedron of manganese atoms. Icosahedrons are connected by octahedra of manganese atoms. The "base element" contains 42 aluminum atoms and 12 manganese atoms. During the solidification process, the rapid formation of “basic elements” occurs, which are quickly connected to each other by rigid octahedral “bridges”. Recall that the faces of the icosahedron are equilateral triangles. In order for an octahedral manganese bridge to form, 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 quasicrystalline structure with “icosahedral” symmetry is formed.

In recent decades, many types of quasicrystalline alloys have been discovered. In addition to those having “icosahedral” symmetry (5th order), there are also alloys with decagonal symmetry (10th order) and dodecagonal symmetry (12th order). The physical properties of quasicrystals have only recently begun to be studied.

What is it like practical significance discovery of quasicrystals? As noted in Gratia's article mentioned above, “the mechanical strength of quasicrystalline alloys increases sharply; 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 fine particles 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 the source of deep and useful scientific ideas. Secondly, quasicrystals destroyed the traditional idea of ​​an insurmountable divide between the world of minerals, in which “pentagonal” symmetry was prohibited, and the world of living nature, 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 living nature and in the world of minerals, that is the main proportion of the Universe.

Penrose Tiles

When Dan Shekhtman 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 icosahedrons, with the help of which the dense filling of three-dimensional space is carried out.

Let's take a closer look at the pentagon in Fig. 8.

Figure 8. Pentagon

After drawing diagonals in it, the original pentagon can be represented as a set of three types of geometric figures. In the center there 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. Yellow triangles are "golden" 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. 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 an acute angle of 36° at the base.

Now let's connect two yellow triangles and two red triangles with their bases. As a result we get two "golden" rhombus. The first of them (yellow) has an acute angle of 36° and an obtuse angle of 144° (Fig. 9).

(A)	(b)

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

Diamond in Fig. We'll call it 9 thin rhombus, and the rhombus in Fig. 9-b – thick rhombus.

The English mathematician and physicist Rogers Penrose used “golden” diamonds in Fig. 9 for the construction of “golden” parquet, which was called Penrose tiles. Penrose tiles are a combination of thick and thin diamonds, shown in Fig. 10.

Figure 10. Penrose tiles

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

Fullerenes

Now let's talk about another outstanding modern discovery in the field of chemistry. This discovery was made in 1985, that is, several years after quasicrystals. We are talking about the so-called “fullerenes”. The term “fullerenes” refers to closed molecules of the type C 60, C 70, C 76, C 84, in which all carbon atoms are located on a spherical or spheroidal surface. In these molecules, the carbon atoms are arranged 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 greatest symmetry and, as a consequence, the greatest stability. In this molecule, which resembles the tire of a soccer ball and has the structure of a regular truncated icosahedron (Fig. 2-e and Fig. 3), the carbon atoms are located on a spherical surface at the vertices of 20 regular hexagons and 12 regular pentagons so that each hexagon is bordered by 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 the domes of buildings (another use of the truncated icosahedron!).

"Fullerenes" are essentially "man-made" structures arising from fundamental physics research. They were first synthesized by scientists G. Croto and R. Smalley (who received the Nobel Prize in 1996 for this discovery). But they were unexpectedly discovered in rocks of the Precambrian period, that is, fullerenes turned out to be not only “man-made”, but natural formations. Fullerenes are now being intensively studied in laboratories. different countries, trying to establish the conditions of 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 evaporating graphite in a helium atmosphere. This produces 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. Fullerenes have unusual chemical and physical properties. Yes, when high blood pressure From 60 it becomes hard like diamond. Its molecules form a crystalline structure, as if consisting of perfectly smooth balls, freely rotating in a face-centered cubic lattice. Thanks to this property, C 60 can be used as a solid lubricant. Fullerenes also have magnetic and superconducting properties.

Russian scientists A.V. Eletsky and B.M. Smirnov in his article “Fullerenes”, published in the journal “Uspekhi Fizicheskikh Nauk” (1993, volume 163, no. 2), note that "fullerenes, the existence of which has been established in the mid-80s, and an effective isolation technology for 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 the forecasts and possible consequences of studying fullerenes in the next decade, but one should be prepared for new surprises."

The artistic world of Slovenian artist Matyushka Teja Krašek

Matjuska Teja Krasek received her BA in Painting from the College of Visual Arts (Ljubljana, Slovenia) and is a freelance artist. Lives and works in Ljubljana. Her theoretical and practical work focuses on symmetry as a bridging concept between art and science. Her artistic works have been presented at many international exhibitions and published in international magazines (Leonardo Journal, Leonardo on-line).

