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Presentation on the topic:"Platonic solids"

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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 240º. 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 240º.

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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 300º. 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 300º.

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Composed 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 324º. Composed 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 324º.

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Regular polyhedra are sometimes called Platonic solids because they figure prominently in the philosophical worldview developed by the great thinker of Ancient Greece, Plato (c. 428 - c. 348 BC). Regular polyhedra are sometimes called Platonic solids because they figure prominently in the philosophical worldview developed by the great thinker of Ancient Greece, Plato (c. 428 - c. 348 BC). Plato believed that the world is built from four “elements” - fire, earth, air and water, and the atoms of these “elements” have the shape of four regular polyhedra. The tetrahedron personified fire, since its apex points upward, like a flaring flame. The icosahedron is like the most streamlined - water. The cube is the most stable of figures - the earth. Octahedron - air. In our time, this system can be compared to the four states of matter - solid, liquid, gaseous and flame. The fifth polyhedron, the dodecahedron, symbolized the whole world and was considered the most important. This was one of the first attempts to introduce the idea of ​​systematization into science.

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Kepler suggested that there was a connection between the five regular polyhedra and the six planets of the solar system discovered by that time. Kepler suggested that there was a connection between the five regular polyhedra and the six planets of the solar system discovered by that time. According to this assumption, a cube can be inscribed into the sphere of Saturn's orbit, into which the sphere of Jupiter's orbit fits. The tetrahedron described near the sphere of the orbit of Mars fits into it, in turn. The dodecahedron fits into the sphere of the orbit of Mars, into which the sphere of the orbit of the Earth fits. And it is described near the icosahedron, into which the sphere of the orbit of Venus is inscribed. The sphere of this planet is described around the octahedron, into which the sphere of Mercury fits. This model of the Solar System (Fig. 6) was called Kepler’s “Cosmic Cup”. The scientist published the results of his calculations in the book “The Mystery of the Universe.” He believed that the secret of the Universe had been revealed. Year after year, the scientist refined his observations, double-checked the data of his colleagues, but finally found the strength to abandon the tempting hypothesis. However, its traces are visible in Kepler's third law, which talks about cubes of average distances from the Sun.

Slide no. 11

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The ideas of Plato and Kepler about the connection of regular polyhedra with the harmonious structure of the world in our time have been continued in an interesting scientific hypothesis, which in the early 80s. expressed by Moscow engineers V. Makarov and V. Morozov. They believe that the Earth's core has the shape and properties of a growing crystal, which influences the development of all natural processes occurring on the planet. The rays of this crystal, or rather, its force field, determine the icosahedron-dodecahedron structure of the Earth (Fig. 7). It manifests itself in the fact that in the earth’s crust projections of regular polyhedra inscribed in the globe appear: the icosahedron and the dodecahedron. The ideas of Plato and Kepler about the connection of regular polyhedra with the harmonious structure of the world in our time have been continued in an interesting scientific hypothesis, which in the early 80s. expressed by Moscow engineers V. Makarov and V. Morozov. They believe that the Earth's core has the shape and properties of a growing crystal, which influences the development of all natural processes occurring on the planet. The rays of this crystal, or rather, its force field, determine the icosahedron-dodecahedron structure of the Earth (Fig. 7). It manifests itself in the fact that in the earth’s crust projections of regular polyhedra inscribed in the globe appear: the icosahedron and the dodecahedron. Many mineral deposits extend along an icosahedron-dodecahedron grid; The 62 vertices and midpoints of the edges of polyhedra, called nodes by the authors, have a number of specific properties that make it possible to explain some incomprehensible phenomena. Here are the centers of ancient cultures and civilizations: Peru, Northern Mongolia, Haiti, Ob culture and others. At these points, maximum and minimum atmospheric pressure and giant eddies of the World Ocean are observed. These nodes contain Loch Ness and the Bermuda Triangle. Further studies of the Earth may determine the attitude towards this scientific hypothesis, in which, as can be seen, regular polyhedra occupy an important place.

