Geometric figures of Plato. § Platonic solids with their detailed description
Polyhedra dual to Archimedean solids. Like the Archimedean solids, there are 13 of them. Rhombic dodecahedron ... Wikipedia
Dodecahedron Regular polyhedron, or Platonic solid is a convex polyhedron with the maximum possible symmetry. A polyhedron is called regular if: it is convex all its faces are equal regular polygons in each of its ... ... Wikipedia
Dodecahedron A regular polyhedron or Platonic solid is a convex polyhedron consisting of identical regular polygons and having spatial symmetry ... Wikipedia
This article is proposed for deletion. An explanation of the reasons and a corresponding discussion can be found on the Wikipedia page: To be deleted / November 22, 2012. While the discussion process ... Wikipedia
A portion of space bounded by a collection of a finite number of planar polygons (see GEOMETRY) connected in such a way that each side of any polygon is a side of exactly one other polygon (called ... ... Collier Encyclopedia
Semi-regular polyhedra in the general case are various convex polyhedra that have certain attributes of regular ones, such as the identity of all faces or the fact that all faces are regular polygons, as well as spatial ... Wikipedia
Or Archimedean solids are convex polyhedra with two properties: All faces are regular polygons of two or more types (if all faces are regular polygons of the same type, it is a regular polyhedron); For any couple ... ... Wikipedia
Type Regular polyhedron Face Regular pentagon Faces 12 Edges 30 Vertices 20 ... Wikipedia
Animation Type Regular polyhedron Face Regular triangle Faces 20 ... Wikipedia
This term has other meanings, see Cube (meanings). Cube Type Regular polyhedron Face square ... Wikipedia
Books
- Sacred geometry, numerology, music, cosmology, or QUADRIVIUM, Martino D., Landi M. and others. “Everywhere you know, as far as possible, the unity of nature” (“Golden Poems” of the Pythagoreans) “The world (cosmos) was not created for you - but you are for him ”(Iamblichus, ancient philosopher) This illustrated ...
- Magic Edges, No. 11, 2015, . Creating models of polyhedrons from cardboard is a very exciting and affordable activity, it is the "magic of turning" a sheet of paper into a three-dimensional figure. The simplest models of polyhedra can be...
annotation
The outstanding Russian philosopher Alexei Losev, a researcher of the aesthetics of antiquity and the Renaissance, formulated the “golden” paradigm of the ancient Greeks in the following words: “From the point of view of Plato, and indeed from the point of view of all ancient cosmology, the world is a kind of proportional whole, subject to the law of harmonic division - the golden section. The latest discoveries modern science, based on Platonic solids, golden ratio, Fibonacci numbers: fullerenes, Nobel Prize- 1996; quasicrystals, Nobel Prize - 2011; experimental proof of the existence of "golden section" harmony in the quantum world; detection of fibonacci patterns in the periodic table; the "hypothesis of Proclus" and a new look at the "Elements" of Euclid and the history of the development of mathematics, starting with Euclid; hyperbolic Fibonacci functions and a new geometric theory of phyllotaxis; Pascal's triangle and generalized Fibonacci numbers; generalized golden proportions and the law of structural harmony of systems; lambda fibonacci numbers as new class integer sequences with unique mathematical properties; "metal proportions" and general theory harmonic hyperbolic functions; solution of Hilbert's fourth problem and search for harmonic hyperbolic worlds of Nature; "golden" matrices, Fibonacci-Lorentz transformations and "golden" interpretation of the special theory of relativity; "golden" genomatrices; algorithmic measurement theory, Fibonacci codes and computers; number systems with irrational bases, ternary mirror-symmetrical arithmetic and "golden" number theory as a new trend in number theory; generalized Fibonacci matrices and new coding theory; finally, the "mathematics of harmony" as a new interdisciplinary direction, going back to the "Principles" of Euclid - all these are the "faces of divine proportion" in modern science, which create a general picture of its movement towards the "Golden" Scientific Revolution, which together reflects one of the most important trends in the development of modern science - a return to Pythagoras, Plato and Euclid.
PartIII
“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.”
Bertrand Russell
Foreword
Each of us more than once had to think about why Nature is able to create such amazing aesthetic structures that delight and delight the eye. Why do artists, poets, composers, architects create amazing works of art from century to century? What is the secret and what laws underlie these harmonious creatures? What is "harmony"? And does it have a mathematical expression? To model the "world of harmony" in the ancient world, primarily in Ancient Greece, was created harmony math, elements of which have been revived in modern science in many books including the book Alexey Stakhov “ The Mathematics of Harmony. From Euclid to Contemporary Mathematics and computer Science” , published in 2009 by one of the most prestigious scientific publishers in the world "World Scientific" .
The purpose of this publication, intended for a wide audience, is to popularly explain the concept of "harmony", which was introduced into science at the dawn of the development of human civilization, to tell about the history of this direction in the ancient period, the Middle Ages, the Renaissance, in the 19th and 20 centuries, as well as to introduce into the circle of ideas and applications the modern “mathematics of harmony”, which is actively developing in the 21st century. . Of course, the "mathematics of harmony" is a branch of mathematics; therefore, the authors failed to completely avoid mathematical formulas in the article devoted to this mathematical discipline. However, “mathematics of harmony” is a fairly simple (one might say, “elementary”) mathematics, which uses mathematical formulas available to high school students. And the authors hope for the indulgence of our readers.
The article consists of 4 parts:
Part III. Platonic solids, "Proclus' hypothesis", a new look at Euclid's "Principles", fullerenes and quasicrystals
Part IV. The role of "mathematics of harmony" in the development of modern science
PartIII. Platonic solids, "Proclus' hypothesis", a new look at Euclid's "Principles", fullerenes and quasicrystals
7. Platonic Solids
Regular polygons and polyhedra
A person shows interest in regular polygons and polyhedra throughout his conscious activity - from a two-year-old child playing wooden cubes, 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 seen with an electron microscope.
