Correct and irregular polyhedra presentation. Dodecahedron properties and interesting facts. Symmetry elements of a regular octahedron

One of the earliest references to regular polyhedra is found in Plato's treatise (BC) "Timus". Therefore, regular polyhedra are also called Platonic solids (although they were known long before Plato). Each of the regular polyhedra, and there are five of them. Plato associated with four "earthly" elements: earth (cube), water (icosahedron), fire (tetrahedron), air (octahedron), and also with the "unearthly" element - the sky (dodecahedron).

A regular polyhedron, or Platonic solid, is a convex polyhedron with the greatest possible symmetry. A polyhedron is called regular if: it is convex all its faces are equal regular polygons at each of its vertices the same number of faces converges all its dihedral angles are equal

Let's note an interesting fact related to the hexahedron (cube) and octahedron. The cube has 6 faces, 12 edges, and 8 vertices, and the octahedron has 8 faces, 12 edges, and 6 vertices. That is, the number of faces of one polyhedron is equal to the number of vertices of the other, and vice versa. The cube and the hexahedron are said to be dual to each other. This also manifests itself in the fact that if you take a cube and build a polyhedron with vertices at the centers of its faces, then, as you can easily see, you get an octahedron. The converse is also true - the centers of the octahedron faces serve as the vertices of the cube. This is precisely the duality of the octahedron and the cube (fig). It is easy to figure out that if we take the centers of the faces of a regular tetrahedron, then we again get a regular tetrahedron (Fig). Thus, the tetrahedron is dual to itself.

The famous mathematician and astronomer Kepler built a model Solar system as a series of sequentially inscribed and described regular polyhedra and spheres. What is the order of the planets' arrangement (in accordance with the "requirements" of regular polyhedra) did Kepler get? A cube was inscribed in the sphere of Saturn's orbit, and the sphere of Jupiter's orbit was inscribed in it; a tetrahedron was inscribed in this sphere, and the sphere of the orbit of Mars was inscribed in it; further: dodecahedron - sphere of the Earth's orbit - icosahedron - sphere of the orbit of Venus - octahedron - sphere of the orbit of Mercury.

Contents: Purpose of the project Purpose of the project Purpose of the project Purpose of the project Term Polyhedra Term Polyhedra Term Polyhedra Term Polyhedra History History History Plato Plato Plato Platonic solids Platonic solids Platonic solids Platonic solids Euclid Euclid Euclid Archimedes Archimedes Archimedes Archimedes Archimedean bodies Archimedesian bodies Archimedesian bodies Archimedesian bodies Archimedesian bodies Johannes Kepler Johannes Kepler cosmological hypothesis of Kepler's cosmological hypothesis of Kepler's cosmological hypothesis of Kepler's cosmological hypothesis Kepler Tetrahedron Tetrahedron Tetrahedron icosahedron icosahedron icosahedron Dodecahedron Dodecahedron dodecahedron hexahedron (cube) hexahedron (cube) hexahedron (cube), octahedron octahedron octahedron A special case A special case A special case A special case of scanning the right of polyhedra Unfolds of regular polytopes Unfolds of regular polytopes Unfolds of regular polytopes Theorem Theorem Theorem Table of char-k Table of char-k Table of char-k Table of char-k Semi-regular many polyhedra Semiregular polyhedra Semiregular polyhedra Semiregular polyhedra Being in nature Being in nature Being in nature Being in nature Historical background Interesting facts Interesting facts Interesting facts Interesting facts

A polyhedron is called regular if all its faces are equal regular polygons, the same number of edges emerge from each of its vertices, and all dihedral angles are equal. A polyhedron is called regular if all its faces are equal regular polygons, the same number of edges emerge from each of its vertices, and all dihedral angles are equal.

The history of regular polyhedra They were studied by scientists, jewelers, priests, architects. These polyhedra were even credited with magical properties. The ancient Greek scientist and philosopher Plato (IV-V century BC) believed that these bodies personify the essence of nature. In his dialogue "Timaeus" Plato says that the atom of fire has the form of a tetrahedron, earth - a hexahedron (cube), air - an octahedron, water - an icosahedron. In this correspondence, there was no place only for the dodecahedron, and Plato suggested the existence of another, fifth entity - the ether, whose atoms just have the shape of a dodecahedron. Plato's disciples continued his work in the study of the listed bodies. Therefore, these polyhedrons are called Platonic solids. They were studied by scientists, jewelers, priests, architects. These polyhedra were even credited with magical properties. The ancient Greek scientist and philosopher Plato (IV-V century BC) believed that these bodies personify the essence of nature. In his dialogue "Timaeus" Plato says that the atom of fire has the form of a tetrahedron, earth - a hexahedron (cube), air - an octahedron, water - an icosahedron. In this correspondence, there was no place only for the dodecahedron, and Plato suggested the existence of another, fifth entity - the ether, whose atoms just have the shape of a dodecahedron. Plato's disciples continued his work in the study of the listed bodies. Therefore, these polyhedrons are called Platonic solids.