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

The artistic creativity of Mother Teia Krashek is associated with various types of symmetry, Penrose tiles and rhombuses, quasicrystals, the golden ratio as the main element of symmetry, Fibonacci numbers, etc. With the help of reflection, imagination and intuition, she tries to select new relationships, new levels of structure, new and different kinds order in these elements and structures. In her works she widely uses computer graphics as a very useful remedy to create artwork that bridges the gap between science, mathematics and art.

In Fig. 11 shows the composition of T.M. Krashek related to Fibonacci numbers. If we choose one of the Fibonacci numbers (for example, 21 cm) for the side length of the Penrose diamond in this palpably unstable composition, we can observe how the lengths of some of the segments in the composition form a Fibonacci sequence.

Figure 11. Mother Teia Krashek “Fibonacci Numbers”, canvas, 1998.

A large number of the artist’s artistic compositions are dedicated to Shechtman quasicrystals and Penrose lattices (Fig. 12).

(A)	(b)

(V)	(G)

Figure 12. The World of Teia Krashek: (a) The World of Quasicrystals. Computer graphics, 1996.
(b) Stars. Computer graphics, 1998 (c) 10/5. Canvas, 1998 (d) Quasi-cube. Canvas, 1999

Mother Theia Krashek and Clifford Pickover's composition Biogenesis, 2005 (Fig. 13) features a decagon composed of Penrose diamonds. The relationships between Petrose's rhombuses can be observed; Every two adjacent Penrose diamonds form a pentagonal star.

Figure 13. Mother Theia Krashek and Clifford Pickover. Biogenesis, 2005.

In the picture Double Star GA(Figure 14) we see how the Penrose tiles combine to form a two-dimensional representation of a potentially hyperdimensional object with a decagonal base. When depicting the painting, the artist used the rigid edge method proposed by Leonardo da Vinci. It is this method of depiction that allows you to see the picture in projection onto a plane. big number pentagons and pentacles, which are formed by 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. Essentially, in this picture, Mother Teia Krashek found a new regular polyhedron, which quite possibly actually exists in nature.

Figure 14. Mother Teia Krashek. Double Star GA

In Krashek’s composition “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 relationships are represented by many different ways on various scales.

Figure 15. Mother Theia Krashek “Stars for Donald”, computer graphics, 2005.

The artistic compositions of Mother Teia Krashek 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 “East 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 Mother Teia Krashek // “Academy of Trinitarianism”, M., El No. 77-6567, pub. 12561, 07.11.2005

Anyone who has studied sacred geometry, or even just ordinary geometry, knows that there are five unique shapes, and they are crucial to understanding both sacred and ordinary geometry. They are called Platonic solids(Fig.6-15>).

The Platonic solid is defined by certain characteristics. First of all, all its faces are the same size. For example, the cube, the most famous of the Platonic solids, has a square on each face, and all its faces are the same size. Second, all the edges of the Platonic solid are the same length; All edges of a cube are the same length. Third: all internal angles between faces are the same size. In the case of a cube, this angle is 90 degrees. And fourth: if the Platonic solid is placed inside a sphere (of regular shape), then all its vertices will touch the surface of the sphere. Such definitions, except Cuba(A), only four forms that have all these characteristics answer. The second one will be tetrahedron(B) (tetra means "four") is a polyhedron with four faces, all equilateral triangles, equal edge lengths and equal angles, and - all vertices touching the surface of a sphere. Another simple form is octahedron(C) (okta means "eight"), all eight faces are equilateral triangles of the same size, the lengths of the edges and corners are the same, and all vertices touch the surface of the sphere.

The other two Platonic solids are a little more complicated. One is called icosahedron(D) - this means that it has 20 faces that look like equilateral triangles with the same length of edges and corners; all its vertices also touch the surface of the sphere. The latter is called pentagonal dodecahedron(E) (dodeka is 12), the faces of which are 12 pentagons (pentagons) with the same length of edges and the same angles; all its vertices touch the surface of the sphere.

If you are an engineer or architect, you studied these five shapes in college, at least superficially, because they are basic structures.

