Regular polyhedra are called convex polyhedra, all of whose faces are identical regular polygons, and the same number of faces meet at each vertex. Such polyhedra are also called Platonic solids.
There are only five regular polyhedra:
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Type of regular polyhedron |
Number of sides on a face |
Number of edges adjacent to a vertex |
Total number of vertices |
Total number of edges |
Total number of faces |
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Tetrahedron |
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Hexahedron or cube |
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Dodecahedron |
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Icosahedron |
The name of each polyhedron comes from the Greek name for the number of its faces and the word "face".
Tetrahedron
A tetrahedron (Greek fefsbedspn - tetrahedron) is a polyhedron with four triangular faces, at each of the vertices of which 3 faces meet. A tetrahedron has 4 faces, 4 vertices and 6 edges.
Properties of the tetrahedron
Parallel planes passing through pairs of intersecting edges of the tetrahedron define the parallelepiped described around the tetrahedron.
The segment connecting the vertex of a tetrahedron with the point of intersection of the medians of the opposite face is called its median, omitted from this vertex.
The segment connecting the midpoints of the intersecting edges of a tetrahedron is called its bimedian connecting these edges.
A segment connecting a vertex to a point on the opposite face and perpendicular to this face is called its height, omitted from the given vertex.
Theorem. All medians and bimedians of a tetrahedron intersect at one point. This point divides the medians in a ratio of 3:1, counting from the apex. This point divides the bimedians in half.
Highlight:
- · an isohedral tetrahedron, in which all faces are equal triangles;
- · an orthocentric tetrahedron in which all heights descending from the vertices to opposite faces intersect at one point;
- · a rectangular tetrahedron in which all edges adjacent to one of the vertices are perpendicular to each other;
- · regular tetrahedron, all of whose faces are equilateral triangles;
- · frame tetrahedron - a tetrahedron that meets any of the conditions:
- · There is a sphere touching all the edges.
- · The sums of the lengths of the crossing edges are equal.
- · The sums of dihedral angles at opposite edges are equal.
- · Circles inscribed in faces touch in pairs.
- · All quadrilaterals resulting from the development of a tetrahedron are described.
- · Perpendiculars, restored to the faces from the centers of the circles inscribed in them, intersect at one point.
- · a commensurate tetrahedron, all biheights of which are equal;
- · an incentric tetrahedron, in which the segments connecting the vertices of the tetrahedron with the centers of circles inscribed in opposite faces intersect at one point.
A cube or regular hexahedron is a regular polyhedron, each face of which is a square. A special case of a parallelepiped and a prism.
Cube properties
- · The four sections of the cube are regular hexagons - these sections pass through the center of the cube perpendicular to its four main diagonals.
- · You can fit a tetrahedron into a cube in two ways. In both cases, the four vertices of the tetrahedron will be aligned with the four vertices of the cube and all six edges of the tetrahedron will belong to the faces of the cube. In the first case, all the vertices of the tetrahedron belong to the faces of a trihedral angle, the vertex of which coincides with one of the vertices of the cube. In the second case, pairwise crossing edges of the tetrahedron belong to pairwise opposite faces of the cube. This tetrahedron is regular.
- · You can fit an octahedron into a cube, and all six vertices of the octahedron will be aligned with the centers of the six faces of the cube.
- · A cube can be inscribed in an octahedron, and all eight vertices of the cube will be located at the centers of the eight faces of the octahedron.
- · An icosahedron can be inscribed into a cube, while six mutually parallel edges of the icosahedron will be located respectively on the six faces of the cube, the remaining 24 edges will be located inside the cube. All twelve vertices of the icosahedron will lie on the six faces of the cube.
The diagonal of a cube is a segment connecting two vertices that are symmetrical about the center of the cube. The diagonal of a cube is found by the formula
polyhedron icosahedron octahedron dodecahedron
where d is the diagonal, and is the edge of the cube.
Octahedron
The octahedron (Greek pkfedspn, from Greek pkfyu, “eight” and Greek Edsb - “base”) is one of the five convex regular polyhedra, the so-called Platonic solids.