Plato circa 429 - 347 BC Platonic solids are called regular homogeneous convex polyhedra, that is, convex polyhedra, all of whose faces and angles are equal, and the faces are regular polygons. Platonic solids are a three-dimensional analogue of flat regular polygons. However, there is an important difference between the two-dimensional and three-dimensional cases: there are infinitely many different regular polygons, but only five different regular polyhedra. The proof of this fact has been known for over two thousand years; with this proof and study of five regular bodies, Euclid's "Beginnings" are completed.

“Beginnings of Euclid. "... in science there is no king's way" about 365 - 300 years. BC. The main work of Euclid - "Beginnings" (in the original "Stoheia". "Beginnings" consist of 13 books, later 2 more were added to them. The first six books are devoted to planimetry. Books VII - X contain the theory of numbers, XI, XII and XIII books "Beginnings" are devoted to stereometry. From the postulates of Euclid it is clear that he represented space as empty, limitless, isotropic and three-dimensional. It is interesting that the "Beginnings" of Euclid open with a description of the construction of a regular triangle and end with the study of five regular polyhedral bodies! Nowadays they are known as platonic solids.

Archimedes of Syracuse circa 287-212 BC. The mathematician, physicist and engineer Archimedes of Syracuse left behind many inventions, thirteen works (such as "On the Sphere and the Cylinder", "Measurement of the Circle", "Equilibrium of Planes", "Stomachion", "Regular Heptagon and others). Archimedes, as a geometer determined the surface of the sphere and its volume, investigated paraboloids and hyperboloids, studied the "Archimedean spiral", defined the number "pi" as being between 3.141 and 3.142. Archimedes' contribution to the theory of polyhedra - description of 13 semiregular convex homogeneous polyhedra (Archimedean solids).

Archimedean solids Many Archimedean solids can be divided into several groups. The first of them will consist of five polyhedra, which are obtained from Platonic solids as a result of their truncation. Thus, five Archimedean bodies can be obtained: a truncated tetrahedron, a truncated hexahedron (cube), a truncated octahedron, a truncated dodecahedron, and a truncated icosahedron. Another group consists of only two bodies, also called quasi-regular polyhedra. These two bodies are named: cuboctahedron and icosidodecahedron, in contrast to the large rhombocuboctahedron and the large rhomboicosidodecahedron. The next two polyhedrons are called rhombocuboctahedron and rhomboicosidodecahedron. Sometimes they are also called "small rhombocubooctahedron" and "small rhomboicosidodecahedron" in contrast to the large rhombocubooctahedron and large rhomboicosidodecahedron. Finally, there are two so-called "snub-nosed" modifications, one for the cube, the other for the dodecahedron. Each of them is characterized by a slightly rotated position of the faces, which makes it possible to build two different versions of the same "snub-nosed" polyhedron (each of them is like a mirror image of the other).

Johannes Kepler 1571 - 1630 German astronomer and mathematician. One of the founders of modern astronomy. German astronomer and mathematician. One of the founders of modern astronomy. Kepler's contribution to the theory of the polyhedron is, first, the restoration of the mathematical content of the lost treatise of Archimedes on semiregular convex homogeneous polyhedra. Kepler's contribution to the theory of the polyhedron is, first, the restoration of the mathematical content of the lost treatise of Archimedes on semiregular convex homogeneous polyhedra. Even more significant was Kepler's proposal to consider non-convex polyhedrons with stellate faces similar to a pentagram and the subsequent discovery of two regular non-convex homogeneous polyhedrons - the small stellated dodecahedron and the large stellated dodecahedron. Even more significant was Kepler's proposal to consider non-convex polyhedra with stellate faces similar to a pentagram and the subsequent discovery of two regular non-convex homogeneous polyhedrons - the small stellated dodecahedron and the large stellated dodecahedron.

Kepler's cosmological hypothesis Kepler tried to connect some properties of the solar system with the properties of regular polyhedra. He suggested that the distances between the six then known planets are expressed in terms of the sizes of five regular convex polyhedrons (Platonic solids). Between each pair of "celestial spheres" along which, according to this hypothesis, the planets revolve, Kepler inscribed one of the Platonic solids. An octahedron is described around the sphere of Mercury, the planet closest to the Sun. This octahedron is inscribed in the sphere of Venus, around which the icosahedron is described. The sphere of the Earth is described around the icosahedron, and around this sphere is the dodecahedron. The dodecahedron is inscribed in the sphere of Mars, around which the tetrahedron is described. The sphere of Jupiter is described around the tetrahedron, inscribed in a cube. Finally, the sphere of Saturn is described around the cube.