What is a polygon and 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 line segments bounding a part of a plane. Such a flat figure is called polygon. It follows that a polyhedron can be defined as a set of polygons bounding a portion of three-dimensional space. The polygons that form a polyhedron are called its faces.
Since ancient times, scientists have been interested in ideal or regular polygons, that is, polygons that have equal sides and equal angles. The simplest regular polygon can be considered equilateral triangle, since it has the smallest number of sides that can bound a portion of the plane. The general picture of regular polygons of interest to us, along with an equilateral triangle, is: 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 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 polyhedra in Euclid's Elements
Many books have been devoted to the theory of polyhedra. One of the most famous is the book of the English mathematician M. Wenninger "Models of polyhedra". 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, named after the ancient Greek philosopher Plato, who used regular polyhedra in his cosmology. We will begin our consideration with regular polyhedra whose faces are equilateral triangles (Fig. 21).
Fig.21. Platonic solids: tetrahedron (tetrahedron), octahedron (octahedron), cube (cube) dodecahedron (dodecaedron), icosahedron (icosahedron)
The first (and simplest) regular polyhedra is tetrahedron. In a tetrahedron, three equilateral triangles meet at one vertex; while their bases form a new equilateral triangle. The tetrahedron has the fewest number of faces among the Platonic solids and is the three-dimensional analog of a flat regular triangle, which has the fewest number of sides among regular polygons.
The next body, which is formed by equilateral triangles, is called octahedron (octahedron). 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 (icosahedron).
The next correct polygon shape is square. If we connect three squares at one point and then add three more, we get a perfect six-sided shape called hexahedron or cube.
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 (dodecahedron).
The next regular polygon is hexagon. However, if we connect three hexagons at one point, then we get a plane, 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. In essence, we have repeated the reasoning that Euclid carried out in Book XIII of his Elements. It is this book that is devoted to the presentation of the completed geometric theory of the Platonic solids. And precisely from these considerations it follows that there are only five convex regular polyhedra, whose faces can only be equilateral triangles, squares and pentagons.
Numerical characteristics of the Platonic solids. The main numerical characteristics of the Platonic solids are the number of sides of a face m, the number of faces n converging at each vertex, the number of faces G, number of vertices IN, number of edges R and the number of flat corners At on the surface of a polyhedron, Euler discovered and proved famous formula:
IN - R+ G = 2 ,
Connecting the number of vertices, edges and faces of any convex polyhedron. The above numerical characteristics are given in Table 2.
table 2. Numerical characteristics of the Platonic solids
It is appropriate to pay attention to the property duality, which connects the Platonic solids. It follows from Table 2 that for a hexahedron (cube) and an octahedron, the number of edges P=12 and the number of plane angles on the surface U=24 coincide. But the number of faces Г=6 of the cube coincides with the number of vertices В=6 of the octahedron, and the number of vertices of the cube В=8 coincides with the number of faces Г=8 of the octahedron. In addition, the number of sides of a cube face m= 4 coincides with the number of faces of the octahedron converging at the vertex, n=4, while the number of faces of the cube converging into n=3, coincides with the number of sides of the face of the octahedron m= 3. A similar situation is observed in the case of the icosahedron and dodcahedron. In such cases, one speaks of dualities corresponding Paid heat, that is, a cube dual octahedron and icosahedron dual dodecahedron. Note that in the property dualities reflects the "hidden" harmony of the Platonic solids.
Golden ratio in dodecahedron and icosahedron. The dodecahedron (dodecahedron) and its dual icosahedron (icosahedron) occupy special place among the Platonic solids. First of all, it must be emphasized that the geometry of the dodecahedron and icosahedron is directly related to the golden ratio. Indeed, the faces of the dodecahedron are pentagons, that is, regular pentagons based on the golden ratio. If you look closely at the icosahedron, you can see that five triangles converge at each of its vertices, the outer sides of which form a pentagon. These facts alone are enough to make sure that golden ratio plays a decisive role in the construction of these two Platonic solids.
But there are deeper confirmations of the deep mathematical connection of the golden ratio with the icosahedron and dodecahedron. And this connection leads to the fact that the dodecahedron and icosahedron express in a "hidden" form the harmony of the golden section.
9. Hypothesis of Proclus: a new look at Euclid's "Elements" and the history of the development of mathematics
Why did Euclid write his Elements?
At first glance, it seems that the answer to this question is very simple: Euclid's main goal was to present the main achievements of Greek mathematics in the 300 years preceding Euclid, using the "axiomatic method" of presentation of the material. Indeed, Euclid's "Elements" is the main work of Greek science, devoted to the axiomatic construction of geometry and mathematics. This view of the "Principles" is most common in modern mathematics.
However, in addition to the "axiomatic" point of view, there is another point of view on the motives that Euclid was guided by when writing the "Beginnings". This point of view was expressed by the Greek philosopher and mathematician Proclus Diadohom(412-485), one of the first commentators on the Principia.
First of all, a few words about Proclus. Proclus was born in Byzantium in the family of a wealthy lawyer from Lycia. Intending to follow in his father's footsteps, as a teenager he left for Alexandria, where he first studied rhetoric, then became interested in philosophy and became a student of the Alexandrian neoplatonist Olympiodorus the Younger. It was from him that Proclus began to study the logical treatises of Aristotle. At the age of 20, Proclus moved to Athens, where the Platonic Academy at that time was headed by Plutarch of Athens. By the age of 28, Proclus wrote one of his major works, a commentary on Plato's Timaeus. Around 450, Proclus becomes the head of the Platonic Academy.
Among the mathematical writings of Proclus, the most famous is his Commentary on the first book of Euclid's Elements. In this Commentary, he puts forward the following unusual hypothesis, which is called the “Proclusian hypothesis”. Its essence is as follows. As you know, the XIIIth, that is, the final book of the "Beginnings", is devoted to the presentation of the theory of five regular polyhedra, which played a dominant role in Plato's Cosmology and are known in modern science as the Platonic Solids. It is to this circumstance that Proclus draws attention. As Eduard Soroko emphasizes, according to Proclus, Euclid "created the Elements, allegedly not for the purpose of presenting geometry as such, but to give a complete systematized theory of the construction of the five" Platonic Solids ", along the way highlighting some of the latest achievements in mathematics."