The octahedron has 8 triangular faces, 12 edges, 6 vertices, and 4 edges converge at each vertex.
If the length of an octahedron edge is equal to a, then the area of its total surface (S) and the volume of the octahedron (V) are calculated using the formulas:
The radius of a sphere circumscribed around an octahedron is equal to:
The radius of a sphere inscribed in an octahedron can be calculated using the formula:
A regular octahedron has Oh symmetry, which coincides with the symmetry of a cube.
The octahedron has a single star shape. The octahedron was discovered by Leonardo da Vinci, then rediscovered almost 100 years later by Johannes Kepler, and he named it Stella octangula - an octagonal star. Hence this form has the second name “Kepler’s stella octangula”.
In essence, it is a combination of two tetrahedrons
Dodecahedron
Dodecahedron (from the Greek dudekb - twelve and edspn - face), dodecahedron - a regular polyhedron made up of twelve regular pentagons. Each vertex of the dodecahedron is the vertex of three regular pentagons.
Thus, the dodecahedron has 12 faces (pentagonal), 30 edges and 20 vertices (3 edges converge at each). The sum of the plane angles at each of the 20 vertices is 324°.
The dodecahedron has 3 stellated shapes: small stellated dodecahedron, large dodecahedron, great stellated dodecahedron (stellated dodecahedron, the final form). The first two of them were discovered by Kepler (1619), the third by Poinsot (1809). Unlike the octahedron, any of the stellated forms of the dodecahedron is not a combination of Platonic solids, but forms a new polyhedron.
All 3 stellated forms of the dodecahedron, together with the great icosahedron, form the family of Kepler-Poinsot solids, that is, regular non-convex (stellate) polyhedra.
The faces of the great dodecahedron are pentagons, which meet five at each vertex. The small stellated and large stellated dodecahedrons have faces of five-pointed stars (pentagrams), which in the first case converge in 5, and in the second in 3. The vertices of the large stellated dodecahedron coincide with the vertices of the described dodecahedron. Each vertex has three faces connected.
Basic formulas:
If we take a to be the edge length, then the surface area of the dodecahedron is:
Dodecahedron volume:
Radius of the described sphere:
Radius of inscribed sphere:
Symmetry elements of the dodecahedron:
· The dodecahedron has a center of symmetry and 15 axes of symmetry.
Each of the axes passes through the midpoints of opposite parallel edges.
· The dodecahedron has 15 planes of symmetry. Any of the planes of symmetry passes in each face through the top and middle of the opposite edge.
Icosahedron
Icosahedron (from the Greek ekpubt - twenty; -edspn - face, face, base) is a regular convex polyhedron, twenty-hedron, 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 area S, volume V of an icosahedron with edge length a, as well as the radii of the inscribed and circumscribed spheres are calculated using the formulas:
radius of the inscribed sphere:
radius of the circumscribed sphere:
Properties
- · The icosahedron can be inscribed in a cube, in this case, six mutually perpendicular edges of the icosahedron will be located respectively on six faces of the cube, the remaining 24 edges 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 combined with the four vertices of the icosahedron.
- · An icosahedron can be inscribed into a dodecahedron, with the vertices of the icosahedron aligned with the centers of the dodecahedron's faces.
- · A dodecahedron can be inscribed into an icosahedron by combining the vertices of the dodecahedron and the centers of the faces of the icosahedron.
- · A truncated icosahedron can be obtained by cutting off 12 vertices to form faces in the form of regular pentagons. In this case, the number of vertices of the new polyhedron increases 5 times (12?5=60), 20 triangular faces turn into regular hexagons (the total number of faces becomes 20+12=32), and the number of edges increases to 30+12?5=90.
The icosahedron has 59 stellated shapes, of which 32 have complete and 27 incomplete icosahedral symmetry. One of these stellations (20th, Wenninger mod. 41), called the great icosahedron, is one of the four regular Kepler-Poinsot stellations. Its faces are regular triangles, which meet at each vertex in fives; This property is common to the great icosahedron with the icosahedron.
Among the stellate forms there are also: a connection of five octahedrons, a connection of five tetrahedrons, a connection of ten tetrahedra.