Their source: Metatron's Cube

If you are studying sacred geometry, it does not matter which book you open: it will show you the five Platonic solids, because they are the ABC of sacred geometry. But if you read all these books - and I read almost all of them - and ask the experts: “Where do Platonic solids come from? What is their source?”, then almost everyone will say that he does not know. The fact is that these five Platonic solids originate from the first information system of the Fruit of Life. Hidden in the lines of Metatron's Cube (see.
Fig.6-14> ), all these five forms exist there. When looking at Metatron's Cube, you are looking at all five Platonic solids at the same time. To see each one better, you'll need to do the trick where you erased some of the lines again. By erasing all the lines except for a few specific ones, you will get this cube ( Fig.6-16 >).

Well, do you see the cube? In reality, it is a cube within a cube. Some of the lines are dotted because they end up behind the front edges. They are invisible if the cube becomes a solid, opaque body. Here is the opaque shape of the larger cube (Fig. 6-16a>). (Make sure you can see it, because it will become more and more difficult to see the next figures as we progress).

By erasing some lines and connecting other centers (
Fig. 6-17>), you get two tetrahedrons inserted into each other, which form a star tetrahedron. As with the cube, you actually get two star tetrahedra, one inside the other. Here is the solid shape of a larger star tetrahedron (Fig. 6-17a>).

Figure 6-18> is an octahedron inside another octahedron, although you are looking at them from a certain special angle. Fig. 6-18a> is an opaque version of the larger octahedron.

Fig.6-19> is one icosahedron inside another, and Fig.6-19a> is an opaque version of the larger one. It becomes somehow easier if you view it this way.

These are three-dimensional objects emanating from the thirteen circles of the Fruit of Life.

This is a painting by Shulamith Wulfing - Christ the Child inside an icosahedron (
Fig. 6-20>), which is very true, since the icosahedron, as you will now see, represents water, and Christ was baptized in water, the beginning of a new consciousness.

This is the fifth and final form - two pentagonal dodecahedrons, one inside the other (Fig. 6-21>) (only the inner dodecahedron is shown here for simplicity).

Rice. 21 is a solid shape.

As we have seen, all five Platonic solids can be found in Metatron's Cube ( Fig.6-22>).

Missing lines

When I was looking for the last Platonic solid in Metatron's Cube, the dodecahedron, it took me about twenty years. After the angels said, “They're all inside,” I started looking, but I couldn't find the dodecahedron. Finally, one day a student said to me: “Hey, Drunvalo, you forgot some of the lines of Metatron's Cube.” When he showed them, I looked and said: “You're right, I forgot.” I thought that I connected all the centers with each other, but it turns out I forgot some. No wonder I couldn't find this dodecahedron because it was defined by these missing lines! For over twenty years I was convinced that I had all the lines drawn, when I had none.

This is one of the great problems of science when a problem is thought to be solved; then it moves on and uses this information to further its construction. Now, for example, science has the same kind of problem around bodies falling in a vacuum. It has always been believed that they fall at the same rate, and much of our advanced science is based on this fundamental “law.” It has been proven that this is not so, but science continues to use it anyway. A spinning ball falls significantly faster than a non-spinning ball. Someday there will come a day of scientific reckoning.

When I was married to McKee, she was also very passionate about sacred geometry. Her work is very interesting to me because it represents the feminine aspect, where the pentagonal energies of the right hemisphere of the brain operate. It shows how emotions, colors and shapes are all interconnected. In fact, she found the dodecahedron in Metatron's Cube before I did. She took it and did something that I would never have thought of. You see, Metatron's Cube is usually drawn on a flat surface, but it is actually a three-dimensional shape. So one day I was holding this three-dimensional shape in my hands and trying to find a dodecahedron there, and McKee said, “Let me look at this thing.” She took the three-dimensional shape and rotated it through the proportion angle f (phi ratio). (What we haven't talked about yet is that the ratio of the Golden Mean, also called the f (phi ratio), is exactly 1.618). Rotating the shape this way was something I would never have thought of. Having done this, she outlined the shadow cast by this form and received the following image (
Fig.6-23>).

McKee first created it herself and then passed it on to me. The center here is in pentagon A. Then, if you take the five pentagons coming out of A (pentagon B) and another pentagon coming out of each of these five (pentagon C), you get expanded dodecahedron. I thought, “Wow, this is my first time finding this here.” actually some kind of dodecahedron." She did this in three days. I couldn't find him for twelve whole years.

One day we spent almost the whole day looking at this picture. She was amazing because every single one the lines in this picture correspond to the proportions of the Golden Mean. And everywhere here are three-dimensional rectangles of the Golden Mean. There is one at point E, where the two diamonds, top and bottom, are the top and bottom of the three-dimensional rectangle of the Golden Mean, and the dotted lines are its edges. This is amazing stuff. I said, "I don't know what it is, but it's probably very important." So, we put it aside to think about it later.