Tetrahedron Tetrahedron (tetra - four, hedra - face). A regular tetrahedron is a regular tetrahedron, that is, a tetrahedron with equal edges, is a regular polyhedron, all faces of which are regular triangles and from each vertex of which exactly three edges emerge. Tetrahedron (tetra is four, hedra is a face). A regular tetrahedron is a regular tetrahedron, that is, a tetrahedron with equal edges, is a regular polyhedron, all faces of which are regular triangles and from each vertex of which exactly three edges emerge It has 4 vertices, 4 faces, 6 edges It has 4 vertices, 4 faces , 6 edges Sum of plane angles at each vertex is 180 degrees Sum of plane angles at each vertex is 180 degrees

Icosahedron (consists of 20 triangles) (consists of 20 triangles) At each vertex of the icosahedron At each vertex of the icosahedron, five faces converge. five faces converge. There is a regular polyhedron in which all faces are regular triangles, and 5 edges come out of each vertex. This polyhedron has 20 faces, 30 edges, 12 vertices and is called an icosahedron (icosi - twenty). There is a regular polyhedron in which all faces are regular triangles, and 5 edges come out of each vertex. This polyhedron has 20 faces, 30 edges, 12 vertices and is called an icosahedron (icosi - twenty). The sum of the plane angles at each vertex is 300 degrees The sum of the plane angles at each vertex is 300 degrees

Dodecahedron There is a regular polyhedron with all faces of regular pentagons and 3 edges coming out of each vertex. This polyhedron has 12 faces, 30 edges and 20 vertices and is called a dodecahedron (dodeka - twelve). There is a regular polyhedron with all faces of regular pentagons and 3 edges coming out of each vertex. This polyhedron has 12 faces, 30 edges and 20 vertices and is called a dodecahedron (dodeka - twelve). The sum of the flat angles at each vertex is 324 degrees The sum of the flat angles at each vertex is 324 degrees

Hexahedron (cube) Hexahedron (cube, hexa - six). A hexahedron is a regular polyhedron, all faces of which are squares, and three edges emerge from each vertex. Hexahedron (cube, hexa - six). A hexahedron is a regular polyhedron, all faces of which are squares, and three edges emerge from each vertex. It has 6 faces, 8 vertices, 12 edges It has 6 faces, 8 vertices, 12 edges The sum of flat angles at each vertex is 270 degrees The sum of flat angles at each vertex is 270 degrees

Octahedron Octahedron. This is a regular polyhedron, all faces of which are regular triangles and four faces of the Octahedron are adjacent to each vertex. This is a regular polyhedron, all the faces of which are regular triangles and four faces adjoin each vertex It has 8 faces, 12 edges, 6 vertices It has 8 faces, 12 edges, 6 vertices

Characteristics of polyhedra. Name: Number of edges at a vertex Number of sides of a face Number of faces Number of edges Number of vertices Tetrahedron33464 Cube Octahedron Dodecahedron Icosahedron

Semiregular polyhedra Snub-nosed cube. This polyhedron can be inscribed into the cube in such a way that the planes of its six square faces coincide with the planes of the cube faces, and these square faces of the snub-nosed cube will turn out to be, as it were, slightly rotated with respect to the corresponding cube faces. Snub-nosed cube. This polyhedron can be inscribed into the cube in such a way that the planes of its six square faces coincide with the planes of the cube faces, and these square faces of the snub-nosed cube will turn out to be, as it were, slightly rotated with respect to the corresponding cube faces. Rhomboicosidodecahedron. This model is one of the most attractive among all other models of Archimedean bodies. The faces are triangles, squares, and pentagons. Rhomboicosidodecahedron. This model is one of the most attractive among all other models of Archimedean bodies. The faces are triangles, squares, and pentagons. Rhombic truncated cuboctahedron. This polyhedron, also known as the truncated cuboctahedron, has squares, hexagons, and octagons on its faces. Rhombic truncated cuboctahedron. This polyhedron, also known as the truncated cuboctahedron, has squares, hexagons, and octagons on its faces. The snub-nosed dodecahedron is the last of a family of convex homogeneous polyhedrons. The faces are triangles and pentagons. The snub-nosed dodecahedron is the last of a family of convex homogeneous polyhedrons. The faces are triangles and pentagons.

Rhombododecahedron. (pro-ruled bodies) It is formed with the help of seven cubes forming a spatial "cross" and a dodecahedron.