The Significance of Proclus' Hypothesis for the Development of Mathematics. The main conclusion from the "Proclus hypothesis" is that Euclid's Elements, the greatest Greek mathematical work, was written by Euclid under the direct influence of the Greek "idea of Harmony", which was associated with the Platonic solids. Thus, the “Proclus hypothesis” allows us to suggest that the well-known in ancient science “Pythagorean doctrine of the numerical harmony of the Universe” and “Plato’s Cosmology”, based on regular polyhedra, were embodied in the greatest mathematical work of Greek mathematics, Euclid’s “Elements”. From this point of view, we can consider the "Beginnings" of Euclid as the first attempt to create a "Mathematical Theory of the Harmony of the Universe", which was associated in ancient science with the Platonic solids. And this was the main idea of Greek science! This is the main secret of the "Beginnings" of Euclid, which leads to a revision of the history of the emergence of mathematics, starting with Euclid.
Unfortunately, Proclus' original hypothesis regarding the true aims of Euclid in writing the Principia has been ignored by many modern historians of mathematics, leading to a distorted view of the structure of mathematics and of all mathematical education. And this is one of the main "strategic mistakes" in the development of mathematics.
"Proclus' Hypothesis" and "Key" Problems of Ancient Mathematics. As you know, academician Kolmogorov in his book identified two main, that is, "key" problems that stimulated the development of mathematics at the stage of its inception - account problem And measurement problem. However, another “key” problem follows from the “Proclus hypothesis” - problem of harmony, which was associated with the "Platonic solids" and the "golden section" - one of the most important mathematical discoveries of ancient mathematics (Proposition II.11 of Euclid's "Beginnings"). It was this problem that was put by Euclid as the basis of the "Beginnings", the main purpose of which was the creation of the geometric theory of "Platonic solids", which in "Plato's cosmology" expressed the harmony of the Universe. This idea leads to a new look at the history of mathematics, presented in Fig.22.
Rice. 22. "Key" problems of ancient mathematics and new trends in mathematics, theoretical physics and computer science
The approach demonstrated with the help of Fig.22 was first described in . It is based on the following reasoning. Already at the stage of the birth of mathematics, a number of important mathematical discoveries were made that fundamentally influenced the development of mathematics and all science in general. The most important of them are:
1. Positional principle of number representation, made by Babylonian mathematicians in the 2nd millennium BC. and embodied by them in the Babylonian 60-ary number system. This important mathematical discovery underlies all subsequent positional number systems, in particular, the decimal system and binary system- basics modern computers. This discovery eventually led to the formation of the concept natural number- the most important concept underlying mathematics.
2. Proof of the existence of incommensurable segments. This discovery, made in the scientific school of Pythagoras, led to a rethinking of early Pythagorean mathematics, which was based on the "principle of commensurability of quantities", and to the introduction irrational numbers- the second (after natural numbers) fundamental concept of mathematics. Ultimately, these two concepts (natural and irrational numbers) formed the basis of "Classical Mathematics".
3. The division of the segment in the extreme and average ratio ("golden section"). A description of this mathematical discovery is given in Euclid's Elements (Proposition II.11). This proposal was introduced by Euclid in order to create a complete geometric theory of the "Platonic solids" (in particular, the dodecahedron), the presentation of which is devoted to the final (XIIIth) book of Euclid's "Elements".
The approach formulated above (Fig. 22) leads to a conclusion that may be unexpected for many mathematicians. It turns out that in parallel with "Classical Mathematics" in science, starting from the ancient Greeks, another mathematical direction began to develop - "Mathematics of Harmony", which, like classical mathematics, goes back to the "Elements" of Euclid, but focuses its attention not on the "axiomatic approach", but on the geometric "problem of dividing a segment in the extreme and average ratio" (Proposition II.11) and on the theory of regular polyhedra set forth in Book XIII of Euclid's Elements. Outstanding thinkers, scientists and mathematicians took part in the development of the "mathematics of harmony" for several millennia: Pythagoras, Plato, Euclid, Fibonacci, Pacioli, Kepler, Cassini, Binet, Lucas, Klein, and in the 20th century - the famous mathematicians Coxeter, Vorobyov, Hoggatt and Vaida. And we cannot ignore this historical fact.
Origins of the doctrine
According to the commentator of the latest edition of Plato's works, he has "all cosmic proportionality rests on the principle of the golden division, or harmonic proportion." As mentioned, Plato's cosmology is based on regular polyhedra called Platonic solids. The idea of a “through” harmony of the universe was invariably associated with its embodiment in these five regular polyhedrons, which expressed the idea of the universal perfection of the world. And the fact that the main "cosmic" figure - the dodecahedron, symbolizing the body of the world and the universal soul, was based on the golden section, gave the latter a special charm, the meaning of the main proportion of the universe.
Plato's cosmology was the beginning of the so-called icosahedral-dodecahedral doctrine, which since antiquity has run like a red thread through all human science. The essence of this doctrine is that the dodecahedron and icosahedron are typical forms of nature in all its manifestations, from the cosmos to the microworld.
earth shape
The question of the shape of the Earth constantly occupied the minds of scientists of ancient times. And when the hypothesis of the spherical shape of the Earth was confirmed, the idea arose that in its shape the Earth is dodecahedron. Thus, Socrates wrote:
"The earth, when viewed from above, looks like a ball sewn from 12 pieces of leather."
This hypothesis of Socrates found further scientific development in the works of physicists, mathematicians and geologists. Yes, a French geologist de Beamon and famous mathematician Poincaré believed that the shape of the Earth is a deformed dodecahedron.
