Geometer. a body bounded by 8 equilateral triangles. Dictionary foreign words included in the Russian language. Pavlenkov F., 1907. OCTAHEDR in Greek. oktaedros, from okto, eight, and hedra, base. Octahedron. Explanation of 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. octaedre m. octaedron. A regular octahedron, a body bounded by eight triangles. SIS 1954. In octahedra. Witt Prom. chem. 1848 2 187. Of the crystalline forms of metals, cubes and especially octahedrons predominate. MB 1900… … Historical Dictionary of Gallicisms of the Russian Language

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

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

octahedron, octahedron, male. (from Greek okto eight and hedra base). A regular octahedron bounded by eight regular triangles. Dictionary Ushakov. D.N. Ushakov. 1935 1940 ... Explanatory Dictionary of Ushakov

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

- [όχτώ (ξwho) eight; έδρα (γhedra) face] a closed octahedron with faces in the form of regular triangles. Symbol O. (111). See Simple crystal forms of the highest (cubic) syngony. ... ... Geological Encyclopedia

octahedron- — [English Russian Gemological Dictionary. Krasnoyarsk, KrasBerry. 2007.] Topics gemology and jewelry production EN octahedron … Technical Translator's Handbook

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

Books

• Magic Edges No. 8. Great cubo-cubo-octahedron,. "Magic Edges" is a magazine for adults and children about models of paper polyhedra. Creating models of polyhedrons from cardboard is a very exciting and affordable activity, this is the "magic of transformation" ...
• Magic Facets #15. Stellated octahedron. Star polyhedron , . Kit for assembling a polyhedron "Star octahedron". Dimensions of the finished polyhedron assembled from the set: 170x180x200 mm. Difficulty level - "Start" (does not require experience and additional ...

An octahedron is one of five regular polyhedra with 8 triangular faces, 12 edges, and 6 vertices. Each of its vertices is a vertex of four triangles. The sum of the plane angles at each vertex is 240 degrees. The octahedron has a center of symmetry - the center of the octahedron, 9 axes of symmetry and 9 planes of symmetry.

In nature, in science, in life, this polyhedron is quite common: it finds application in explaining the structure and forms of the Universe, in the structure of DNA and nanotechnology, in creating puzzle games.

But most often it is found, perhaps, in the first - in nature. Namely, in the structure of crystals. The shape of the octahedron has crystals of diamond, perovskite, olivine, fluorite, spinel, aluminum-potassium alum, copper sulphate and even sodium chloride and gold!

Polyhedra are also used in painting. The clearest example of the artistic depiction of polyhedra in the 20th century is, of course, the graphic fantasies of Maurits Cornelis Escher (1898-1972), a Dutch artist born in Leeuwarden. Maurits Escher in his drawings, as it were, discovered and intuitively illustrated the laws of combination of symmetry elements, i.e. those laws that govern crystals, determining both their external form, and their atomic structure, and their physical properties.

Regular geometric bodies - polyhedra - had a special charm for Escher. In many of his works, polyhedra are the main figure, and in many more works they appear as auxiliary elements.

Rice. 7. Engraving "Stars" by Escher

Escher's most interesting work is the engraving "Stars", on which you can see the bodies obtained by combining tetrahedra, cubes and octahedrons.

Conclusion

In the course of this work, the concept of regular polyhedra was considered, we learned that a polyhedron is called regular if: 1) it is convex; 2) all its faces are regular polygons equal to each other; 3) all its dihedrals are equal; 4) the same number of edges converge at each of its vertices.

Having considered the history of the emergence of the Platonic solids, we learned that there are five regular polyhedra in total: a tetrahedron, a cube, an octahedron, a dodecahedron and an icosahedron. Their names from Ancient Greece. In literal translation from Greek "tetrahedron", "octahedron", "hexahedron", "dodecahedron", "icosahedron" mean: "tetrahedron", "octahedron", "hexahedron", "dodecahedron", "twenty-sided".