Being in nature In crystalline bodies, particles are arranged in a strict order, forming spatial periodically repeating structures throughout the body. For a visual representation of such structures, spatial crystal lattices are used, in the nodes of which the centers of atoms or molecules of a given substance are located. Most often, the crystal lattice is built from ions (positively and negatively charged) atoms that are part of the molecule of a given substance. For example, the lattice of table salt contains Na + and Cl– ions, which are not combined in pairs to form NaCl molecules. Such crystals are called ionic. In crystalline bodies, particles are arranged in a strict order, forming spatial periodically repeating structures throughout the body. For a visual representation of such structures, spatial crystal lattices are used, in the nodes of which the centers of atoms or molecules of a given substance are located. Most often, the crystal lattice is built from ions (positively and negatively charged) atoms that are part of the molecule of a given substance. For example, the lattice of table salt contains Na + and Cl– ions, which are not combined in pairs to form NaCl molecules. Such crystals are called ionic.

Crystals The crystal lattices of metals are often in the form of a hexahedral prism (zinc, magnesium), a face-centered cube (copper, gold) or a body-centered cube (iron). Crystalline lattices of metals are often in the form of a hexagonal prism (zinc, magnesium), a face-centered cube (copper, gold), or a body-centered cube (iron). Crystalline bodies can be monocrystals and polycrystals. Polycrystalline bodies consist of many chaotically oriented small crystals fused together, which are called crystallites. Large single crystals are rarely found in nature and technology. Most often, crystalline solids, including those that are obtained artificially, are polycrystals. Crystalline bodies can be monocrystals and polycrystals. Polycrystalline bodies consist of many chaotically oriented small crystals fused together, which are called crystallites. Large single crystals are rarely found in nature and technology. Most often crystalline solids, including those that are obtained artificially, are polycrystals. Simple crystal lattices: 1 - simple cubic lattice; 2 - face-centered cubic lattice; 3 - body-centered cubic lattice; 4 - hexagonal lattice.

Crystals - polyhedrons Calcium. When struck, calcite crystals break into regular figures, each face of which has the shape of a parallelogram. Calcium forms a variety of crystals from plastic to elongated-prismatic shape. Calcium. When struck, calcite crystals break into regular figures, each face of which has the shape of a parallelogram. Calcium forms a variety of crystals from plastic to elongated-prismatic shape. Apatite. They form crystals in the shape of a rectangular prism. Apatite. They form crystals in the shape of a rectangular prism. Beryllium. Usually occurs as columnar hexagonal crystals. Beryllium. Commonly found as columnar hexagonal crystals.

The history of regular polyhedra goes back to ancient times. Starting from the 7th century BC, philosophical schools were created in Ancient Greece, in which a gradual transition from practical to philosophical geometry took place. Of great importance in these schools are the reasoning with the help of which it was possible to obtain new geometric properties. Historical background One of the first and most famous schools was Pythagorean, named after its founder Pythagoras. A distinctive sign of the Pythagoreans was the pentagram, in the language of mathematics, it is a regular non-convex or star-shaped pentagon. The pentagram was assigned the ability to protect a person from evil spirits.

Earth earth hexahedron hexahedron (cube) (cube) universe universe Dodecahedron The Pythagoreans and later Plato believed that matter consists of four basic elements: fire, earth, air and water. They attributed the existence of five regular polyhedra to the structure of matter and the Universe. According to this opinion, the atoms of the main elements should have the form of various Platonic solids:

Artists About Regular Polyhedra During the Renaissance, sculptors, architects, and ARTISTS showed great interest in the forms of regular polyhedra. Leonardo da Vinci was fond of the theory of polyhedra and often depicted them in his canvases. He illustrated the book of his friend, monk Luca Pacioli "On Divine Proportion" with images of regular and semi-regular polyhedra. During the Renaissance, sculptors, architects, and ARTISTS showed great interest in the forms of regular polyhedra. Leonardo da Vinci was fond of the theory of polyhedra and often depicted them in his canvases. He illustrated the book of his friend, monk Luca Pacioli "On Divine Proportion" with images of regular and semi-regular polyhedra

In the painting "The Last Supper" by the artist Salvador Dali, Christ and his disciples are depicted against the background of a huge transparent dodecahedron. The form of a dodecahedron, according to the ancients, had the UNIVERSE, i.e. they believed that we live inside a vault, which has the shape of the surface of a regular dodecahedron.

Egyptian Pyramids Among egyptian pyramids a special place is occupied by the pyramid of Pharaoh Cheops. The length of the side of its base L \u003d 233.16 m; height H \u003d 146.6; 148.2 m. Initially, the height was not accurately estimated. This is due to the settlement of the seams, the deformation of the blocks, the supposed partial dismantling of the summit from S 66 to 1010 m. Among the Egyptian pyramids, the pyramid of Pharaoh Cheops occupies a special place. The length of the side of its base L \u003d 233.16 m; height H \u003d 146.6; 148.2 m. Initially, the height was not accurately estimated. This is due to the settlement of the seams, the deformation of the blocks, the assumed partial dismantling of the summit from S 66 to 1010 m.