Slide 2

There are five unique shapes that are of paramount importance to understanding both sacred and ordinary geometry. They are called Platonic solids, although Pythagoras used them long before Plato, calling them ideal geometric solids. Any Platonic solid has some special characteristics.

Slide 3

Firstly, all faces of such a body are equal in size. For example, the cube, the most famous of all Platonic solids, has each face in the form of a square, and they are all the same size. Secondly, the edges of the Platonic solid are the same length: all the edges of the cube are the same. Third, the interior angles between its adjacent faces are equal. For a cube, this angle is 90 degrees.

Slide 4

Fourthly, each of the Platonic solids can be inscribed in a sphere, each of its vertices touching the surface of this sphere. There are only four shapes other than the cube (A) that meet all these characteristics: the tetrahedron - B (tetra means "four"), which has four faces in the form of equilateral triangles; octahedron - (octa means "eight"), the eight faces of which are equilateral triangles of the same size; icosahedron - D; dodecahedron - E. The last two Platonic solids are somewhat more complicated. The icosahedron has 20 faces, represented by equilateral triangles. The dodecahedron (dodeca is “twelve”) has 12 faces in the form of regular pentagons. In fact, there is an original form - this is the sphere from which everything begins, which is considered the sixth body. Ancient alchemists believed that these six shapes were associated with certain elements: cube - earth, tetrahedron - fire, octahedron - air, icosahedron - water, dodecahedron - ether (ether, prana and tachyon energy are the same; they spread everywhere and are at any point in space - time - dimensions). And the sphere is emptiness. These six elements are the building stones of the Universe. They create the qualities of the Universe.

Slide 5

The six elements - the primary forms as they are represented inscribed in the spheres - can be correlated with the three pillars corresponding to the Tree of Life and the three primary energies of the Universe. On the left is a male pillar, on the right is a female one, the central pillar, the creator, is a child. Or, if we turn to the matter of the Universe, we get a proton on the left, an electron on the right and a neutron in the center.

Slide 6

Cube The cube differs from the other Platonic solids in one feature that no one except the sphere has: the cube and the sphere can completely contain the four other Platonic solids and each other, covering them with their surface. While the sphere is the Mother, the most important female form, the cube is the Father, the most important male form. In all of reality, the sphere and the cube are the two most important shapes, and they almost always dominate when it comes to the original connections in creation. Symbolically, the cube is identical to the square - four, the number of matter, the number of four elements. The cube is ideal stability, a stable base - a symbol of the earth itself. Therefore, monarchs (for example, Egyptian pharaohs) are often depicted sitting on a cubic stone, a symbol of the stability of their kingdom. A cube is a square in three dimensions, each face of which has the same characteristics as the others, which is why it has become an emblem of truth. In iconography it is often used as a pedestal for allegorical figures of Truth and History. According to Mayan legend, the Tree of Life grew from a cube. In both Judaism and Islam, the cube represents the center of faith. Pilgrims in Mecca circle the cubic structure of the Kaaba, the most revered Muslim shrine. The development of a cube into space represents a cross, and if Christian churches are usually built in such a way that they have the shape of a cross in plan, this is precisely because the cross is an expansion into the plane of a cubic stone: the church should represent the establishment of the religion of Christ on earth for a long time. The cube, being a completely closed figure, symbolizes limitation. Therefore, the cross generated by the unfolding of the cube also means limitation, suffering.

Slide 7

Tetrahedron. This figure consists of four regular triangles. If you unfold them on a plane, they form an equilateral triangle - a symbol of God. Like an equilateral triangle, the tetrahedron represents the embodiment of harmony and balance itself. The corner points of a cube, like a square, are at different distances from each other, which means that there is constant tension in these figures.

Slide 8

Octahedron. Strictly speaking, the octahedron is the “double” of the cube: if you connect the centers of adjacent faces of the cube, you get an octahedron.