Quasi-crystals

Later I learned about a completely new science. This new science will completely change the world of technology. With the new technology, metallurgists will probably be able to create a metal ten times harder than diamond, if you can imagine that. It will be incredibly durable.

For a long time, metals were studied using a technique called X-ray diffraction to see where atoms were located. I'll show you an X-ray diffraction photo soon. Certain special models have been discovered that determine the existence of only certain atomic structures. It seemed that this was all that could be known, because that was all that could be discovered. This limited the ability to make metals.

Then, Scientific American magazine ran a game that was based on the Penrose model. There was a British mathematician and relativist, Roger Penrose, who figured out how to lay pentagon-shaped tiles so that they completely covered a flat surface. It is impossible to completely cover a flat surface with tiles in the shape of just pentagons - there is no way to make it work. He then proposed two diamond shapes derived from the pentagon, and using these two shapes he was able to create many different patterns covering a flat surface. In the eighties, the magazine Scientific American proposed a game, the essence of which was to fold these given models into new forms; this subsequently enabled metallurgical scientists watching the game to suggest the existence of something new in physics.

Eventually, they discovered a new model of the atomic lattice. It has always existed; they just discovered it. These lattice patterns are now called quasi-crystals; It's a New Phenomenon (1991). Through metals they figure out what shapes and patterns are possible. Scientists are finding ways to use these shapes and patterns to make new metal products. I'm willing to bet that the model McKee got from Metatron's Cube is the most remarkable of all, and that any Penrose model is a derivative of it. Why? Because it is all subject to the law of the Golden Section, it is basic - it came directly from the main model in Metatron's Cube. Although it is none of my business, someday I will probably determine whether this is true. I see that instead of using two Penrose models and a pentagon, it uses only one of these models and a pentagon (I was just thinking that I would suggest this option). What's happening in this new science now is interesting.

Latest Information: According to David Adair, NASA has just produced a metal in space that is 500 times stronger than titanium, as light as foam and as transparent as glass. Is it based on these laws?

As the events in this book unfold, you will discover that sacred geometry can explain any subject in detail. There is not a single thing that you could say with your voice that could not be described entirely, completely and perfectly, taking into account all possible knowledge, sacred geometry. (We distinguish between the concepts of “knowledge” and “wisdom”: wisdom needs experience). However, the more important purpose of this work is to remind you that you yourself have the potential of a living Mer-Ka-Ba field around your body and to teach you how to use it. I will constantly come to places where I go into all sorts of roots and branches and talk about all kinds of topics imaginable and unimaginable. But I will always get back on track, because I am leading everything in one specific direction, towards the Mer-Ka-Ba, the light body of man.

I spent many years studying sacred geometry, and I am sure that you can learn everything that is generally possible to know, anything you want about any subject, you just have to focus your attention on the geometry hidden behind this subject. All you need is a compass and a ruler - you don't even need a computer, although it helps. All the knowledge you already have within you, and all you have to do is reveal it. You are simply exploring the map of the movement of the spirit in the Great Void, that's all. You can unravel the mystery of any object.

Let's summarize: first Information system emerges from the Fruit of Life through Metatron's Cube. By connecting the centers of all the spheres you get five figures - actually six, because there is still a central sphere from which it all began. So, you have six original shapes - tetrahedron, cube, octahedron, icosahedron, dodecahedron and sphere.

Latest information: In 1998 we begin to develop another new science: nanotechnology. We have created microscopic “machines” that can go inside metal or crystalline matrices and rearrange atoms. In 1996 or 1997, a diamond was created from graphite using nanotechnology in Europe. It's a diamond about three feet across, and it's real. When the science of quasi-crystals and nanotechnology come together, our understanding of life will also change. Look at the late 1800s compared to today.

Platonic Solids and Elements

Such ancient alchemists and great souls as Pythagoras, the father of Greece, believed that each of these six figures was a model of the corresponding element (Fig.6-24>).

The tetrahedron was considered a model of the element of fire, the cube - of earth, the octahedron - of air, the icosahedron - of water, and the dodecahedron - of ether. (Aether, prana and tachyon energy) are all one and the same; it is ubiquitous and accessible at any point in space/time/dimension. This is the great mystery of zero point technology. And the sphere represents the Void. These six elements are the building blocks of the universe. They create the qualities of the universe.