The Russian geologist S. Kislitsin also shared the opinion about the dodecahedral shape of the Earth. He hypothesized that 400-500 million years ago the dodecahedral geosphere turned into a geo-icosahedron. However, such a transition turned out to be incomplete and incomplete, as a result of which the geo-dodecahedron turned out to be inscribed in the structure of the icosahedron. More detailed information this hypothesis is described in the book.
Mystery 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 length of the calendar year does not correspond to astronomical data. And then the Egyptians added 5 more days to the calendar year, which, however, were not considered days of the months. It was 5 public holidays connecting adjacent calendar years. Thus, the Egyptian calendar year had the following structure: 365=12 x 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 did each month have exactly 30 days (more precisely, days)? Some questions can be raised about the system of measuring time, which, perhaps, was formed in later times. 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 (2 x 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°=12 x 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?
Analyzing the Egyptian calendar, as well as systems for measuring time and angular values, we find that four numbers are repeated with amazing constancy: 12, 30, 60 and the number 360 = 12´30 derived from them. The question arises: is there not some fundamental scientific idea that could give a simple and logical explanation for the use of these numbers in the Egyptian calendar and systems?
Let's turn to the dodecahedron (Fig.21). From Table 1 it follows that the dodecahedron has 12 faces, 30 edges and 60 flat corners on its surface. What was the surprise of the ancient Egyptians when they discovered that the cycles are expressed by the same numbers solar system, namely the 12-year cycle of Jupiter, the 30-year cycle of Saturn, and finally the 60-year cycle of the solar system. Thus, between such a perfect spatial figure as dodecahedron, and the 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. Since, according to 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 amazingly 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°. Later, the same scientific idea was used by the creators of the time measurement system. The division of each half of the day into 12 parts (12 faces dodecahedron) led to the introduction hours- the most important unit of time. Division of an hour into 60 minutes (60 flat corners on the surface dodecahedron) led to the introduction minutes- the next important unit of time. It was also introduced second(1 minute = 60 seconds).
Thus, choosing dodecahedron as the main "harmonic" figure of the universe, and strictly following the numerical characteristics of the dodecahedron 12, 30, 60, scientists managed to build an extremely harmonious calendar, as well as systems for measuring time and angular values.
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 section" (which underlies the dodcahedron)! And every time we look at the dial of our watch, which is also built on the use of the numerical characteristics of the dodecahedron 12, 30 and 60, we touch the main "Mystery of the Universe" - golden ratio without knowing it! Apparently, such a hypothesis of the Egyptian calendar concerns some "hidden" secret of the solar system, connected with the "golden section".
Johannes Kepler and Felix Klein
"Misterium Cosmographicum". Johannes Kepler began his scientific career in the small Austrian city of Graz, where, after graduating from the Tübingen Academy, he was sent as a teacher of mathematics at a gymnasium.
Let's make one "lyrical digression". From 15 to 19 July 1996, the 7th international Conference Fibonacci numbers and their applications. Alexey Stakhov made a presentation at this conference “ TheGoldenSectionandModernHarmonyMathematics” , from which, in essence, the development of modern "mathematics of harmony" as a new interdisciplinary direction of modern science began. The report aroused great interest among Fibonacci mathematicians and was selected for publication in the collection "Applications of Fibonacci Numbers" (1998) . During his stay in Graz, Prof. Alexey Stakhov was photographed near the monument to Johannes Kepler, installed in one of the parks of Graz.
Alexey Stakhov next to the monument to Johannes Kepler
(Graz, July 1996)
Kepler's first astronomical work, written in Graz, was a small book with the following title: "The harbinger of cosmographic research, containing the secret of the universe regarding the wonderful proportions between the celestial circles and true reasons, the number and size of the celestial spheres, as well as the periodic motions set forth with the help of five regular bodies by Johannes Kepler of Württemberg, a mathematician from the illustrious province of Styria. He himself called this book, published in 1597, "Misterium Cosmographicum" ("The Mystery of Cosmography").
Reading Kepler's first work, Misterium Cosmographicum (The Mystery of Cosmography), one never ceases to be amazed at his imagination. A deep conviction in the existence of harmony in the world left an imprint on all of Kepler's thinking. The purpose of his research, outlined in "The Secret of Cosmography", Kepler formulated in the preface:
« Dear reader! In this book, I set out to prove that the all-good and all-powerful God, when creating our moving world and when arranging the celestial orbits, chose as a basis five regular bodies, which from the time of Pythagoras and Plato to the present day have gained such loud fame, chose the number and proportions of the celestial orbits , as well as the relationship between the movements chose in accordance with the nature of the right bodies. The essence of three things - why they are arranged this way and not otherwise - was of particular interest to me, namely: the number, size and movements of the celestial orbits.
To reveal the secret of the universe meant, according to Kepler, to answer the question that he posed to himself for the first time in the history of astronomy. It was in the book "The Secret of Cosmography" that Kepler managed, as it seemed to him, to reveal this secret. Its essence, according to Kepler, is as follows:
“The earth (orbit of the earth) is the measure of all orbits. We describe a dodecahedron around it. The sphere circumscribed around the dodecahedron is the sphere of Mars. Let us describe a tetrahedron around the sphere of Mars. The sphere circumscribed around the tetrahedron is the sphere of Jupiter. Let us describe a cube around the sphere of Jupiter. The sphere circumscribed around the tetrahedron is the sphere of Saturn. Let's put an icosahedron into the sphere of the Earth. The sphere inscribed in it is the sphere of Venus. Let's put an octahedron into the sphere of Venus. The sphere inscribed in it is the sphere of Mercury.
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A circle divided into equal parts allows us to construct "ideal" or regular polygons. There can be infinitely many regular polygons obtained.
The simplest regular polygon can be considered an equilateral triangle.
But, polyhedra, geometric bodies, it cannot be infinitely many, since polyhedra are figures obtained by connecting polygons, in such a way that each side of one polygon is also a side of another polygon (called adjacent). Moreover, each vertex of the resulting body forms connections of the faces of polygons with edges - sides and vertices.