Slide 9

Dodecahedron and icosahedron. The dodecahedron is such a sacred shape that in the time of Pythagoras, if someone had uttered this word outside the Pythagorean school, he would have been killed on the spot. Two hundred years later, when Plato lived, he could already talk about it, but very carefully. “This was partly explained by the fact that the fifth element was associated with the dodecahedron - ether, or prana. Alchemy usually deals with only four elements: fire, earth, air and water, and prana is rarely talked about because it is considered very sacred. Another reason is that in those days the ancient knowledge was carefully hidden, according to which the dodecahedron is close to the outer edge of the human energy field and is the highest form of consciousness... The dodecahedron is the end point of geometry, and it is very important. At the microscopic level, the dodecahedron and icosahedron are interconnected parameters of DNA, a blueprint for all life” (DrunvaloMelchizedek). If you connect the centers of the dodecahedron faces with straight lines, you get an icosahedron. By connecting the centers of the faces of the icosahedron, we again obtain a dodecahedron. Many polyhedra have “doubles”. In general, a polyhedron is one of the three-dimensional geometric figures. At all times they have been given magical significance.

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Presentation on the topic "Platonic solids - the key to the structure of the Earth and the Universe" in algebra in powerpoint format. This presentation for schoolchildren tells about what the Platonic solid is and its role in entertaining mathematics. Author of the presentation: mathematics teacher Artamonova L. IN.

Fragments from the presentation

The earth, if you look at it from above, looks like a ball sewn from twelve pieces of leather... (c) Plato, "Phaedo"

Study one. Spherical frying pan

• The idea of ​​a dodecahedral Earth was revived in 1829 by the French geologist, member of the Paris Academy, Elie de Beaumont. He hypothesized that the initially liquid planet, when solidified, took the shape of a dodecahedron. De Beaumont built a network consisting of the edges of the dodecahedron and its dual icosahedron, and then began to move it around the globe. So he looked for a position that would best reflect the topography of our planet. And he found an option when the faces of the icosahedron more or less coincided with the most stable areas of the earth’s crust, and its thirty edges coincided with mountain ranges and places where its fractures and crumples occurred.
• A hundred years later, the idea was picked up by our compatriot S.I. Kislitsyn, who proposed combining the two opposite vertices of the icosahedron with the Earth’s poles, while the largest diamond deposits seemed to be at some of its other vertices. And in the last third of the last century, de Beaumont’s model with Kislitsyn’s orientation began to be developed in our country by N.F. Goncharov, V.A. Makarov and V.S. Morozov.
• Goncharov, Makarov and Morozov believed that a solid core in the form of a dodecahedron arose inside the Earth, which directed flows of matter to the surface; as a result, a kind of power frame of the planet was formed, repeating the structure of the core. However, according to our famous crystallographer and mineralogist I.I. Shafranovsky, the dodecahedron and icosahedron with their fifth-order symmetry axes do not have crystallographic symmetry, and therefore the assumption about the formation of such bodies in the core of the planet is incorrect.
• Tessellation of a sphere with hexagons alone is impossible, since it contradicts Euler’s theorem, which relates the numbers of vertices, edges and faces in any polyhedron. Ivanyuk and Goryainov believe that the sphere will be covered with a grid of pentagons, since they are closest to hexagons, but they can be used to pave the surface of the sphere. So, you get a dodecahedron! The same conclusion will remain valid if the liquid layer on the surface of the sphere becomes thicker and the radius of the sphere becomes smaller, so that the liquid fills almost the entire volume of the ball.
• In relation to the Earth, this means that if for billions of years it was a hot core surrounded by a viscous liquid, then pentagonal convective cells (the side of which is commensurate with the radius of the planet) could arise in it. And then the flows of matter in them, cooling and hardening, would form that dodecahedral frame that de Beaumont and his followers spoke about