Alchemy usually speaks only of these elements: fire, earth, air and water; Rarely is ether or prana mentioned because it is so sacred. In the Pythagorean school, if you just mentioned the word “dodecahedron” outside the school walls, you would be killed on the spot. This figure was considered so sacred. They didn't even talk about her. Two hundred years later, during Plato’s lifetime, they talked about it, but only very carefully.

Why? Because the dodecahedron is located at the outer edge of your energy field and is the highest form of consciousness. When you reach the 55-foot limit of your energy field, it will be shaped like a sphere. But the inner figure closest to a sphere is the dodecahedron (actually a dodecahedral-icosahedral relationship). In addition to this, we live inside a large dodecahedron that contains the universe. When your mind reaches the limit of space space - and the limit is here There is– then he stumbles upon a dodecahedron closed in a sphere. I can say this because human body is a hologram of the universe and contains the same principles and laws. The twelve constellations of the zodiac are included here. The dodecahedron is the final figure of geometry and it is very important. At the microscopic level, the dodecahedron and icosahedron are relative parameters of DNA, the plans on which all life is built.

You can relate the three bars in this image ( Fig.6-24>) with the Tree of Life and the three primary energies of the universe: male (left), female (right) and childish (center). Or, if you delve directly into the structure of the universe, you have a proton on the left, an electron on the right and a neutron in the middle. This central column, which is creative, is the baby. Remember, to begin the process of exiting the Void, we went from the octahedron to the sphere. This is the beginning of the process of creation, and is found in the baby, or central column.

The left column, containing a tetrahedron and a cube, represents the male component of consciousness, the left hemisphere of the brain. The faces of these polygons are triangles or squares. The central column is the corpus callosum, connecting the left and right sides. The right column containing the dodecahedron and icosahedron represents the female component of consciousness, the right hemisphere of the brain, and the faces of these polygons are made up of triangles and pentagons. Thus, the polygons on the left have three- and four-edge faces, and the shapes on the right have three- and five-edge faces.

In the language of Earthly consciousness, the right column is the missing component. We have created the male (left) side of the Earth's consciousness, and now, in order to achieve integrity and balance, we are completing the creation of the female component. The right side is also associated with Christ consciousness or unity consciousness. The dodecahedron is the basic shape of the grid of Christ consciousness around the Earth. The two shapes of the right column represent relative to each other what are called paired figures, that is, if you connect the centers of the dodecahedron faces with straight lines, you get an icosahedron, but if you connect the centers of an icosahedron, you get a dodecahedron again. Many polyhedra have pairs.

Sacred 72

Dan Winter's book, Heartmath, shows that the DNA molecule is made up of the dual relationships of dodecahedrons and icosahedrons. You can also see that the DNA molecule is a rotating cube. When the cube is rotated sequentially by 72 degrees according to a certain model, an icosahedron is obtained, which, in turn, forms a pair with a dodecahedron. Thus, the double strand of the DNA helix is ​​built on the principle of two-way correspondence: the icosahedron is followed by the dodecahedron, then the icosahedron again, and so on. This rotation through the cube creates a DNA molecule. It has already been determined that the structure of DNA is based on sacred geometry, although other hidden relationships may be discovered.

This 72 degree angle spinning in our DNA is associated with the plan/purpose of the Great White Brotherhood. As you may know, there are 72 orders associated with the Great White Brotherhood. Many speak of 72 orders of angels, and the Jews mention 72 names of God. The reason why it is 72 has to do with the structure of the Platonic solids, which is also connected with the grid of Christ consciousness around the Earth.

If you take two tetrahedrons and put them on top of each other (but in different positions), you will get a star tetrahedron, which, when viewed from a certain angle, will look nothing more than a cube ( Fig.6-25>). You can see how they are interconnected. In the same way, five tetrahedra can be added together to form an icosahedral cap (Fig. 6-26).

If you create twelve icosahedral caps and place one on each face of the dodecahedron (it will take 5 times 12 or 60 tetrahedra to create a dodecahedron), then it will be a star - stellated- a dodecahedron, because each of its vertices is exactly above the center of each face of the dodecahedron. The figure paired with it will be composed of 12 vertices in the center of each face of the dodecahedron and will turn out to be an icosahedron. These 60 tetrahedrons plus 12 points in the centers will add up to 72 - again the number of orders associated with the White Brotherhood. The Brotherhood actually operates through the physical relationships of this dodecahedron/icosahedron star shape, which is the basis of the grid of Christ consciousness around the world. In other words, the Brotherhood is making attempts to identify the consciousness of the right hemisphere of the planet's brain.