Slope angle \u003d 5151. It was first measured by the English Colonel G. Vaizov in 1837 tg \u003d 1.27306 \u003d vd \u003d 1, The angle of inclination of the edges \u003d 5151. It was first measured by the English colonel G. Vaisov in 1837 tg \u003d 1.27306 \u003d vd \u003d 1.27202.

The Royal Tomb The Great Pyramid was built as the tomb of Khufu, known to the Greeks as Cheops. He was one of the pharaohs, or kings of ancient Egypt, and his tomb was completed in 2580 BC. Later, two more pyramids were built in Giza, for the son and grandson of Khufu, as well as smaller pyramids for their queens. The pyramid of Khufu, the farthest in the figure, is the largest. His son's pyramid is in the middle and looks higher because it stands on a higher place.

In the III century BC. a lighthouse was built so that ships could safely pass the reefs on their way to the bay of Alexandria. At night they were helped in this by the reflection of the flames, and during the day - by a column of smoke. It was the first lighthouse in the world, and it stood for 1500 years. The Pharos lighthouse consisted of three marble towers, standing on a base of massive stone blocks. The first tower was rectangular, it contained rooms in which workers and soldiers lived. Above this tower was a smaller, octagonal tower with a spiral ramp leading up to the upper tower. The upper tower was shaped like a cylinder, in which a fire was burning, which helped the ships safely reach the bay. At the top of the tower stood a statue of Zeus the Savior. The total height of the lighthouse was 117 meters. Alexandrian lighthouse

The simplest animal The skeleton of the unicellular organism of Feodaria (Circogonia icosahedra) resembles an icosahedron in shape. The skeleton of the unicellular organism of Feodaria (Circogonia icosahedra) resembles an icosahedron in shape. Most of the feudariums live in the deep sea and serve as prey for coral fish. But the simplest animal protects itself with twelve needles extending from the 12 vertices of the skeleton. It looks more like a stellated polyhedron. Most of the feudariums live in the deep sea and serve as prey for coral fish. But the simplest animal protects itself with twelve needles extending from the 12 vertices of the skeleton. It looks more like a stellated polyhedron. Of all polyhedrons with the same number of faces, the icosahedron has the largest volume with the smallest surface area. This property helps the marine organism to overcome the pressure of the water column.

Interestingly, the icosahedron has become the focus of biologists' debate over the shape of viruses. The icosahedron has become the focus of biologists' debate over the shape of viruses. The virus cannot be perfectly round as previously thought. To establish its shape, they took various polyhedrons, directed light at them at the same angles as the stream of atoms onto the virus. It turned out that only one polyhedron gives exactly the same shadow - the icosahedron.

Performed by a student of group G 2-9 N.Yu. Koblyuk

Head E.V. Morozova

Tula 2010

"Mathematics possesses not only the truth, but also the highest beauty - a beauty that is perfected and strict, sublimely pure and striving for true perfection, which is characteristic only of the greatest examples of art."

Bertrand Russell

The polyhedron is called correct , if a:

• It is convex.
• All its faces are equal regular polygons.
• The same number of faces converges at each of its vertices.
• All its dihedral angles are equal.

There are only five regular polyhedra :

• Tetrahedron (tetrahedron)
• Cube (hexagon)
• Octahedron (octahedron)
• Dodecahedron (dodecahedron)
• Icosahedron (twenty-sided)

Regular polyhedron is a convex polyhedron with the greatest possible symmetry.

Since ancient times, our ideas about beauty have been associated with symmetry. Perhaps this explains the human interest in polyhedrons - amazing symbols of symmetry that attracted the attention of prominent thinkers.

The history of regular polyhedra goes back to ancient times. Pythagoras and his students were engaged in the study of regular polyhedra. They were amazed by the beauty, perfection, harmony of these figures. The Pythagoreans considered regular polyhedra to be divine figures and used them in their philosophical writings.

One of the oldest references to regular polyhedra is found in the treatise of Plato (427-347 BC) "Timus".

Therefore, regular polyhedra are also called Platonic solids. Each of the regular polyhedrons, and there are five of them in total, Plato associated with four "earthly" elements: earth (cube), water (icosahedron), fire (tetrahedron), air (octahedron), as well as with the "unearthly" element - the sky (dodecahedron ).

By the time of Plato in ancient philosophy, the concept of four elements (elements) matured - the fundamental principles of the material world: fire , air , water and land .

The shape of the cube is the atoms of the earth, because both the earth and the cube are immobile and stable.

The shape of the icosahedron is water atoms, because water is distinguished by its fluidity, and of all regular bodies, the icosahedron is the most "rolling" one.