There can be only five polyhedra in a circle (that is, three-dimensional geometric shapes). Plato correlated the obtained bodies with the Elements as follows.
1. FIRE - Tetrahedron. Consists of four equilateral triangles. Each vertex is a vertex three triangles. Therefore, the sum of the plane angles at each vertex is 180;.
Number of faces - 4, vertices - 4, edges - 6
Volume - V= (a;;2)/12.
Surface area - S= a;;3
In terms of astrology, 180 degrees is the aspect of opposition. In which one beginning transforms the other, at its discretion.
The elements of Fire tend to show their potential in an established environment and achieve their goals. The Yang, external element manifests itself as an internal contradiction of individuality with the whole, Yin qualities inherent in the Earth element.
2.AIR - Octahedron. It looks like two combined triangles connected at the base. Each vertex of the octahedron is a vertex of four triangles. Therefore, the sum of plane angles at each vertex is 240;.
Number of faces - 8, vertices - 6, edges - 12
Volume - V= (a;;2)/3.
Surface area - S= 2a;;3
In terms of astrology, 240 degrees is an aspect of a trine.
The air expands unhindered. Quickly or slowly, but without overcoming and transforming the environment into which it enters. It is perceived as desirable and favorable. Yang external element, shows the qualities inherent in the elements of Water.
3. EARTH - A cube or a regular hexahedron is a regular polyhedron, each face of which is a square.
The cube consists 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;.
Number of faces - 6, vertices - 8, edges - 12
Volume - V= a;.
Surface area - S= 6a;
From the point of view of ASTROLOGY, 270 gr represents the dynamic aspect of quadrature.
The superficial contradiction between the Element and the property of the aspect is easily resolved, given that there is an external and internal level. Yin and Yang.
So - Fire, has a stable and static aspect. The Yang element manifests itself in a Yin way.
The potential of Fire is so great that after its manifestation the reality cannot remain the same. It has to build new centers of gravity, look for new ways of existence and adapt to the transformations caused by Fire.
After the manifestation of Fire, the contradiction cannot be eliminated, it is permanent. It does not affect the Fire element itself, only the environment in which the Element manifests itself, experiences its influence and tries on it, adapts to it. The manifested element Fire has Yin - long-term consequences.
The manifested element of the Earth, with its stable and static potential, due to slow movement, does not damage the environment, but makes it adapt and look for ways of interaction in which the environment manifests Yang qualities.
4. SPACE (Ether) - Dodecahedron - dodecahedron - a regular polyhedron, composed of twelve regular pentagons. The dodecahedron has a center of symmetry and 15 axes and 15 planes of symmetry.
Each vertex of the dodecahedron is a vertex of three regular pentagons. Therefore, the sum of the plane angles at each vertex is 324;.
Number of faces - 12, vertices - 20, edges - 30
Volume - V \u003d a; (15 + 7; 5) / 4.
Surface area - S= 3a;;5(5+2;5)
From the point of view of astrology, Space gives rise to a creative minor, discrete aspect with an angle of 36 (72, 144) degrees - Decile / Semi-quintile, which has the nature of unexpected, creative dynamics that affects the environment. It is believed that this is an aspect of "humanity", proportionality and appropriateness of initiatives.
He tactfully integrates the individual into the whole.
5. WATER - Icosahedron - twenty-sided. Each of the 20 faces is an equilateral triangle. 30 edges, 20 faces and 12 vertices. The icosahedron has 59 star shapes.
Each vertex of the icosahedron is a vertex of five triangles, the sum of the plane angles at each vertex is 300;.
Number of faces - 20, vertices - 12, edges - 30
Volume - V= 5a;(3+;5)/12.
Surface area - S= 5a;;3
From the point of view of astrology, this is an aspect of the sextile, characterized by a short-term intense interaction between the environment and the individual.
(The shorter the "edge", the longer the interaction, the more vertices, the more activity peaks.)
The Yin, hidden, internal Element gives rise to the Yang way of interaction on the external level, the qualities of manifestation are more appropriate for the Air element.
_____________________________
“The day science begins to study more than physical phenomena, it will make more progress in one decade than in all the previous centuries of its existence.” - Nikola Tesla.
There are many examples of random coincidences.
But, coincidences cannot be by nature, since only that which is in resonance, symmetry, multiplicity - in interaction can happen.
There are so many numerical "Coincidences" that it becomes obvious that they are not random.
Everyone can find them on their own, here are a few examples of this
entertaining abstraction:
The dynamics of the interaction of the Elements in degrees:
Water - Fire 300-180=120;
Air - Fire 270-180=90;
Water - Air 300-240=60;
Water - Earth 300-270=30;
Air-Earth 270-240=30;
We add the sums of the plane angles of the resulting polyhedra
FIRE, Tetrahedron 180;
AIR, Octahedron 240;
EARTH, Cube 270;
WATER, Icosahedron 300;
Space, Dodecahedron 324;
180+240+270+300+324=1314;. Divide by 360; circles.
1314:360=3,65
365 days a year.
The human body temperature is 36.5 degrees.
324-180=144
Multiply 24 hours by 60 minutes = 1440.
60 minutes times 60 seconds = 3600, 360 degrees in a circle.
Let's add the vertices of the polygons: 4+6+8+12+ 20=50
360:50=72
72 hours in three days.
72 beats per minute is the average heart rate of a healthy adult.
The angle of rotation of the DNA chain =72.
72 - the result of adding all the letters inscribed in the tetragrammaton.
72 is the maximum number of spheres touching one densely packed sphere in 6-dimensional space.
In Islam and Judaism there is a concept of 72 names of God.
72 degrees - outer corner regular pentagon
If we exclude Space from the calculations, then 360:30=12.
12 signs of the zodiac
12 months a year and so on.
180+240+270+300=990;
990:360=2,75
The average gestation period is 275 days.
Numerology believes that the number 275 is the union of God with man in the name of creativity.
Regular polyhedra can be inscribed into each other.