The original order was the Alpha and Omega Order of Melchizedek, which was founded by Machiventa Melchizedek about 200,200 years ago. Since then, other orders have been founded, 71 in total. The youngest is the Brotherhood of the Seven Rays in Peru/Bolivia, the seventy-second order.

Each of the 72 orders has a rhythm of life similar to a sine wave, where some of them appear for a certain period of time, then disappear for a while. They have biorhythms just like their human body does. The cycle of the Rosicrucian Order, for example, lasts a century. They appear for a hundred years, then for the next hundred years they disappear completely - they literally disappear from the face of the Earth. After a hundred years, they reappear in this world and act for the next hundred years.

They are all in different cycles and all working together to achieve one goal - to bring the Christ consciousness back to this planet, to restore this lost feminine component of consciousness and bring balance to the left and right hemispheres of the planet's brain. There is another way of looking at this phenomenon, which is truly unusual. I will come to this when we talk about England.

Using Bombs and Understanding the Basic Model of Creation

Question: What happens to the elements when an atomic bomb is detonated?

As for the elements, they are converted into energy and other elements. But it's not only that. There are two types of bombs: decay and melt - thermonuclear. Decay splits matter into pieces, and thermonuclear reaction fuses it together. The fusing together is all right – no one is complaining about that. All known suns in the universe are fusion reactors. I am aware that what I am saying now is not yet recognized by science, but the tearing of matter into pieces here on Earth affects the corresponding area in outer space - both above and below. In other words, microcosm and macrocosm are interconnected. This is why the decay reaction is illegal throughout the universe.

The explosion of atomic bombs also causes a monstrous imbalance on Earth. For example, if we take into account that creation balances earth, air, fire, water and ether, then an atomic bomb causes the manifestation of a huge amount of fire in one place. This leads to an imbalance and the Earth must react to this.

If you pour 80 billion tons of water on a city, that will also be an unbalanced situation. If somewhere there is too much air, too much water, too much whatever, then it upsets the balance. Alchemy is the knowledge of how to keep all these phenomena in balance. If you understand the meaning of these geometric shapes and know their relationships, then you can create what you want. The whole idea is to understand the underlying cards. Remember, the map shows the path the spirit moves in the Void. If you know the underlying map, then you have the knowledge and understanding necessary to co-create with God.

Figure 6-27> shows the relationship of all these figures. Each vertex is connected to the next and they are all in certain mathematical relationships associated with the proportion f (phi ratio).

PLATONIAN SOLIDS [P. - from Greek Plato (427–347 BC / T. - origin see BODY), the totality of all regular polyhedra [i.e. e. volumetric (three-dimensional) bodies bounded by equal regular polygons] of the three-dimensional World, first described by Plato (the final, XIII book of the “Elements” of Plato’s student Euclid is also dedicated to them); // with all the infinite variety of regular polygons (two-dimensional geometric figures limited by equal sides, adjacent pairs of which form equal angles in pairs), there are only five volumetric polygons. (see Table 6), according to which, since the time of Plato, the five elements of the Universe have been placed; a curious connection exists between the hexahedron and the octahedron, as well as between the dodecahedron and the icosahedron: the geometric centers of the faces of each first are the vertices of each second.

A person shows interest in polyhedra throughout his entire conscious activity - from a two-year-old child playing with wooden blocks to a mature mathematician. Some of the regular and semi-regular bodies occur in nature in the form of crystals, others - in the form of viruses that can be examined using an electron microscope. What is a polyhedron? To answer this question, let us recall that geometry itself is sometimes defined as the science of space and spatial figures - two-dimensional and three-dimensional. A two-dimensional figure can be defined as a set of straight segments that bound a part of a plane. Such a flat figure is called a polygon. It follows that a polyhedron can be defined as a set of polygons that bound a portion of three-dimensional space. The polygons that form a polyhedron are called its faces.

Scientists have long been interested in “ideal” or regular polygons, that is, polygons with equal sides and equal angles. The simplest regular polygon can be considered an equilateral triangle, since it has the smallest number of sides that can limit part of the plane. The big picture The regular polygons of interest to us, along with the equilateral triangle, are: square (four sides), pentagon (five sides), hexagon (six sides), octagon (eight sides), decagon (ten sides), etc. Obviously, theoretically there are no restrictions on the number of sides of a regular polygon, that is, the number of regular polygons is infinite.