The shape of the octahedron is air atoms, because the air moves back and forth, and the octahedron seems to be directed simultaneously in different directions.

The shape of the tetrahedron is the atoms of fire, because the tetrahedron is the sharpest, it seems that it rushes in different directions.

Plato introduces the fifth element - "the fifth essence" - the world ether, the atoms of which are given the form of a dodecahedron as the closest to the ball.

Platonic solids are called regular homogeneous convex polyhedra, that is, convex polyhedra, all of whose faces and angles are equal, and the faces are regular polygons.

Platonic solids are a three-dimensional analogue of flat regular polygons. However, there is an important difference between the two-dimensional and three-dimensional cases: there are infinitely many different regular polygons, but only five different regular polyhedra.

about 429 - 347 BC

convex polyhedron whose faces are regular

polygons with the same number of sides and in each

whose vertex has the same number of edges.

Icosahedron

Tetrahedron

Octahedron

Hexahedron

Dodecahedron

Plato's body

Face geometry

Number

Tetrahedron

Icosahedron

Hexahedron

Dodecahedron

Euler's formula G + B - P \u003d 2

The surface of a tetrahedron consists of four equilateral triangles, three converging at each vertex.

Have regular tetrahedron all faces are equilateral triangles, all dihedral angles at edges and all trihedral angles at vertices are equal.

Tetrahedron properties :

• An octahedron can be inscribed into a tetrahedron, moreover, four (out of eight) faces of the octahedron will be aligned with the four faces of the tetrahedron, all six vertices of the octahedron will be aligned with the centers of the six edges of the tetrahedron.
• A tetrahedron with an edge x consists of one inscribed octahedron (in the center) with an edge x / 2 and four tetrahedra (along the vertices) with an edge x / 2.
• The tetrahedron can be inscribed in the cube in two ways, with the four vertices of the tetrahedron being aligned with the four vertices of the cube.

All six edges of the tetrahedron will lie on all six faces of the cube and are equal to the diagonal of the face-square.

• The tetrahedron can be inscribed in the icosahedron, moreover, the four vertices of the tetrahedron will be aligned with the four vertices of the icosahedron.

Regular polyhedron

Regular triangle

Faces at the top

Rib length

Surface area

Symmetry elements:

The tetrahedron has no center of symmetry,

but has 3 axes of symmetry and 6 planes of symmetry

Radius of the circumscribed sphere:

Inscribed sphere radius:

Surface area:

Tetrahedron volume:

Cube or hexahedron is a regular polyhedron, each face of which is a square. A special case of a parallelepiped and a prism. The cube has six square faces, three converging at each vertex.

Cube properties :

• You can inscribe a tetrahedron into a cube in two ways, moreover, the four vertices of the tetrahedron will be aligned with the four vertices of the cube. All six edges of the tetrahedron will lie on all six faces of the cube and are equal to the diagonal of the face-square.
• The four sections of the cube are regular hexagons - these sections pass through the center of the cube perpendicular to its four diagonals.
• An octahedron can be inscribed into a cube, moreover, all six vertices of the octahedron will be aligned with the centers of the six faces of the cube.
• The cube can be inscribed in an octahedron, moreover, all eight vertices of the cube will be located in the centers of the eight faces of the octahedron.
• An icosahedron can be inscribed into a cube, while the six mutually parallel edges of the icosahedron will be located respectively on the six faces of the cube, the remaining 24 edges are inside the cube, all twelve vertices of the icosahedron will lie on the six faces of the cube.

Regular polyhedron

Faces at the top

Rib length

Surface area

Symmetry elements:

The cube has a center of symmetry - the center of the cube, 9 axes

symmetry and 9 planes of symmetry .

Radius of the circumscribed sphere:

Inscribed sphere radius:

Cube surface area:

Cube volume:

S \u003d 6 a 2

V \u003d a 3

Octahedron is one of five regular polyhedra.

The octahedron has 8 faces (triangular),

12 edges, 6 vertices (4 edges converge at each vertex).

The octahedron has eight triangular faces, four converging at each vertex .

Octahedron properties :

• An octahedron can be inscribed in a tetrahedron, moreover, four (out of eight) faces of the octahedron will be aligned with the four faces of the tetrahedron, all six vertices of the octahedron will be aligned with the centers of the six edges of the tetrahedron.
• An octahedron with an edge y consists of 6 octahedra (along the vertices) with an edge y: 2 and 8 tetrahedra (along the edges) with an edge y: 2
• An octahedron can be inscribed into a cube, and all six vertices of the octahedron will be aligned with the centers of the six faces of the cube.
• You can inscribe a cube into the octahedron, and all eight vertices of the cube will be located at the centers of the eight faces of the octahedron.