Therefore, all the Elements can manifest both on the external and on the internal level.
The dodecahedron, SPACE, contains all figures.
A tetrahedron - FIRE - fits into a cube, and a cube fits into a tetrahedron in a similar way.
The element Fire resides in the bowels of the planet Earth, and also, Fire can manifest itself above the Earth in the form of light, lightning and heat.
Octahedron - AIR, can be inscribed in a cube, and also, a cube can be inscribed in an Octahedron.
The element Air is contained in the empty cavities of the planet Earth, as well as around the Earth.
An icosahedron can be inscribed in a cube. Water tends to fill the empty cavities of the Earth.
A dodecahedron can be inscribed in an icosahedron, and therefore a cube and a tetrahedron.
The element of Water is able to connect all the Elements with each other.
She resides both on the surface of the Earth and in the Air, is released from the Air in the process of burning, just like all figures, she is able to reside in Space, Ether.
The names of the five convex regular polyhedra are tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The polyhedra are named after Plato, who in Op. Timaeus (4th century BC) gave them mystical. meaning; were known before Plato... Mathematical Encyclopedia
The same as the regular Polyhedra ... Great Soviet Encyclopedia
- ... Wikipedia
Phaedo, or On the Immortality of the Soul, named after the student of Socrates, Phaedo (see), Plato's dialogue, one of the most outstanding. This is the only dialogue of Plato that Aristotle names, and one of the few that are recognized as authentic by ... ...
encyclopedic Dictionary F. Brockhaus and I.A. Efron
One of the best in the artistic and philosophical sense of the dialogues of Plato, recognized as authentic by the unanimous verdict of both antiquity and modern science. In the latest Platonic criticism, they argued only about the time of its writing: some put ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
Philosophical ideas in the writings of Plato- Briefly Plato's philosophical heritage is extensive, It consists of 34 works, which are almost entirely preserved and have come down to us. These works are written mainly in the form of a dialogue, and the main character in them for the most part is ... ... Small Thesaurus of World Philosophy
Dodecahedron Regular polyhedron, or Platonic solid is a convex polyhedron with the maximum possible symmetry. A polyhedron is called regular if: it is convex all its faces are equal regular polygons in each of its ... ... Wikipedia
Solids of Plato, convex polyhedra, all faces of which are identical regular polygons and all polyhedral angles at vertices are regular and equal (Fig. 1a 1e). In the Euclidean space E 3, there are five P. m., data on which are given in ... Mathematical Encyclopedia
SOUL- [Greek. ψυχή], together with the body, forms the composition of a person (see the articles Dichotomism, Anthropology), while being an independent beginning; D. man contains the image of God (according to some of the fathers of the Church; according to others, the image of God is contained in everything ... ... Orthodox Encyclopedia
Books
- Timaeus (2011 ed.), Plato. Plato's Timaeus is the only systematic outline of Plato's cosmology, which until now has appeared in him only in a scattered and random form. This created glory for Timaeus by ...
- Controversial questions about the soul. Studies 6, Aquinas F.. The genre of `debatable questions` (quaestiones disputatae) is a special scholastic genre used in medieval universities. `Debatable questions about the soul` are one of ...
Anyone who has studied sacred geometry, or even 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 have the same size. For example, the cube, the most famous of the Platonic solids, has a square on each of its faces, and all its faces are of 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 the faces have the same value. In the case of a cube, this angle is 90 degrees. And fourth: if the Platonic solid is placed inside a sphere (of the correct form), then all its vertices will touch the surface of the sphere. Such definitions, apart from Cuba(A), meet only four forms that have all these characteristics. The second will be tetrahedron(B) (tetra means "four") is a polyhedron that has four faces, all equilateral triangles, the same edge length and the same angle, and - all vertices touch the surface of the sphere. Other simple form- This octahedron(C) (octa means "eight"), all eight faces are equilateral triangles of the same size, the length of the edges and corners is 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) - it 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 edge length and the same angles; all its vertices touch the surface of the sphere.
If you are an engineer or an architect, then you studied these five forms in college, at least superficially, because they are the basic structures.
Their Source: Metatron's Cube
If you are studying sacred geometry, no matter what 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've read almost all of them - and ask the experts, "Where do the 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 of them better, you need to redo the trick where you erased some of the lines. By erasing all but a few specific lines, you get this cube ( Fig.6-16 >).
Well, see the cube? It's really a cube within a cube. Some of the lines are dashed because they are behind the front faces. They are invisible if the cube becomes a solid, opaque body. Here is the opaque shape of the larger cube (Figure 6-16a>). (Make sure you can see it, because it will get harder and harder to see the next figures as we go.)
Erasing some lines and connecting other centers ( 
Fig.6-17>), you get two nested tetrahedra, 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 the larger star tetrahedron (Figure 6-17a>).
Figure 6-18> is an octahedron within 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 that way.
These are three-dimensional objects emanating from the thirteen circles of the Fruit of Life.
This is a picture of Shulamith Wulfing - the Christ Child inside the 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 last 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 searched 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 here," 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 Metatron Cube lines.” 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 some I, it turns out, forgot. No wonder I couldn't find that dodecahedron, because those missing lines defined it! For over twenty years I was convinced that I had all the lines drawn when I didn't.
This is one of big problems science, when it is considered that the problem is solved; then she moves on and uses this information for her further constructions. Now, for example, science has the same kind of problem around bodies falling in a vacuum. They have always been thought to 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 the case, but science still continues to use it. A spinning ball falls much faster than a non-spinning one. Someday there will come a day of scientific reckoning.
When I was married to Mackie, she was also very passionate about sacred geometry. Her work is very interesting for me, because it represents the feminine aspect, where the pentagonal energies of the right hemisphere of the brain operate. She shows how emotions, colors and shapes are all interconnected. She actually 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's 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 take a look at this thing." She took a 3D shape and rotated it through an f (phi ratio) angle. (What we have not yet talked about is that the ratio (ratio) of the Golden Mean, also called the proportion f (phi ratio), is exactly 1.618) . Rotating the shape in this way was something I would never have thought of. Having done this, she outlined the shadow cast by this form and received such an 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 (pentagons B) and one more pentagon coming out of each of these five (pentagons C), you get deployed dodecahedron. I thought: “Wow, this is the first time I find here some sort of dodecahedron." She did it in three days. I couldn't find him for twelve whole years.