Study two. Frozen music

• At first glance at the globe, the distribution of continents and oceans seems poorly ordered, but some patterns, as has long been noted, still exist.
• Firstly, the two hemispheres separated by the equator are very different: the Northern hemisphere is dominated by land, and the Southern hemisphere is dominated by the sea.
• Secondly, the shapes of the continents and oceans are close to triangular, with continental triangles with their bases facing north and tapering ends to the south; oceanic - on the contrary.
• Thirdly, diameters drawn through land, in the vast majority of cases, will pass on the other side of the globe through water, that is, the antipodality of continents and oceans is observed.
• The latter fact means that the earth's surface does not have a center of symmetry, but there is a center of antisymmetry, or two-color symmetry, the idea of ​​which was developed by our largest crystallographer, Academician A.V. Shubnikov. The point is that the initially equal centrally symmetrical elements of a certain figure are divided into two classes, which are conventionally marked with two colors. And then the operation of reflection from the center transforms an element of one color into an element of another - into an anti-element.
• Shafranovsky noted that the above properties of the Earth's topography can be, to a first approximation, covered by the geometric model proposed in the 50s by the prominent Soviet geologist B.L. Lichkov. It is based on an octahedron, the eight faces of which are painted in two colors so that adjacent faces are of different colors. It is clear that the “chess” coloring corresponds to antisymmetry: opposite each face lies a face of a different color.
• Let the white edges represent the continents, and the blue ones the oceans. Let's put the octahedron on the white face, which will be Antarctica. Then the upper blue edge will depict the Arctic Ocean, and the three triangular white edges surrounding it will become the triangles that are visible on the globe - North and South America, Europe plus Africa and Asia. Turning the octahedron over, we get a different picture: around the white edge (Antarctica) there are three blue oceans.

Conclusion

• In both studies, the basic ideas are similar: some physical process breaks the continuous symmetry of the sphere and as a result, a discrete symmetry of one of the Platonic solids arises. It is possible that at a time when the Earth “was formless and empty,” such effects determined the main features of its surface. And since many other factors were also at work in different geological eras, the final picture turned out to be much more complex and confusing.
• Apparently, regular polyhedra will play an increasingly important role in various fields of knowledge. And here it’s not just ludi mathematici (mathematical games) - these figures are internally connected with natural phenomena. As Plato said, of all visible bodies they are the most wonderful, and each of them is beautiful in its own way. This is probably the case when beauty and truth are one.

Slide 1

Regular convex polyhedra
Platonic solids

Slide 2

There are a shockingly small number of regular polyhedra, but this very modest squad managed to get into the very depths of various sciences. L. Carroll

Slide 3

Regular tetrahedron
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 180º.
Rice. 1

Slide 4

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 240º.
Regular octahedron
Rice. 2

Slide 5

Regular icosahedron
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 300º.
Rice. 3

Slide 6

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 270º.
Cube (hexahedron)
Rice. 4

Slide 7

Regular dodecahedron
Composed 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 324º.
Rice. 5

Slide 8

came from Ancient Greece, they indicate the number of faces: “edra” - face; “tetra” – 4; “hexa” – 6; "okta" - 8; “Ikosa” – 20; "dodeka" - 12.
Names of polyhedra

Slide 9

Regular polyhedra are sometimes called Platonic solids because they figure prominently in the philosophical worldview developed by the great thinker of Ancient Greece, Plato (c. 428 - c. 348 BC). Plato believed that the world is built from four “elements” - fire, earth, air and water, and the atoms of these “elements” have the shape of four regular polyhedra. The tetrahedron personified fire, since its apex points upward, like a flaring flame. The icosahedron is like the most streamlined - water. The cube is the most stable of figures - the earth. Octahedron - air. In our time, this system can be compared to the four states of matter - solid, liquid, gaseous and flame. The fifth polyhedron, the dodecahedron, symbolized the whole world and was considered the most important. This was one of the first attempts to introduce the idea of ​​systematization into science.
Regular polyhedra in Plato's philosophical picture of the world

Slide 10

Kepler's "Cosmic Cup"
Kepler suggested that there was a connection between the five regular polyhedra and the six planets of the solar system discovered by that time. According to this assumption, a cube can be inscribed into the sphere of Saturn's orbit, into which the sphere of Jupiter's orbit fits. The tetrahedron described near the sphere of the orbit of Mars fits into it, in turn. The dodecahedron fits into the sphere of the orbit of Mars, into which the sphere of the orbit of the Earth fits. And it is described near the icosahedron, into which the sphere of the orbit of Venus is inscribed. The sphere of this planet is described around the octahedron, into which the sphere of Mercury fits. This model of the Solar System (Fig. 6) was called Kepler’s “Cosmic Cup”. The scientist published the results of his calculations in the book “The Mystery of the Universe.” He believed that the secret of the Universe had been revealed. Year after year, the scientist refined his observations, double-checked the data of his colleagues, but finally found the strength to abandon the tempting hypothesis. However, its traces are visible in Kepler's third law, which talks about cubes of average distances from the Sun.
Model of the Solar System by I. Kepler
Rice. 6