Regular polyhedron

triangle

Faces at the top

The dual polytope

Symmetry elements:

The octahedron has a center of symmetry - the center of the octahedron, 9 axes of symmetry and 9 planes of symmetry.

Radius of the circumscribed sphere:

Inscribed sphere radius:

Surface area:

Octahedron volume:

Icosahedron - a regular convex polyhedron, a 20-sided polyhedron, one of the Platonic solids. Each of the 20 faces is an equilateral triangle. The number of edges is 30, the number of vertices is 12. The surface of the icosahedron consists of twenty equilateral triangles, converging at each vertex by five.

Properties :

• An icosahedron can be inscribed into a cube, while six mutually parallel edges of the icosahedron will be located respectively on six faces of the cube, the remaining 24 edges are inside the cube, all twelve vertices of the icosahedron will lie on six faces of the cube
• A tetrahedron can be inscribed in an icosahedron, moreover, the four vertices of the tetrahedron will be aligned with the four vertices of the icosahedron.
• The icosahedron can be inscribed in the dodecahedron, moreover, the vertices of the icosahedron will be aligned with the centers of the faces of the dodecahedron.
• A dodecahedron can be inscribed into an icosahedron, moreover, the vertices of the dodecahedron will be aligned with the centers of the faces of the icosahedron.

Regular polyhedron

Regular triangle

Faces at the top

The dual polytope

dodecahedron

Symmetry elements:

The icosahedron has a center of symmetry - the center of the icosahedron, 15 axes of symmetry and 15 planes of symmetry.

Radius of the circumscribed sphere:

Inscribed sphere radius:

Surface area:

Icosahedron volume:

Dodecahedron (dodecahedron) - a regular polyhedron, a three-dimensional geometric figure made up of twelve regular pentagons. Each vertex of the dodecahedron is the vertex of three regular pentagons. It has twelve pentagonal faces, converging at the vertices of three.

Thus, the dodecahedron has 12 faces (pentagonal), 30 edges and 20 vertices (3 edges converge in each. The sum of the plane angles at each of the 20 vertices is 324 °.

The dodecahedron is used as a random number generator (along with other dice) in tabletop role-playing games.

Regular polyhedron

Regular pentagon

Faces at the top

The dual polytope

icosahedron

Symmetry elements:

The dodecahedron has a center of symmetry - the center of the dodecahedron, 15 axes of symmetry and 15 planes of symmetry.

Radius of the circumscribed sphere:

Inscribed sphere radius:

Surface area:

Dodecahedron volume:

Regular polyhedra are found in nature. For example, the skeleton of a single-celled organism of a feudaria ( Circjgjnia icosahtdra ) resembles an icosahedron in shape.

What caused such a natural geometrization of the feudariums? Apparently, because of all 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 to overcome the pressure of the water column.

Regular polyhedra are the most "advantageous" shapes. And nature makes wide use of this. This is confirmed by the shape of some crystals.

Take, for example, table salt, which we cannot do without. It is known that it is soluble in water and serves as a conductor of electric current. And the crystals of table salt ( NaCl ) have the shape of a cube.

Aluminum-potassium quartz is used in the production of aluminum ( K [ Al ( SO 4 ) 2 ]  12 H 2 O ), the single crystal of which has the shape of a regular octahedron.

Obtaining sulfuric acid, iron, special grades of cement is not complete without pyrite ( FeS ). Crystals of this chemical have the shape of a dodecahedron.

Antimony sodium sulfate is used in various chemical reactions. ( Na 5 ( SbO 4 ( SO 4 )) - a substance synthesized by scientists. The crystal of antimony sodium sulfate has the shape of a tetrahedron.

The last regular polyhedron - the icosahedron gives the shape of boron crystals (IN). At one time, boron was used to create the first generation of semiconductors.

Feodaria

( Circjgjnia icosahtdra )

"There are defiantly few regular polyhedra, but this very modest detachment managed to get into the very depths of various sciences."

L. Carroll

Materials used:

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Used

A polyhedron is a surface made up of polygons and bounding some geometric body. Polyhedra are convex and non-convex polygons A polyhedron is called convex if it is located on one side of the plane of each polygon on its surface

Octahedron Octahedron (Greek οκτάεδρον, from Greek οκτώ, "eight" and Greek οδρα "base") is one of five convex regular polyhedrons, the so-called Platonic solids. of regular polyhedra Platonic Octahedron has 8 triangular faces, 12 edges, 6 vertices, 4 edges converge at each vertex.

Icosahedron Icosahedron (from the Greek εικοσάς twenty; -εδρον face, face, base) is a regular convex polyhedron, a twenty-sided one, one of the Platonic solids. Each of the 20 faces is an equilateral triangle. The number of edges is 30, the number of vertices is 12. The icosahedron has 59 star-shaped shapes.