The used literature and sources made it possible to consider this topic in more depth.

Having analyzed in more detail the icosahedron and octahedron, as well as their application in various fields, we saw that the study of the Platonic solids and the figures associated with them continues to this day. And although beauty and symmetry are the main motives of modern research, they also have some scientific significance, especially in crystallography. Common salt, sodium thioantimonide, and chromic alum crystals occur naturally in the form of a cube, tetrahedron, and octahedron, respectively. The icosahedron is not found among crystalline forms, but it can be observed among the forms of microscopic marine organisms known as radiolarians.

The ideas of Plato and Kepler about the connection of regular polyhedra with the harmonious structure of the world have found their continuation in our time in an interesting scientific hypothesis that the core of the Earth has the shape and properties of a growing crystal that affects the development of all natural processes taking place on the planet. The rays of this crystal, or rather, its force field, determine the icosahedral-dodecahedral structure of the Earth. It manifests itself in the fact that in the earth's crust, as it were, the projections of regular polyhedra inscribed in the globe appear: the icosahedron and the dodecahedron.

Sculptors, architects, and artists also showed great interest in the forms of regular polyhedra. They were all amazed by the perfection, the harmony of polyhedrons.

Bibliography

1. Aleksandrov A. D. et al. Geometry for grades 10-11: Proc. A guide for school students. and classes with deepening. study Mathematics / A. D. Aleksandrov, A. L. Werner, V. I. Ryzhik. - 3rd ed., revised. - M.: Enlightenment, 1992 - 464 p.

2. Atanasyan L.S. and others. Geometry 10 - 11.- M .: Education, 2003.

3. Vasilevsky A.B. Parallel projections. - Moscow, 2012.

4. Voloshinov A.V. Mathematics and Art. - M .: Education, 2002.

5. Gonchar VV Models of polyhedra. - M.: Akim, 1997. - 64 p.

6. Dityatkin V.G. Leonardo da Vinci.- M.: Moscow, 2002.

7. Euclid. Beginning. - In 3 vols. M.; L.; 1948 - 1950.

8. Math: School Encyclopedia/ Ch. ed. Nikolsky S. M. - M .: Scientific ed.. "Great Russian Encyclopedia", 1996

9. Pidow D. Geometry and art. - Moscow, 1999.

TEXT EXPLANATION OF THE LESSON:

Our acquaintance with polyhedrons continues.

Recall that a polyhedron is called regular if the following conditions are met:

1. the polyhedron is convex;

2. all its faces are equal regular polygons;

3. the same number of faces converge at each of its vertices;

4. all its dihedral angles are equal.

In previous lessons, you learned about the uniqueness of the existence of five types of regular polyhedra:

tetrahedron, octahedron, icosahedron, hexahedron (cube) and dodecahedron.

Today we will consider the symmetry elements of the studied regular polyhedra.

A regular tetrahedron has no center of symmetry.

Its axis of symmetry is a straight line passing through the midpoints of opposite edges.

The plane of symmetry is the plane passing through any edge perpendicular to the opposite edge.

A regular tetrahedron has three axes of symmetry and six planes of symmetry.

The cube has one center of symmetry - this is the point of intersection of its diagonals.

The axes of symmetry are straight lines passing through the centers of opposite faces and the midpoints of two opposite edges that do not belong to the same face.

The cube has nine axes of symmetry that pass through the center of symmetry.

The plane passing through any two axes of symmetry is the plane of symmetry.

The cube has nine planes of symmetry.

A regular octahedron has a center of symmetry - the center of the octahedron, 9 axes of symmetry and 9 planes of symmetry: three axes of symmetry pass through opposite vertices, six through the middle of the edges.

The center of symmetry of an octahedron is the point of intersection of its axes of symmetry.

Three of the 9 planes of symmetry of the tetrahedron pass through every 4 vertices of the octahedron that lie in the same plane.