Slide 11

The ideas of Plato and Kepler about the connection of regular polyhedra with the harmonious structure of the world in our time have been continued in an interesting scientific hypothesis, which in the early 80s. expressed by Moscow engineers V. Makarov and V. Morozov. They believe that the Earth's core has the shape and properties of a growing crystal, which influences the development of all natural processes occurring on the planet. The rays of this crystal, or rather, its force field, determine the icosahedron-dodecahedron structure of the Earth (Fig. 7). It manifests itself in the fact that in the earth’s crust projections of regular polyhedra inscribed in the globe appear: the icosahedron and the dodecahedron. Many mineral deposits extend along an icosahedron-dodecahedron grid; The 62 vertices and midpoints of the edges of polyhedra, called nodes by the authors, have a number of specific properties that make it possible to explain some incomprehensible phenomena. Here are the centers of ancient cultures and civilizations: Peru, Northern Mongolia, Haiti, Ob culture and others. At these points, maximum and minimum atmospheric pressure and giant eddies of the World Ocean are observed. These nodes contain Loch Ness and the Bermuda Triangle. Further studies of the Earth may determine the attitude towards this scientific hypothesis, in which, as can be seen, regular polyhedra occupy an important place.
Icosahedral-dodecahedral structure of the Earth
Icosahedral-dodecahedral structure of the Earth
Rice. 7

Slide 12

Regular polyhedron Number Number Number
Regular polyhedron of edge vertex faces
Tetrahedron 4 4 6
Cube 6 8 12
Octahedron 8 6 12
Dodecahedron 12 20 30
Icosahedron 20 12 30
Table No. 1

Slide 13

Regular polyhedron Number Number
Regular polyhedron of faces and vertices (G + V) of edges (P)
Tetrahedron 4 + 4 = 8 6
Cube 6 + 8 = 14 12
Octahedron 8 + 6 = 14 12
Dodecahedron 12 + 20 = 32 30
Icosahedron 20 + 12 = 32 30
Table No. 2

Slide 14

The sum of the number of faces and vertices of any polyhedron is equal to the number of edges increased by 2. Г + В = Р + 2
Euler's formula
The number of faces plus the number of vertices minus the number of edges in any polyhedron is 2. Г + В  Р = 2

Slide 15

Salvador Dali
"The Last Supper"

Slide 16

Regular polyhedra and nature
Regular polyhedra are found in living nature. For example, the skeleton of the single-celled organism Feodaria (Circjgjnia icosahtdra) is shaped like an icosahedron (Fig. 8). What caused this natural geometrization of feodaria? Apparently, because of all the polyhedra with the same number of faces, it is the icosahedron that has the largest volume with the smallest surface area. This property helps the marine organism overcome the pressure of the water column. Regular polyhedra are the most “profitable” figures. And nature makes extensive use of this. This is confirmed by the shape of some crystals. Take table salt, for example, which we cannot do without. It is known that it is soluble in water and serves as a conductor of electric current. And crystals of table salt (NaCl) have the shape of a cube. In the production of aluminum, aluminum-potassium quartz (K  12H2O) is used, the single crystal of which has the shape of a regular octahedron. The production of sulfuric acid, iron, and special types of cement is not complete without pyrite sulfur (FeS). The crystals of this chemical are dodecahedron shaped. In various chemical reactions, sodium antimony sulfate (Na5(SbO4(SO4))) is used - a substance synthesized by scientists. The crystal of sodium antimony sulfate has the shape of a tetrahedron. The last regular polyhedron - the icosahedron - conveys the shape of boron crystals (B). At one time, boron was used to create first generation semiconductors.
Feodaria (Circjgjnia icosahtdra)
Rice. 8

