Harmony of regular polyhedra. Theoretical material

The correct many-grand-ni-ki in-te-re-so-va-whether there are many great scientists. And this in-te-res you-ho-dyl da-le-ko for pre-de-la ma-te-ma-ti-ki. Plato (427 BC - 347 BC) considered them as the basis of the structure of the All-Lena, Kepler (1571-1630) py - began to connect the correct many-grand-ni-ki with the movement of the planets of the Solar system (which-ryh in his time it was known to be five). Perhaps, it is precisely the beauty and the gar-mony of the correct many-grand-niks for -good-go-to-lay-gat something more deeper in their knowledge than just geo-metric objects Comrade

The right many-face-no-one for-zy-va-there is a lot of-face, all the face-no-one-that-ro-th are the right-handed many go-coal-ni-ki, all flat corners of ko-to-ro-go are equal between themselves and two-sided corners of ko-to-ro-go are equal between themselves. (Plos-ki-mi ug-la-mi a lot-gran-ni-ka-zy-va-yut-Xia angles many-coal-ni-kov-gran-nei, two-facet us-mi-la-mi a lot of-grand-ni-ka na-zy-va-yut-Xia angles between the gra-ny-mi, having a common reb- ro.)

Note that from this definition of the av-to-ma-ti-che-ski it follows you-bulging of the right granny, some paradise in some books is included in the definition.

In a three-dimensional space, there are exactly five correct many-facets: tet-ra-edr, ok-ta-edr, cube (gek-sa-edr), iko-sa-edr, do-de-ka-edr. The fact that there are many other right-handed many-grand-no-covs did not exist, was-lo-ka-za-but Ev-kli-dom (about 300 g BC) in his great Na-cha-lah.

An analogous construction is used in a more general case. Ras-smot-rim pro-out-of-free you-bunch-a-lot-granny and take-points in the se-re-di-nah of his gran-ny. Co-one-nim between oneself the points of the neighboring faces from-cut-ka-mi. Then the points-ki appear-la-are-sya ver-shi-na-mi, from-cut-ki - rib-ra-mi, and a lot of coal-ni-ki, some ogre -ni-chi-va-yut these cuts, gran-ny-mi one more vy-bunch-lo-th-many-gran-no-ka. This many-sided nickname is na-zy-va-is-Xia dual-ny-ny-ny-ny-ny-ny-ny-ny-mu.

As if by-ka-za-but higher, dual to tet-ra-ed-yav-la-et-sya tet-ra-edr.

Uwe-li-chim size-mer tet-ra-ed-ra, ver-shi-na-mi ko-to-ro-go yav-la-yut-sya-re-di-us gran-nei ex-move -th tet-ra-ed-ra, to the size of the next. Eight vertices of so ras-lo-feminine tet-ra-ed-dov are ver-shi-na-mi ku-ba.

Pe-re-se-che-ni these tet-ra-ed-dov yav-la-et-sya one more right-handed many-grand-nickname - ok-ta-edr (from the Greek. οκτώ - eight-seven). Ok-ta-hedron has 8 triangular faces, 6 vertices, 12 edges. The flat corners of the ok-ta-ed-ra are equal to $ \ pi / 3 $, since its edges are right-wicked triangles no-ka-mi, dihedral angles are equal to $ \ arccos (–1/3) ≈ 109 (,) 47 ^ \ circ $.

From-me-tim se-re-di-us gran-nei ok-ta-ed-ra and pe-re-dyom to dual-n-mo to ok-ta-ed-ru many-gran- no-ku. This is a cube or gek-sa-hedr (from the Greek εξά - six). At ku-ba grani yav-la-yut-sya kvad-ra-ta-mi. It has 6 edges, 8 vertices, 12 edges. Flat ku-ba angles are equal to $ \ pi / 2 $, dihedral angles are also equal to $ \ pi / 2 $.

If you take points on the se-re-di-nah gran-ney ku-ba and look at the multi-facet dual to it, then you can convince Xia, that they will again-va-det ok-ta-edr. It is also true that a more general statement is made: if, for you-a-fart, there are many-sided-nik-ka-to-build-it-duality, and the dual-ness to the dual-ness, then they will be the initial multi-facet (with the accuracy up to the do-biy).