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Polyhedra. Vertices, edges, faces of a polyhedron. EULER'S THEOREM. 10th grade Completed by: Kaygorodova S.V.
A polyhedron is called regular if all its faces are regular polygons and all polyhedral angles at its vertices are equal.
Five amazing polyhedra have been known to man since ancient times.
Based on the number of faces they are called a regular tetrahedron
hexahedron (hexagon) or cube
octahedron (octahedron)
dodecahedron (dodecahedron)
icosahedron (twenty-hedron)
Developments of regular polyhedra
Historical background Four essences of nature were known to mankind: fire, water, earth and air. According to Plato, their atoms had the form of regular polyhedra. The great ancient Greek philosopher Plato, who lived in the 4th – 5th centuries. BC, believed that these bodies personify the essence of nature.
the atom of fire had the form of a tetrahedron, earth - a hexahedron (cube) of air - an octahedron of water - an icosahedron
But there remained a dodecahedron, which had no correspondence. Plato suggested that there was another (fifth) entity. He called it the world ether. The atoms of this fifth essence had the shape of a dodecahedron. Plato and his students paid great attention to the listed polyhedra in their works. Therefore, these polyhedra are also called Platonic solids.
For any convex polyhedron the following relation is true: Г+В-Р=2, where Г is the number of faces, В is the number of vertices, Р is the number of edges of the given polyhedron. Faces + Vertices - Edges = 2. Euler's Theorem
Characteristics of regular polyhedra Polyhedron Number of sides of a face Number of faces meeting at each vertex Number of faces (G) Number of edges (P) Number of vertices (V) Tetrahedron 3 3 4 6 4 Hexahedron 4 3 6 12 8 Octahedron 3 4 8 12 6 Icosahedron 3 5 20 30 12 Dodecahedron 5 3 12 30 20
Duality of regular polyhedra Hexahedron (cube) and octahedron form a dual pair of polyhedra. The number of faces of one polyhedron is equal to the number of vertices of another and vice versa.
Let's take any cube and consider a polyhedron with vertices at the centers of its faces. As you can easily see, we get an octahedron.
The centers of the octahedron's faces serve as the vertices of the cube.
Antimony sodium sulfate is a tetrahedron. Polyhedra in nature, chemistry and biology Crystals of some substances familiar to us have the shape of regular polyhedra. Pyrite crystal is a natural dodecahedron model. Table salt crystals give the shape of a cube. The single crystal of aluminum-potassium alum has the shape of an octahedron. Crystal (prism) The icosahedron has become the focus of biologists' attention in their disputes regarding the shape of viruses. The virus cannot be perfectly round, as previously thought. To establish its shape, they took various polyhedra and directed light at them at the same angles as the flow of atoms at the virus. It turned out that only one polyhedron gives exactly the same shadow - the icosahedron. During the process of egg division, a tetrahedron of four cells is first formed, then an octahedron, a cube and, finally, a dodecahedral-icosahedral gastrula structure. And finally, perhaps most importantly, the DNA structure of the genetic code of life is a four-dimensional development (along the time axis) of a rotating dodecahedron! The methane molecule has the shape of a regular tetrahedron.
Polyhedra in art “Portrait of Monna Lisa” The composition of the picture is based on golden triangles, which are parts of a regular star-shaped pentagon. engraving “Melancholy” In the foreground of the picture there is a dodecahedron. “The Last Supper” Christ and his disciples are depicted against the background of a huge transparent dodecahedron.
Polyhedra in Architecture The Yamanashi Fruit Museum was created using 3D modeling. The four-tiered Spasskaya Tower with the Church of the Savior Not Made by Hands is the main entrance to the Kazan Kremlin. It was erected in the 16th century by Pskov architects Ivan Shiryai and Postnik Yakovlev, nicknamed “Barma”. The four tiers of the tower are a cube, polyhedra and a pyramid. Spasskaya Tower of the Kremlin. Alexandria Lighthouse Pyramids Fruit Museums
Unfortunately, spherical geometry and Lobachevsky geometry are not studied in the school curriculum. Meanwhile, their study together with Euclidean geometry allows us to better understand what is happening with objects. For example, understand the connection of regular polyhedra with partitions of the sphere, partitions of the Euclidean plane and partitions of the Lobachevsky plane.