Slide 17

Determine the number of faces, vertices and edges of the polyhedron shown in Figure 9. Check the feasibility of Euler's formula for this polyhedron.
Task
Rice. 9

18.03.2018 04:55

The presentation was made for the research work, which was presented at the Regional Scientific and Production Complex “Step into Science” and the All-Russian “Youth.Science.Culture - Siberia”. The main part of the work examines the concepts of regular polyhedra, their types and developments, kusudama balls and their types, and conducts a study of kusudama balls. Regular polyhedra are made using reamers and kusudama balls are made using modular origami. The implementation of Euler's formula is checked. A comparison is made of regular polyhedra with kusudama balls. Similarities and differences were found. The work has great practical and theoretical value; it can be used in mathematics, technology lessons, and extracurricular activities. The methods used are modeling, design, search method, analysis and comparison of data. The work was awarded a 3rd degree diploma at the All-Russian Scientific and Practical Conference. Published on the research site "Trainer"

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"Presentation for the research work "Platonic and Archimedean solids as the main forms of kusudama balls""

“Youth, science, culture - Siberia”

MBOU "Duldurga Secondary School"

All-Russian Scientific and Practical Conference

Duldurginsky district 7-a class Supervisor: Kibireva Irina Valerievna mathematics teacher of the highest qualification category

Honorary Worker of General Education of the Russian Federation

MBOU "Duldurga Secondary School"

Platonic and Archimedean solids as the main forms of kusudama balls

Pythagoras (570 - 497 BC) Plato (real name Aristocles,

427-347 BC)

Euclid (365-300 BC)

Leonhard Euler (1707-1783)

In the artist's painting Salvador Dali "The Last Supper" Christ and his disciples are depicted against the background of a huge transparent dodecahedron.

According to the ancients, the UNIVERSE had the shape of a dodecahedron, i.e. they believed that we live inside a vault shaped like the surface of a regular dodecahedron.

Polyhedra in Moscow architecture

Cathedral of the Immaculate Conception

Virgin Mary

on Malaya Gruzinskaya

Historical Museum

Geological finds

Garnets: Andradite and Grossular (found in the Akhtaranda River basin, Yakutia)

Goal of the work:

Find out which polyhedra belong to the Platonic and Archimedean solids and how they are related to kusudama balls. Do kusudama balls really have their shape?

Object of study: Platonic and Archimedean solids, kusudama balls

Subject of study: origametry

Hypothesis:

If you study regular, semi-regular polyhedra and kusudama balls, you can see similarities in them and give a description of kusudama balls from a geometric point of view.

Research objectives:

• Collect and study literature on the topics “Platonic and Archimedean solids”, “Kusudama balls”.
• Using developments to make regular polyhedra
• 3. Make kusudama balls
• 4. Check the fulfillment of Euler’s formula for regular and semiregular polyhedra.
• 4. Find the relationship between polyhedra and kusudama balls.

Methods and means:

• modeling
• design
• search method
• data analysis and comparison

Research stages:

• Studying the literature on regular polyhedra (Platonic solids), semiregular polyhedra (Archimedean solids), kusudama balls.
• Modeling polyhedra and kusudama balls.
• Comparing and contrasting kusudama balls with regular polyhedra.
• Description of the data received.

Polyhedron

• A polyhedron is a closed surface made up of polygons.

• It is called convex , if it is all located on one side of the plane of each of its faces.

Execution of Euler's formula for regular polyhedra

Tetrahedron

Peaks

Ribs

Edges

Euler's formula

Dodecahedron

Icosahedron

Star shapes

The stellation of the octahedron is an octagonal star.

Small stellated dodecahedron

Kusudama balls

• Kusudama are ancient decorative traditional Japanese products using the origami technique.
• Kusudama is a type of origami; paper craft resembling a flower ball.

Cube

Analog of a cube

Gyroscope

The faces are triangles that are not explicitly visible. If you put a triangle on every three vertices, you get an octahedron. Which one:

The total number of vertices is 8;

total number of vertices – 6,

total number of ribs – 12,

Has the shape of an octahedron

total number of faces – 6.

total number of ribs – 12,

the total number of faces is 8.