Take-me on the edges ok-ta-ed-ra to the point, so that each de-li-la rib-ro in the co-relation is $ 1 : (\ sqrt5 + 1) / 2 $ (golden cross-section) and at the same time the points that come over one face, appeared ver-shi-na-mi right-vil-no-go tri-coal-no-ka. By-receive-chen-nye 12 points are-a-la-yut-sya ver-shi-na-mi one more right-vil-no-th-many-grand-ni-ka - iko- sa-ed-ra (from the Greek είκοσι - two-twenty). Iko-sa-edr is a right-sided multi-facet, which has 20 triangular faces. It has 12 vertices, 30 ribs. Flat angles of iko-sa-ed-ra are equal to $ \ pi / 3 $, dihedral ones are equal to $ \ arccos (–1/3 \ cdot \ sqrt5) ≈ 138 (,) 19 ^ \ circ $.

Iko-sa-edr can be entered into a cube. At the same time, on each side of the ku-ba, there will be two peaks of iko-sa-ed-ra.

In-ver-him iko-sa-edr, “putting-viv” it on top-shi-well, and getting it more accustomed form: two hats from the heel ti tre-coals-nikov at the south-and-north-no-lu-sov and the middle layer, consisting of ten tri-coals no-kov.

Se-re-di-us gran-nei iko-sa-ed-ra yav-la-yut-sya ver-shi-na-mi one more right-vile-th-many-grand- no-ka - do-de-ka-ed-ra (from the Greek δώδεκα - two-at-twenty). Gra-ni-do-de-ka-ed-ra are the right-handed five-coal-ni-ki. Thus, its flat angles are equal to $ 3 \ pi / 5 $. Do-de-ka-ed-ra has 12 grains, 20 vertices, 30 ribs. Do-de-ca-ed-ra dihedral angles are equal to $ \ arccos (–1/5 \ cdot \ sqrt5) ≈116 (,) 57 ^ \ circ $.

Taking se-re-di-us gra-nei do-de-ka-ed-ra, and pe-re-dya to the dual-n-th him a lot of-grand-ni-ku, in- chim again iko-sa-edr. So, iko-sa-edr and do-de-ka-edr are dual-us-gu. This is once again il-lu-stri-ru-u-e the fact that dual to dual to dual will be the original multi-grand nickname.

Note that with the pe-re-ho-de to the dual-n-th, many-grand-ni-ku, the ver-shi-us of the ex-move-but-ni-ni-ni-ku -ni-ka-ot-vet-stvu-gi-nam duality, rib-ra - rib-ram of duality, and grani - ver-shi-us double -that-but-th-many-grand-ni-ka. If iko-sa-ed-ra has 20 grains, it means that he has 20 vertices in his dual-ness, and they have one the whole number of ribs, if the ku-ba has 8 vertices, then the dual-n-th to it ok-ta-ed-ra has 8 faces.

There are different ways of typing the right many-grand-niks into each other, which to many of them-me-cha-tel constructions. In-te-res-ny and beautiful many-grand-ni-ki on-lo-cha-yut-Xia as well with un-unity-not-nii and pe-re-se-che -nies of correct many-grand-nikov.

In do-de-ka-edr we write a cube so that all 8 vertices of ku-ba sov-pa-da-li with ver-shi-na-mi do-de-ka-ed-ra. Around the do-de-ka-ed-ra, we describe the iko-sa-edr so that its tops-shi-ny-be-in-se-re-di-nah gran-nei iko-sa-ed -ra. Around iko-sa-ed-ra, describe ok-ta-edr, so that the vertices of iko-sa-ed-ra le-zha-li on the edges of ok-ta-ed-ra ... At the end, around ok-ta-ed-ra, describe the tet-ra-edr so that the ver-shi-us ok-ta-ed-ra po-pa-li na se-re-di -we ryo-ber tet-ra-ed-ra.

Such a construction from ku-soch-kovs of layered de-vian-ny ski plots was made by another re-byon-com be-blowing ve-li cue ma-te-ma-tik XX century V. I. Ar-nold. Vla-di-mir Igo-re-vich kept her for a long time, and then he gave it to la-bo-ra-to-rya by bullet-ri-zation and pro pa-gan-dy ma-te-ma-ti-ki Ma-te-ma-ti-che-go institute-tu-ta them. V.A. Stek-lo-va.

Literature

G. S. M. Koks-ter. Introduction to geo-metry. - M .: Na-u-ka, 1966.

J. Ada-Mar. Ele-men-tar-naya geo-met-ria. Part 2. Ste-rheo-met-ria. - M .: Pro-sveshchenie, 1951.

Euclid. Na-cha-la Ev-kli-da. Books XXI-XXV. - M.-L .: GITTL, 1950.

Pyramid is a polyhedron with one face - the base of the pyramid - an arbitrary polygon, and the rest - side faces - triangles with a common vertex, called the apex of the pyramid. The perpendicular dropped from the top of the pyramid to its base is called pyramid height. A pyramid is called triangular, quadrangular, etc., if the base of the pyramid is a triangle, quadrangle, etc. A triangular pyramid is a tetrahedron - a tetrahedron. Quadrangular - pentahedron, etc.