Dodecahedron The dodecahedron (from the Greek δώδεκα twelve and εδρον face), a dodecahedron is a regular polyhedron, composed of twelve regular pentagons. Each vertex of a dodecahedron is a vertex of three regular pentagons. Greek regular polyhedron of regular pentagons is a vertex Thus, a dodecahedron has 12 faces (pentagonal), 30 edges and 20 vertices (3 edges converge in each). The sum of the flat angles at each of the 20 vertices is 324 °.

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A polyhedron is a body whose surface consists of a finite number of flat polygons.

Regular polyhedra

How many regular polyhedra are there? - How are they determined, what properties they have? -Where are they found, do they have a practical application?

A convex polyhedron is called regular if all of its faces are equal regular polygons and the same number of edges converge at each of its vertices.

"Hedra" - facet "tetra" - four hexes "- six" octas "- eight" dodeca "- twelve" icos "- twenty The names of these polyhedra came from Ancient Greece and they indicate the number of faces.

Name of a regular polyhedron Face type Number of vertices of edges of faces of faces converging at one vertex Tetrahedron Regular triangle 4 6 4 3 Octahedron Regular triangle 6 12 8 4 Icosahedron Regular triangle 12 30 20 5 Cube (hexahedron) Square 8 12 6 3 Dodecahedron Regular pentagon 20 30 12 3 Data on regular polyhedra

Question (problem): How many regular polyhedra are there? How do you establish their number?

α n \u003d (180 ° (n -2)): n At each vertex of the polyhedron there are at least three plane angles, and their sum must be less than 360 °. Shape of faces Number of faces at one vertex Sum of plane angles at the vertex of a polyhedron Conclusion about the existence of a polyhedron α \u003d 3 α \u003d 4 α \u003d 5 α \u003d 6 α \u003d 3 α \u003d 4 α \u003d 3 α \u003d 4 α \u003d 3

L. Carroll

The great mathematicians of antiquity Archimedes Euclid Pythagoras

The ancient Greek scientist Plato described in detail the properties of regular polyhedra. That is why regular polyhedra are called Plato's bodies

tetrahedron - fire cube - earth octahedron - air icosahedron - water dodecahedron - universe

Polyhedrons in space and earth sciences

Johannes Kepler (1571-1630) - German astronomer and mathematician. One of the founders of modern astronomy - discovered the laws of planetary motion (Kepler's laws)

kepler Cup Space

"Ecosahedron - dodecahedron structure of the Earth"

Polyhedrons in art and architecture

Albrecht Durer (1471-1528) "Melancholy"

Salvador Dali "The Last Supper"

Modern architectural structures in the form of polyhedrons

Alexandrian lighthouse

Brick polyhedron of a Swiss architect

Modern building in England

Polyhedra in nature FEODARIA

Pyrite (sulphide pyrite) Single crystal of potassium alum Crystals of red copper ore NATURAL CRYSTALS

Table salt consists of cube-shaped crystals Mineral sylvin also has crystal lattice in the form of a cube. Water molecules are tetrahedral. The mineral cuprite forms crystals in the form of octahedrons. Pyrite crystals have the shape of a dodecahedron

Diamond In the form of an octahedron, diamond, sodium chloride, fluorite, olivine and other substances crystallize.

Historically, the octahedron was the first cut to appear in the 14th century. Shah diamond Diamond weight 88.7 carats

Problem The Queen of England gave instructions to cut along the edges of the diamond with a gold thread. But the cut was not done, since the jeweler was unable to calculate the maximum length of the gold thread, and the diamond itself was not shown to him. The jeweler was given the following data: the number of vertices B \u003d 54, the number of faces G \u003d 48, the length of the largest edge L \u003d 4mm. Find the maximum length of a golden thread.

Regular polyhedron Number of Faces Vertices Edges Tetrahedron 4 4 6 Cube 6 8 12 Octahedron 8 6 12 Dodecahedron 12 20 30 Icosahedron 20 12 30 Research "Euler's formula"

Euler's theorem. For any convex polyhedron B + G - 2 \u003d P where B is the number of vertices, G is the number of faces, P is the number of edges of this polyhedron.

FIZMINUTKA!

Problem Find the angle between two edges of a regular octahedron that have a common vertex but do not belong to the same face.

Task Find the height of a regular tetrahedron with an edge of 12 cm.

The crystal has the shape of an octahedron, consisting of two correct pyramids from common ground, the edge of the pyramid base is 6 cm. octahedron height is 8 cm. Find the area of \u200b\u200bthe lateral surface of the crystal

Surface area Tetrahedron Icosahedron Dodecahedron Hexahedron Octahedron

Assignment at home: mnogogranniki.ru Using the reamers, make models of the 1st regular polyhedron with a side of 15 cm, the 1st semi-regular polyhedron

Thanks for your work!