Six planes of symmetry pass through two vertices that do not belong to the same face and the midpoints of opposite edges.

A regular icosahedron has 12 vertices. The icosahedron has a center of symmetry - the center of the icosahedron, 15 axes of symmetry and 15 planes of symmetry: Five planes of symmetry pass through the first pair of opposite vertices (each of them passes through the edge containing the vertex, perpendicular to the opposite corner).

For the third pair, we get - 3 new planes, and for the fourth - two planes, and for the fifth pair, only one new plane.

Not a single new plane of symmetry will pass through the sixth pair of vertices.

A regular dodecahedron is made up of twelve regular pentagons. The dodecahedron has a center of symmetry - the center of the dodecahedron, 15 axes of symmetry and 15 planes of symmetry: the planes of symmetry pass through the edge containing the vertex, perpendicular to the opposite edge. Therefore, 5 planes pass through the first pair of opposite pentagons, 4 through the second pair, 3 through the third, 2 through the fourth, and 1 through the fifth.

Let's solve some problems using the acquired knowledge.

Prove that in a regular tetrahedron the line segments connecting the centers of its faces are equal.

Since all edges regular tetrahedron are equal and any of them can be considered the base, and the other three - side faces, it will be enough to prove the equality of the segments OM and ON.

Proof:

1.Additional construction: draw the line DN to the intersection with the AC side, we get the point F;

draw the line DM to the intersection with the side AB, we get the point E.

Then we connect vertex A with point F;

vertex C with point E.

2. Consider the triangles DEO and DOP they

rectangular, because TO the height of the tetrahedron, then they are equal in hypotenuse and leg: TO-general, DE \u003d DF (heights of equal faces of the tetrahedron)).

It follows from the equality of these triangles that OE=OF, ME=NF(midpoints of equal sides),

angle DEO equal to the angle DFO.

3. It follows from the above that the triangles OEM and OFN are equal in two sides and the angle between them (see item 2).

And from the equality of these triangles it follows that OM = ON.

Q.E.D.

Is there a quadrangular pyramid whose opposite sides are perpendicular to the base?

Let us prove that such a pyramid does not exist by contradiction.

Proof:

1. Let the edge RA1 be perpendicular to the base of the pyramid and the edge RA2 also be perpendicular to the base.

2. Then, according to the theorem (two lines perpendicular to the third are parallel), we get that the edge RA1 is parallel to the edge RA2.

3. But the pyramid has a common point for all side edges (and hence the faces) - the top of the pyramid.

We have obtained a contradiction, thus there is no quadrangular pyramid, the opposite faces of which are perpendicular to the base.

Pra-vil-nye many-grand-ni-ki in-te-re-co-va-li of many great scientists. And this in-te-res you-ho-dil yes-le-ko for pre-de-ly ma-te-ma-ti-ki. Plato (427 BC - 347 BC) ras-smat-ri-val them as the basis of the construction of the All-len-noy, Kepler (1571-1630) py -tal-sya-to-connect the right-vil-m-go-grand-ni-ki with the movement of the planets of the Solar system (some of them in his time- I would-lo from-west-but five). Possibly, it’s just the beauty and gar-mo-niya of the right-of-the-grand-ni-kov for-becoming-la-la-ve-li-scientists about -it was-to-pre-launch some-more-deeper-bo-something-of-their-meaning than just geo-met-ri-che-objects- comrade

The right-for-m-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o, all the o-o-o-o-go are the right-o-o-o-o-o go-coal-no-ki, all flat corners of something-ro-go are equal between each other and two-sided corners of someone-ro-go are equal between themselves. (Flat-ki-mi-corner-la-mi-many-grand-no-ka on-zy-va-yut-sya corners of many-coal-no-kov-faces, two-sided- us-mi corner-la-mi many-go-no-ka-na-zy-va-yut-sya corners between gra-ny-mi, having-u-schi-mi in common reb- ro.)