PyramidTruncated pyramid

SO - the height of the pyramid OO1 - the height of the pyramid

SF - apothem of the pyramid Ff - apothem of the pyramid

Pyramid properties:

If all side edges are equal , then:

• a circle can be described around the base of the pyramid, and the top of the pyramid is projected into its center;
• lateral ribs form equal angles with the base plane;
• the opposite is also true, that is, if the side edges form equal angles with the base plane, or if a circle can be described near the base of the pyramid, and the top of the pyramid is projected into its center, then all the side edges of the pyramid are equal.

If the side faces are inclined to the plane of the base at the same angle , then:

• a circle can be inscribed at the base of the pyramid, and the top of the pyramid is projected into its center;
• the heights of the side faces are equal.

Tetrahedron - regular polyhedron, has 4 faces, which are regular triangles. The tetrahedron has 4 vertices, 3 edges converge to each vertex, and there are 6 edges in total. Also, a tetrahedron is a pyramid.

Tetrahedron properties:

• Parallel planes passing through pairs of crossing edges of a tetrahedron define a parallelepiped described around the tetrahedron.
• The plane passing through the midpoints of two intersecting edges of the tetrahedron divides it into two parts of equal volume.

Prism is a polyhedron with two faces (prism bases) - equal polygons with correspondingly parallel sides. The rest of the faces are parallelograms, the planes of which are parallel to one straight line.

Oblique prism Straight prism

If the lateral edges of the prism are perpendicular to the plane of the base, then - straight prism... If not - oblique prism. If in a straight prism the base is a regular polygon - correct prism.

Prism properties:

• The bases of the prism are equal polygons.
• The side faces of the prism are parallelograms.
• The side edges of the prism are parallel and equal.
• The perpendicular section is perpendicular to all lateral edges of the prism.
• The perpendicular section angles are the linear angles of the dihedral angles at the corresponding side edges.
• Perpendicular section perpendicular to all side faces

Parallelepiped is a prism whose base is a parallelogram. The parallelepiped has six faces, and they are all parallelograms. Opposite faces are pairwise equal and parallel. The parallelepiped has four diagonals. All the diagonals of the Box intersect at one point and are halved by this point. The base of the box can be any face.

A parallelepiped, the four side faces of which are rectangles, is called straight. A straight parallelepiped with all six faces of a rectangle is called rectangular. A rectangular parallelepiped, all faces of which are squares, is called cube or regular hexahedron ... All edges of a cube are equal.

Box properties:

• The parallelepiped is symmetrical about the middle of its diagonal.
• Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is halved by it; in particular, all the diagonals of the parallelepiped meet at one point and are bisected by it.
• Opposite faces of the box are parallel and equal.
• The square of the length of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Octahedron

The octahedron has 8 triangular faces, 12 edges, 6 vertices, 4 edges converge to each vertex.

Octahedron properties:

• An octahedron can be inscribed in a tetrahedron, moreover, four of the 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.
• The 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 in the octahedron, and all eight vertices of the cube will be located at the centers of the eight faces of the octahedron.
• A regular octahedron has symmetry O h coinciding with the symmetry of the cube.

Dodecahedron composed of twelve regular pentagons, which are its faces. Each vertex of the dodecahedron is the vertex of three regular pentagons. Thus, a dodecahedron has 12 faces (pentagonal), 30 edges and 20 vertices (3 edges converge in each).

The dodecahedron has a center of symmetry and 15 axes of symmetry. Each of the axes passes through the midpoints of opposite parallel ribs. The dodecahedron has 15 planes of symmetry. Any of the planes of symmetry passes in each face through the vertex and midpoint of the opposite edge.

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 icosahedron has 59 stellate shapes.

Geometer. body limited 8 equilateral triangles... Dictionary foreign words included in the Russian language. Pavlenkov F., 1907. OCTAEDR Greek. oktaedros, from okto, eight, and hedra, bottom. Octahedron. Explanation 25000 ... ... Dictionary of foreign words of the Russian language

Polyhedron, octahedron Dictionary of Russian synonyms. octahedron n., number of synonyms: 2 octahedron (2) ... Synonym dictionary

octahedron- a, m. octaèdre m. octaedron. Regular octahedron, body bounded by eight triangles. SIS 1954. In octaedra. Witt Prom. chem. 1848 2 187. Of the crystalline forms of metals, cubes and especially octahedrons predominate. MB 1900 ... ... Historical Dictionary of Russian Gallicisms