Once we spent almost the whole day looking at this picture. She was amazing because one and all the lines in this picture correspond to the proportions of the Golden Mean. And everywhere there are three-dimensional rectangles of the Golden Mean. One is 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 reflect on later.
Quasi-crystals
Later I learned about a completely new science. This new science will completely change the world of technology. Using the new technology, metallurgists will surely 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, in the study of metals, in order to see where the atoms are located, they used a method called X-ray diffraction. I'll show you an x-ray diffraction photo soon. Some special models have been discovered that determine the existence of only certain atomic structures. That seemed to be all there was to know, because that was all there was to be found. This limited the ability to manufacture metals.
Then, in Scientific American magazine, there was a game that was based on the Penrose model. There was a British mathematician and relativist, Roger Penrose, who figured out how to lay tiles, whose tiles are shaped like a pentagon, so that they completely cover a flat surface. It is impossible to completely cover a flat surface with tiles in the form of only pentagons - there is no way to make it work. Then he proposed two forms of rhombus, which are derivatives of the pentagon, and using these two forms, he managed to create a set various models covering a flat surface. In the 1980s, Scientific American magazine proposed a game whose essence was to put these given models into new forms; this subsequently enabled metallurgical scientists who watched the game to surmise the existence of something new in physics.
In the end, they discovered a new model of the atomic lattice. It has always existed; they just discovered it. These lattice patterns are now referred to as quasi-crystals; this is a new phenomenon (1991). Through metals, they figure out what shapes and patterns are possible. Scientists are finding ways to use these molds and patterns to make new ones. metal products. I'm willing to bet that the Metatron Cube Mackey model 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 the main one - it came directly from the main model in Metatron's Cube. Although it's none of my business, someday I'll probably find out if it's true. I see that instead of using two Penrose models and a pentagon, it only uses one of these models and a pentagon (I was just thinking that I would suggest this option). What is happening in this new science now is interesting.
Update: According to David Adair, NASA has just made a metal in space that is 500 times stronger than titanium, light as foam, and clear as glass. Is it based on these laws?
As the events in this book unfold, you will find that sacred geometry can explain any subject in detail. There is not a single phenomenon that you can utter with your voice that cannot be described in its entirety, completely and to perfection, taking into account all possible knowledge, sacred geometry. (We distinguish between 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 deviate to all sorts of roots and branches and speak on all sorts of topics imaginable and unimaginable. But I will always get back into the groove because I lead everything in one specific direction, towards the Mer-Ka-Ba, the human light body.
I have spent many years in the study of sacred geometry, and I am sure that one can learn everything that is possible to know at all, anything about any subject, one has only to focus one's 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 does help. All the knowledge you already have within you, and all you have to do is unfold it. You are just exploring the map of the movement of the spirit in the Great Void, that's all. You can unravel the mystery of any subject.
To summarize: the first information system comes out of 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, graphite diamond was created in Europe using nanotechnology. It's a diamond about three feet across, and it's real. When the science of quasi-crystals and nanotechnology merge, our understanding of life will also change. Look at the end of the 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 represented a model of the corresponding element (
Fig.6-24>).
The tetrahedron was considered a model of the element of fire, the cube - of the earth, the octahedron - of air, the icosahedron - of water, and the dodecahedron - of ether. (Ether, prana, and tachyon energy) are all one and the same; it is ubiquitous and available at any point in space/time/dimension. This is the great secret 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.
In alchemy, only these elements are usually spoken of: fire, earth, air, and water; ether or prana is rarely mentioned because it is so sacred. In the Pythagorean school, if you just mentioned the word "dodecahedron" outside the walls of the school, 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 the life of Plato, they talked about her, 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 in the shape of a sphere. But the inner figure closest to the sphere is the dodecahedron (actually, the dodecahedron-icosahedral relationship). In addition to this, we live inside a large dodecahedron that contains the universe. When your mind reaches the limit of the space of the cosmos - and the limit is here There is- then he stumbles upon a dodecahedron closed in a sphere. I can say this because the human body is a hologram of the universe and contains the same foundations and laws. The twelve constellations of the zodiac are included here. The dodecahedron is the final figure of geometry and it is very important. On a microscopic level, the dodecahedron and icosahedron are the relative dimensions of DNA, the plans upon which all life is built.
You can match the three columns in this image ( Fig.6-24>) with the Tree of Life and the three primary energies of the universe: masculine (left), feminine (right) and childish (center). Or, if you go 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 pillar, which is creative, is the baby. Remember, to begin the process of exiting the Void, we went from an octahedron to a sphere. This is the beginning of the process of creation and is found in the baby or the 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 (corpus callosum), which connects the left and right sides. The right column containing the dodecahedron and icosahedron represents the female component of consciousness, right hemisphere brain, and the faces of these polygons are composed of triangles and pentagons. So the polygons on the left have 3- and 4-edge faces, while the shapes on the right have 3- and 5-edge faces.
In the language of the Earth consciousness, the right column is the missing component. We have created the male (left) side of the Earth consciousness, and now, in order to achieve integrity and balance, we are completing the creation of the female component. Right side is also associated with Christ consciousness or unity consciousness. The dodecahedron is the main form of the Christ consciousness grid around the Earth. The two forms of the right column represent relative to each other what are called paired figures, that is, if you connect the centers of the faces of the dodecahedron with straight lines, you will get an icosahedron, but if you connect the centers of the icosahedron, you will get a dodecahedron again. Many polyhedra have pairs.
Sacred 72
Dan Winter's book Heartmath shows that the DNA molecule is composed of the duality 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, is a pair of 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 again the icosahedron, 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 revealed.