What is a regular polyhedron? A regular polyhedron is such a polyhedron, all of whose 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 - there are as many regular polygons as there are. However, it is not. In Euclid's Elements we find a rigorous proof that there are only five regular polyhedra, and their faces can only be three types of regular polygons: triangles, squares and pentagons.

Name Number of faces Element
Tetrahedron 4 Fire
Hexahedron/Cube 6 Earth
Octahedron 8 Air
Icosahedron 10 Water
Dodecahedron 12 Ether

Platonic solids

The world of star polyhedra

Our world is full of symmetry. Since ancient times, our ideas about beauty have been associated with it. This probably explains man's enduring interest in the amazing symbols of symmetry, which attracted the attention of many outstanding thinkers, from Plato and Euclid to Euler and Cauchy.

However, polyhedra are by no means only an object of scientific research. Their forms are complete and whimsical, and are widely used in decorative arts.

Star-shaped polyhedra are very decorative, which allows them to be widely used in the jewelry industry in the manufacture of all kinds of jewelry. They are also used in architecture. Many forms of stellate polyhedra are suggested by nature itself. Snowflakes are star-shaped polyhedra. Since ancient times, people have tried to describe all possible types of snowflakes and compiled special atlases. Several thousand different types of snowflakes are now known.

Stellated dodecahedron

The great stellated dodecahedron belongs to the family of Kepler-Poinsot solids, that is, regular non-convex polyhedra. The faces of the large stellated dodecahedron are pentagrams, like those of the small stellated dodecahedron. Each vertex has three faces connected. The vertices of the great stellated dodecahedron coincide with the vertices of the described dodecahedron.

The great stellated dodecahedron was first described by Kepler in 1619. It is the last stellated form of the regular dodecahedron.

Suvorov Mikhail, 10th grade student

This work is devoted to describing the views of the ancient Greek philosopher Plato on the structure of the Universe, through the use of regular polygons, such as tetrahedron, octahedron, hexahedron (cube), dodecahedron and icosahedron. In modern mathematics, these bodies are called Platonic bodies.

The work also reflects the question of how Platonic solids are used in modern natural science theories.

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Research work on geometry. Topic: “Platonic solids” Prepared by: Suvorov student Mikhail Suvorov Mathematics teacher Marina Valerievna Kharkov

Plato (427–347 BC) - the great ancient Greek philosopher, student of Socrates, founder of the Academy. Plato's main merit in the history of mathematics is that he recognized that knowledge of mathematics is necessary for every educated person. Plato's contribution to mathematics is insignificant. However, his ideas regarding the structure and methods of mathematics are extremely valuable. He introduced the tradition of giving impeccable definitions and determining which positions in mathematical considerations can be accepted without proof. Plato was the first to substantiate the method of proof by contradiction, which is now widely used in geometry. In Plato's school Special attention focused on solving construction problems. Thanks to this, she formed the concept of the geometric location of points, and also developed a technique for solving construction problems. Convex regular polyhedra - tetrahedron, octahedron, hexahedron (cube), dodecahedron and icosahedron - are usually called Platonic solids.

Definition: PLATONIAN SOLIDS - from Greek. Plato 427-347 BC. – the totality of all regular polyhedra [i.e., volumetric bodies bounded by equal regular polygons] of the three-dimensional World, first described by Plato.

A regular polygon is a flat figure bounded by straight lines with equal sides and equal interior angles. An analogue of a regular polygon in three-dimensional space is a regular polyhedron: a spatial figure with identical faces in the shape of regular polygons and identical polyhedral angles at the vertices. There are only five regular convex polyhedra: regular tetrahedron, cube, octahedron, dodecahedron and icosahedron.

The history of the creation of Platonic solids. Four polyhedrons personified four essences or “elements” in it. The tetrahedron symbolized Fire, since its apex points upward; Icosahedron - Water, since it is the most “streamlined” polyhedron; Cube - Earth, as the most “stable” polyhedron; Octahedron - Air, as the most “airy” polyhedron. The fifth polyhedron, the Dodecahedron, embodied "all that exists"

Tetrahedron The ancient Greeks gave the polyhedron a name based on the number of faces. “Tetra” means four, “hedra” means face (tetrahedron is a tetrahedron). A polyhedron refers to regular polyhedra and is one of the five Platonic solids. The tetrahedron has the following characteristics: Face type - regular triangle; The number of sides on a face is 3; Total number faces – 4; The number of edges adjacent to the vertex is 3; The total number of vertices is 4; The total number of ribs is 6; A regular tetrahedron is made up of four equilateral triangles. Each of its vertices is the vertex of three triangles. Therefore, the sum of the plane angles at each vertex is 180°. The tetrahedron has no center of symmetry, but has 3 axes of symmetry and 6 planes of symmetry.