For-me-tim that from this definition-de-le-niya av-to-ma-ti-che-ski follows you-fart-of-p-vil-no-go-much- gran-no-ka, some-paradise in some-ry books is included in the definition-de-le-nie.

In a three-dimensional space, there is a moat-but five right-for-many-grand-ni-kov: tetra-hedron, oc-ta-hedron, cube (hex-sa-hedron), ico-sa-hedron, do-de-ca-hedron. The fact that there are no other right-of-the-grand-ni-kovs, it was before-ka-for-but Ev-kli-dom (about 300 g BC) in his great Na-cha-lah.

Ana-logic in-stro-e-nie with-me-ni-mo and in a more general case. Ras-look at the pro-out-of-the-free convex multi-grand-nick and take the points in the se-re-di-nah of its faces. Connect it between the points of the neighbors of the neighboring faces from the cut. Then the points are yav-la-yut-sya tops-shi-na-mi, from-cut-ki - ribs-ra-mi, and many-coal-no-ki, some-rye ogres -no-chi-va-yut these from-cuts, gra-nya-mi still one-but-you-bunch-lo-go-much-grand-no-ka. This multi-faceted nickname is na-zy-va-et-sya is dual-us-us-mi to is-go-no-mu.

As it were, in-for-but higher, dual to tet-ra-ed-ru yav-la-et-sya tet-ra-hedron.

Increase the size of the tet-ra-ed-ra, top-shi-na-mi-ko-ro-th-y-y-yut-se-re-di-ny faces is-move -no-go tet-ra-ed-ra, up to the size of the next-not-go. In-seven peaks-tires are so-ra-la-women-nyh tet-ra-ed-ditch are-la-yut-xia tops-shi-na-mi ku-ba.

Pe-re-se-che-ni-em of these tet-ra-ed-ditch yav-la-et-sya is another right-vil-many-grand-nickname - ok-ta-hedron (from the Greek. οκτώ - at seven). Ok-ta-hedron has 8 triangular faces, 6 vertices, 12 edges. The flat angles of ok-ta-ed-ra are equal to $\pi/3$, since its faces are right-angled triangles no, dihedral angles are $\arccos(–1/3) ≈ 109(,)47^\circ$.

From-me-tim se-re-di-ny faces ok-ta-ed-ra and re-rey-dem to dual-no-mu to ok-ta-ed-ru many-gran- no way. This is a cube or hex-sa-hedron (from the Greek εξά - six). At the ku-ba gra-no yav-la-yut-sya quad-ra-ta-mi. It has 6 faces, 8 vertices, 12 edges. The flat angles of the ku-ba are equal to $\pi/2$, the two-faced angles are also equal to $\pi/2$.

If we take points on the se-re-di-nah of the faces of the ku-ba and consider the multi-face-nick dual to it, then we can convince Xia, that they will again be an oc-ta-hedron. A more general statement is also true: if for you there are many the dual to the dual-no-mu, then it will be an out-going multi-faceted nickname (with accuracy to the be-to-biya).

Take on the edges of ok-ta-ed-ra on the dot, with the condition that each de-li-la reb-ro in co-from-no-she-nii $ 1 :(\sqrt5+1)/2$ (golden se-che) top-shi-on-mi right-vil-no-th triangle-no-ka. Po-lu-chen-nye 12 to-check is-la-yut-sya ver-shi-on-mi is still one-of-the-right-vil-no-go-many-gran-no-ka - iko- sa-ed-ra (from Greek είκοσι - twenty). Ico-sa-hedron is a right-handed multi-faceted nickname, someone has 20 triangular faces. It has 12 vertices, 30 edges. Flat angles of iko-sa-ed-ra are equal to $\pi/3$, two-faced equals are $\arccos(–1/3\cdot\sqrt5) ≈ 138(,)19 ^\circ$.