- (from the Greek okto eight and hedra seat, plane, face), one of five types of regular polyhedra; has 8 faces (triangular), 12 edges, 6 vertices (4 edges converge in each) ... Modern encyclopedia

- (from the Greek okto eight and hedra facet) one of five types of regular polyhedra; has 8 faces (triangular), 12 edges, 6 vertices (4 edges converge in each) ... Big Encyclopedic Dictionary

OCTAHEDRON, octahedron, husband. (from the Greek okto eight and hedra base). Regular octahedron bounded by eight regular triangles. Ushakov's Explanatory Dictionary. D.N. Ushakov. 1935 1940 ... Ushakov's Explanatory Dictionary

One of the forms of the structural organization of viruses (bacteriophages), the virions of which are a regular polyhedron with 8 faces and 6 vertices. (Source: "Microbiology: glossary of terms", Firsov N.N., M: Bustard, 2006) ... Microbiology Dictionary

- [όχτώ (ξ who) eight; έδρα (γhedra) face] is a closed octahedron with faces in the form of regular triangles. O. symbol (111). See Simple forms of crystals of the highest (cubic) system. ... ... Geological encyclopedia

octahedron- - [English Russian gemological dictionary. Krasnoyarsk, KrasBerry. 2007.] Topics gemology and jewelry production EN octahedron ... Technical translator's guide

Octahedron- (from the Greek okto eight and hedra seat, plane, face), one of five types of regular polyhedra; has 8 faces (triangular), 12 edges, 6 vertices (4 edges converge in each). ... Illustrated encyclopedic Dictionary

Books

• Magic facets number 15. Star octahedron. Star polyhedron,. A set for assembling a polyhedron "Stellated octahedron". Dimensions of the finished polyhedron assembled from the set: 170x180x200 mm. Difficulty level - "Start" (does not require experience and additional ...
• Magic facets number 21. Archimedean bodies. Part 2 , . Special issue. Archimedean bodies. Two polyhedra in one set. Sets for assembling polyhedra: 171; Archimedean solids: rhombo-cubo-octahedron, rhombo-truncated cubo-octahedron 187;. ...

A regular octahedron has 8 triangular faces, 12 edges, 6 vertices, and 4 edges converge at each vertex.

Dimensions (edit)

If the edge length of the octahedron is a, then the radius of the sphere described around the octahedron is:

r u = a 2 2 ≈ 0.7071067 ⋅ a (\ displaystyle r_ (u) = (\ frac (a) (2)) (\ sqrt (2)) \ approx 0.7071067 \ cdot a),

the radius of the sphere inscribed in the octahedron can be calculated by the formula:

r i = a 6 6 ≈ 0.4082482 ⋅ a. (\ displaystyle r_ (i) = (\ frac (a) (6)) (\ sqrt (6)) \ approx 0.4082482 \ cdot a.) Orthographic projections

Centered	Rib	Normal to the brink	Pinnacle	By the edge
Image
Projective symmetry

Spherical mosaic

An octahedron can be thought of as a spherical mosaic and projected onto a plane using a stereographic projection. This projection is conformal, preserving angles, but not length and area. The lines on the sphere are mapped to circular arcs on the plane.

	triangular-centered
Orthographic projection	Stereographic projection

Cartesian coordinates

Octahedron with edge length 2 (\ displaystyle (\ sqrt (2))) can be placed at the origin, so that its vertices lie on the coordinate axes. The Cartesian coordinates of the vertices will then be

(± 1, 0, 0); (0, ± 1, 0); (0, 0, ± 1).

The octahedron is unique among Platonic solids in that it alone has an even number of faces at each vertex. In addition, it is the only member of this group that has planes of symmetry that do not intersect any face.

Using standard Johnson polytope terminology, the octahedron can be called square bipyramid... Truncating two opposite vertices results in truncated bipyramid.

• An octahedron can be inscribed in a tetrahedron, moreover, four of the 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.
• The 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 in the octahedron, and all eight vertices of the cube will be located in the centers of the eight faces of the octahedron.

Uniform coloring and symmetry

There are 3 uniform coloring octahedrons, named for their face colors: 1212, 1112, 1111.

The symmetry group of the octahedron is O h with order 48, three-dimensional hyperoctahedral group... The subgroups of this group include D 3d (of order 12), the symmetry group of the triangular antiprism, D 4h(order 16), the symmetry group of a square bipyramid, and T d (order 24), the symmetry group. These symmetries can be emphasized by different coloring of the faces.

Name	Octahedron	Fully truncated tetrahedron (Tetratetrahedron)	Triangular antiprism	Square bipyramid	Rhombic bipyramid
Drawing (Face painting)	(1111)	(1212)	(1112)	(1111)	(1111)
Coxeter diagram		=