Hexahedron (the more common name is cube) The ancient Greeks gave the polyhedron a name based on the number of faces. “Hexo” means six, “hedra” means face (Hexahedron is a hexagon). A polyhedron refers to regular polyhedra and is one of the five Platonic solids. The hexahedron has the following characteristics: The number of sides on the face is 4; The total number of faces is 6; The number of edges adjacent to the vertex is 3; The total number of vertices is 8; The total number of ribs is 12; The hexahedron is made up of six squares. Each vertex of the cube is the vertex of three squares. Therefore, the sum of the plane angles at each vertex is 270°. The hexahedron has no center of symmetry, but has 3 axes of symmetry and 6 planes of symmetry.

Icosahedron The ancient Greeks gave the polyhedron a name based on the number of faces. “Ikosi” means twenty, “hedra” means face (Icosahedron - twenty-sided). The polyhedron belongs to the regular polyhedra and is one of the five Platonic solids. The icosahedron has the following characteristics: Face type - regular triangle; The number of sides on a face is 3; The total number of faces is 20; The number of edges adjacent to the vertex is 5; The total number of vertices is 12; The total number of ribs is 30; The regular icosahedron is made up of twenty equilateral triangles. Each vertex of the icosahedron is the vertex of five triangles. Therefore, the sum of the plane angles at each vertex is 270°. The icosahedron has a center of symmetry - the center of the icosahedron, 15 axes of symmetry and 15 planes of symmetry.

Octahedron The ancient Greeks gave the polyhedron a name based on the number of faces. “Octo” means eight, “hedra” means face (octahedron is an octahedron). A polyhedron refers to regular polyhedra and is one of the five Platonic solids. The octahedron has the following characteristics: Face type - regular triangle; The number of sides on a face is 3; The total number of faces is 8; The number of edges adjacent to the vertex is 4; The total number of vertices is 6; The total number of ribs is 12; A regular octahedron is made up of eight equilateral triangles. Each vertex of the octahedron is the vertex of four triangles. Therefore, the sum of the plane angles at each vertex is 240°. The octahedron has a center of symmetry - the center of the octahedron, 9 axes of symmetry and 9 planes of symmetry.

Dodecahedron The ancient Greeks gave the polyhedron a name based on the number of faces. “Dodeca” means twelve, “hedra” means face (dodecahedron - dodecahedron). The polyhedron belongs to the regular polyhedra and is one of the five Platonic solids. The dodecahedron has the following characteristics: Face type – regular pentagon; The number of sides on a face is 5; The total number of faces is 12; The number of edges adjacent to the vertex is 3; The total number of vertices is 20; The total number of ribs is 30; The regular dodecahedron is made up of twelve regular pentagons. Each vertex of the dodecahedron is the vertex of three regular pentagons. Therefore, the sum of the plane angles at each vertex is 324°. The dodecahedron has a center of symmetry - the center of the dodecahedron, 15 axes of symmetry and 15 planes of symmetry.

Application of Platonic solids in science Johannes Kepler (1571-1630) - German astronomer. Discovered the laws of planetary motion. In 1596, Kepler proposed a rule according to which a dodecahedron is described around the sphere of the Earth, and an icosahedron fits into it. The distance between the orbits of the planets can be obtained based on the Platonic solids nested within each other. The distances calculated using this model were quite close to the true ones.

V. Makarov and V. Morozov believe that the Earth’s core has the shape and properties of a growing crystal, which influences the development of all natural interactions and processes taking place on the planet. The force field of this growing crystal determines the icosahedron - the dodecahedral structure of the Earth (IDSZ). These polyhedra are inscribed into each other. All natural anomalies, as well as centers of development of civilizations, correspond to the vertices and edges of these figures.

Examples: Some of the regular polyhedra occur in nature as crystalline viruses. The polio virus is shaped like a dodecahedron. It can live and reproduce only in human or primate cells. At the microscopic level, the dodecahedron and icosahedron are the relative parameters of DNA on which all life is built. You can see that the DNA molecule is rotating into a cube.

Application in crystallography Platonic solids are widely used in crystallography, since many crystals have the shape of regular polyhedra. For example, a cube is a single crystal of table salt (NaCl), an octahedron is a single crystal of potassium alum, one of the forms of diamond crystals is an octahedron.

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