Triangular icosahedron

Has the shape of an icosahedron

Flower ball

It is one of the stellated forms of the icosahedron - the small triambic icosahedron.

It has the shape of a dodecahedron, in which:

Has the shape of an icosahedron

Has the shape of a dodecahedron

total number of vertices – 20,

For which:

total number of vertices – 32;

total number of ribs – 30,

total number of ribs – 60,

the total number of faces is 12.

the total number of faces is 20.

It has the shape of a dodecahedron, in which:

total number of vertices – 20,

Has the shape of a dodecahedron

If you bend the ears of the kusudama, you can clearly see that it has the shape of a cube. Therefore, apart from the ears, we can say that she has:

total number of ribs – 30,

total number of vertices – 8;

Shaped like a cube

the total number of faces is 12.

total number of ribs – 12,

total number of faces – 6.

Flexi ball

It has the shape of an icosahedron, in which:

total number of vertices – 12,

Has the shape of an icosahedron

total number of ribs – 30,

the total number of faces is 20.

Cube without corners

Classic kusudama

Has the shape of a truncated cube

It has the shape of a truncated cube. Which one:

total number of vertices – 24,

total number of ribs – 36,

total number of vertices – 24,

Has the shape of a truncated cube

the total number of faces is 14.

total number of ribs – 36,

the total number of faces is 14.

Faces: 8 – triangles (not visible),

6 - octagons

6 - octagons

Has the shape of a truncated cube

Kusudama rose

Has the shape of a truncated cube

It has the shape of a truncated cube. Which one:

Which one:

total number of vertices – 24,

total number of vertices – 24,

Has the shape of a truncated cube

total number of ribs – 36,

total number of ribs – 36,

the total number of faces is 14.

the total number of faces is 14.

Faces: 8 – triangles (not visible),

6 – octagons (if you bend the ears

6 - octagons

Star octahedron

Is the intersection of two tetrahedrons. He has:

Star Baskets

Has the shape of a stellated octahedron

This is an analogue of the great stellated dodecahedron. He has:

total number of vertices – 14,

total number of ribs – 36,

total number of vertices – 32,

Shaped like a large stellated dodecahedron

the total number of faces is 24.

total number of ribs – 90,

the total number of faces is 60.

Kusudama curler

It is difficult to determine the total number of vertices, edges and faces of this kusudama. But we can definitely say that it has a star shape. This may be the seventeenth stellation of the icosahedron.

Execution of Euler's formula for Archimedean solids and kusudama balls

Polyhedron name

Truncated tetrahedron

Peaks

Ribs

Truncated octahedron

Truncated cube

Edges

Euler's formula

Truncated icosahedron

Truncated dodecahedron

24 + 14 = 36 + 2

cuboctahedron

24 + 14 = 36 + 2

Icosidodecahedron

60 + 32 = 90 + 2

Rhombicuboctahedron

60 + 32 = 90 + 2

Rhombicosidodecahedron

Rhombic truncated cuboctahedron

12 + 14 = 24 + 2

30 + 32 = 60 + 2

Rhombic truncated icosidodecahedron

24 + 26 = 48 + 2

Snub cube

Snub dodecahedron

60 + 62 = 120 + 2

48 + 26 = 72 + 2

120 + 62 = 180 + 2

24 + 38 = 60 + 2

60 + 92 = 150 + 2

Conclusion:

• Kusudama are similar to polyhedra in many ways. They mostly consist of a large number of parts and have a clear geometric shape. Folding the parts is usually not difficult, but assembling the whole product sometimes requires some effort.
• The basis of kusudama, as a rule, is some regular polyhedron (most often a cube, dodecahedron or icosahedron). Somewhat less often, a semi-regular polyhedron is taken as a basis.
• Models of kusudama balls in the shape of polyhedrons produce an aesthetic impression on a person and can be used as decorative ornaments.
• Such amazing and perfect objects of the modern world as kusudama have been little studied.