Ico-sa-hedron can be inscribed in a cube. At the same time, on each gra-ni-ku-ba, there will be two peaks of iko-sa-ed-ra.

Let’s turn it back to the iko-sa-hedron, “stand-up” it on top-shi-nu, and get it more familiar look: two caps from five ty triangles near the south and north of the south and north-poly-owls and the middle layer, consisting of de-s-ti triangles no-kov.

Se-re-di-ny gra-ney iko-sa-ed-ra yav-la-yut-sya ver-shi-na-mi one more right-vil-no-go-many-gran- no-ka - do-de-ka-ed-ra (from the Greek δώδεκα - two-twenty). Gra-no to-de-ka-ed-ra are the right-for-vil-ny five-coal-ni-ki. In such a way, its flat angles are equal to $3\pi/5$. Do-de-ka-ed-ra has 12 faces, 20 vertices, 30 edges. The dihedral angles to-de-ka-ed-ra are equal to $\arccos(–1/5\cdot\sqrt5) ≈116(,)57^\circ$.

Taking se-re-di-ny faces to-de-ka-ed-ra, and re-rei-dya to dual-stven-no-mu him a lot-gran-ni-ku, in-lu- chim again iko-sa-hedron. So, the iko-sa-hedron and the do-de-ka-hedron are dual to each other. This once again il-lu-stri-ru-is the fact that dual-to-dual-no-mu will be an out-going multi-grand-nick.

For-me-tim that when you re-re-ho-de to a dual-many-grand-no-ku, the tops of the is-move-no-go-many-grand -no-so-correspond-to-the-reply-to-yut-yum-dual-no-go, ribs-ra - ribs-dual-no-go, and gra-no - tops-shi-we are two -stven-but-go-many-grand-no-ka. If Iko-sa-ed-ra has 20 faces, then it means that the dual has 20 vertices to-de-ca-ed-ra and they have one-to- the first number of edges, if the cube has 8 vertices, then the dual oc-ta-ed-ra has 8 faces.

There are different-personal ways-to-inscribe-sy-va-niya of the right-for-many-grand-no-kovs into each other, pri-dya- to many for-me-cha-tel-ny construct-structures-ci-pits. In-te-res-nye and beautiful-so-much-grand-ni-ki in-lu-cha-yut-sya the same with united-non-nii and re-re-se-che -nii pra-vil-nyh many-grand-ni-kov.

Inscribe a cube in the do-de-ka-hedron so that all 8 vertices of the ku-ba are owl-pa-da-li with the top-shi-on-mi to-de-ka-ed-ra. In the circle do-de-ka-ed-ra, describe the ico-sa-hedron so that its tops-shi-we-eye-are in the se-re-di-nah faces of the ico-sa-edr -ra. In the circle of Iko-sa-ed-ra, describe the ok-ta-hedron, so that the tops of the iko-sa-ed-ra are left-zha-li on the edges of the ok-ta-ed-ra . Finally, in the circle of ok-ta-ed-ra, describe the tetra-hedron so that the vertices of ok-ta-ed-ra are pa-whether on se-re-di -ny ryo-ber tet-ra-ed-ra.

Such a construction from ku-soch-kov slo-man-ny de-re-vyan-ny ski pa-loks was made by re-byon-kom bu-du-ve-li- cue ma-te-ma-tic XX century V. I. Ar-nold. Vla-di-mir Igo-re-vich kept her for many years, and then gave her to la-bo-ra-to-riya in a bullet-ri-za-tion and pro- pa-gan-dy ma-te-ma-ti-ki Ma-te-ma-ti-che-sko-go in-sti-tu-ta them. V. A. Stek-lo-va.

Literature

G. S. M. Cox-ter. Introduction to geometries. - M.: Na-at-ka, 1966.

J. Ada-mar. Element-men-tar-naya geo-met-riya. Part 2. Ste-reo-met-riya. - M .: Pro-sve-shche-tion, 1951.

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