This 72 degree angle spinning in our DNA is related to the plan/purpose of the Great White Brotherhood. As you may know, there are 72 orders associated with the Great White Brotherhood. Many people talk about 72 angelic orders, and the Jews mention 72 names of God. The reason why exactly 72 has to do with the structure of the Platonic solids, which is also connected with the grid of the Christ consciousness around the Earth.
If you take two tetrahedra and put them on top of each other (but in different positions), you get a star tetrahedron, which, when viewed from a certain angle, will look nothing but a cube ( 
Fig.6-25>). You can see how they are related. Five tetrahedra can be added together in the same way to form an icosahedral cap (Figure 6-26).
If you create twelve icosahedral caps and put one on each face of the dodecahedron (it takes 5 times 12 or 60 tetrahedra to create a dodecahedron), then it will be a star - stellated- 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 tetrahedra plus the 12 points in the centers 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 stellar form that is the basis of the Christ consciousness grid around the world. In other words, the Brotherhood is undertaking attempts to reveal the consciousness of the right hemisphere of the brain of the planet.
The original order was Alpha and Omega, the Order of Melchizedek, which was founded by Machiventa Melchizedek about 200,200 years ago. Since then, other orders have been founded, 71 in all. The youngest is the Brotherhood of the Seven Rays in Peru/Bolivia, the seventy-second order.
Each of the 72 orders has a sinusoid-like rhythm of life, where some of them appear for a certain period of time, then disappear for a while. They have biorhythms just like their human body has. The cycle of the Rosicrucian Order, for example, is 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 work together to achieve one goal - to bring the Christ consciousness back to this planet in order to restore this lost female component of consciousness and bring the left and right hemispheres of the planet's brain into balance. There is another way of looking at this phenomenon that is really 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 turn 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 a thermonuclear reaction fuses it together. Fusion together is fine - no one complains 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 - tearing matter apart here on Earth affects the corresponding area in outer space - both above and below. In other words, the microcosm and the macrocosm are interconnected. This is why the decay reaction is outlawed 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 a huge amount of fire to manifest in one place. This leads to an imbalance and the Earth must respond to this.
If you pour 80 billion tons of water on the city, this will also be an unbalanced situation. If there is too much air somewhere, too much water, too much of anything, 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 takes in the Void. If you know the underlying map, then you have the knowledge and understanding necessary to co-create with God.
Fig.6-27> shows the relationship of all these figures. Each vertex is connected to the next one, and all of them are in certain mathematical ratios related to the proportion f (phi ratio).
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Question: Why is the achievement of the goal of creation such a complex and multi-stage process, including descent, ascent, breaking?
Answer: Because initially it consists of two opposite components: light and desire (Kli), which build a connection between themselves on the basis of their opposite nature.
The process of development proceeds from 2 conditions and determines one law of development: two opposites - at the beginning must come to full similarity at the end.
From these initial and final conditions the whole process inevitably follows. We cannot replace anything in it and act according to a different logic.
There are strict conditions - initial and final.
Condition 1: Infinite distance between them.
Condition 2: Their full connection, to the very end.
Before that: one is all giving, and the other is completely receiving. On the one hand, there is an endless abyss between them, and on the other hand, there is an end point where they merge into one.
Now try to write a formula according to which from the 1st condition - the complete opposite, you can come to the 2nd condition - complete similarity.
In the World of Infinity, both the separation of these two components and their union already exist. Although there is light and desire, they are merged and complement each other. This fusion is held together by the power of the upper light.
That is, the final condition is first provided by the power of light, the Creator. This allows us to reveal the formula - how to make this transition from one state to another?
Potential conditions have been created - now we need to start implementation. And the whole reality is now sequentially revealed scene by scene before our eyes - those very creations that are in the world of Infinity. We never get out of there.
In this single place (desire) created by the Creator, there is light, desire, and the condition for their merging – and all of this is now taking place.
Therefore, we ourselves are called the actions of the Creator, the results of His work: creations, works, creatures.
It is impossible to change something here - the conditions themselves: the initial and final ("the end of the action, contained in the initial plan") already determine everything that will happen - including the split.
Bill Gates is preparing a vaccine to reduce the birth rate
An unexpected war broke out in the Russian blogosphere after Bill Gates spoke at the closed TED2010 (back in 2010) Conference in Long Beach, California. In his speech titled "Renewing to Zero!" The founder of Microsoft said that for quite a long time his charitable foundation has been supporting the development of a vaccine that leads to a decrease in human reproduction. It is assumed that residents of Third World countries will undergo such vaccination. As you might expect, some bloggers support the charity of the richest man in the world. Indeed, the so-called third world countries suffer from poverty, hunger and overpopulation. First, "rich white people" brought these people civilization and modern medicine, and then it turned out that fertile land can not provide everyone with food, and the state - with work. The road is paved with good intentions, you know where...There is also a radical part of bloggers, acting under the slogan "Kill Bill!" and hinting at Hitler's eugenics. Some sarcastically remark that in developed countries, even without a vaccine, the birth rate is not good. Like, if we ourselves can’t be fruitful, then let’s at least feed those who don’t know how and don’t want anything else. The problems in the world do not arise because they are idle somewhere, cannot use contraceptives, or eat too much. It's all about the wrong distribution of world forces and means. The world is like a big house, and we, like an exhausted housewife, rush to stir the soup, then shake the baby, then milk the cow - and as a result we do not have time to do anything. Therefore, it turns out that somewhere the crop is dying and because of too low prices producers are forced to pour milk on the ground, and somewhere mothers trade their own children to feed other children.
At the same time, the mechanical distribution of funds, charitable assistance and transport with food will lead to the same success as vaccination against fertility. Any action is doomed to failure in advance, just like the once “disinterested” help. Soviet Union. The earth can feed large quantity of people. We have the ability to make ourselves happy. We can fill our lives with meaning. To do this, you just really need to understand that we are one family.