Knowledge of the geometry of spaces of constant curvature helps to rise above three dimensions and identify polyhedra in spaces of dimension 4 and higher. The issues of finding polyhedra, finding partitions of spaces of constant curvature, deriving the formula for the dihedral angle of a regular polyhedron in n-dimensional space are so closely intertwined that it turned out to be problematic to include all this in the title of the article. Let the focus be on regular polyhedra, understandable to everyone, although they are not only the result of all conclusions, but also, at the same time, a tool for comprehending spaces of higher dimensions and uniformly curved spaces.
For those who don’t know (forgot), I inform (remind) that in the three-dimensional Euclidean space we are used to, there are only five regular polyhedra:
| 1. Tetrahedron: | 2. Cube: | 3. Octahedron: | 4. Dodecahedron: | 5. Icosahedron: |
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In three-dimensional space, a regular polyhedron is a convex polyhedron in which all vertices are equal to each other, all edges are equal to each other, all faces are equal to each other and the faces are regular polygons.
A regular polygon is a convex polygon in which all sides are equal and all angles are equal.
The vertices are equal to each other means that the number of edges and the number of faces approaching each vertex are the same and they approach at the same angles at each vertex.
In this notation, our polyhedra will receive the following designations:
1. Tetrahedron (3, 3),
2. Cube (4, 3),
3. Octahedron (3, 4),
4. Dodecahedron (5, 3),
5. Icosahedron (3, 5)
For example, (4, 3) - a cube has 4 corner faces, and 3 such faces meet at each vertex.
The octahedron (3, 4), on the contrary, has 3 carbon faces, 4 of which converge at the apex.
Thus, the Schläfli symbol completely determines the combinatorial structure of the polyhedron.
Why are there only 5 regular polyhedra? Maybe there are more of them?
To fully answer this question, you must first obtain an intuitive understanding of geometry on the sphere and on the Lobachevsky plane. For those who do not yet have such an idea, I will try to give the necessary explanations.
Sphere
1. What is a point on a sphere? I think it’s intuitively clear to everyone. It is not difficult to mentally imagine a point on a sphere.
2. What is a segment on a sphere? We take two points and connect them by the shortest distance on the sphere; we get an arc if we look at the sphere from the side.
3. If you continue this segment in both directions, it will close and you will get a circle. In this case, the plane of the circle contains the center of the sphere; this follows from the fact that we connected the two starting points by the shortest, and not arbitrary, distance. From the side it looks like a circle, but in terms of spherical geometry it is a straight line, since it was obtained from a segment, extended to infinity in both directions.
4. And finally, what is a triangle on a sphere? We take three points on the sphere and connect them with segments.
By analogy with a triangle, you can draw an arbitrary polygon on a sphere. For us, the property of a spherical triangle is fundamentally important, namely that the sum of the angles of such a triangle is greater than 180 degrees, which we are accustomed to in the Euclidean triangle. Moreover, the sum of the angles of two different spherical triangles is different. The larger the triangle, the HIGHER the sum of its angles.
Accordingly, the 4th sign of equality of triangles on a sphere appears - at three angles: two spherical triangles are equal to each other if their corresponding angles are equal.
For simplicity, it’s easier not to draw the sphere itself, then the triangle will look a little bloated:
A sphere is also called a space of constant positive curvature. The curvature of space precisely leads to the fact that the shortest distance is an arc, and not the straight line segment we are used to. The segment seems to be bent.
Lobachevsky
Now that we have become acquainted with the geometry on the sphere, it will not be difficult to understand the geometry on the hyperbolic plane, discovered by the great Russian scientist Nikolai Ivanovich Lobachevsky, since everything happens here similarly to the sphere, only “inside out”, “in reverse”. If we drew arcs on a sphere in circles with a center inside the sphere, now arcs must be drawn in circles with a center outside the sphere.Let's get started. We will represent the Lobachevsky plane in the interpretation of Poincaré II (Jules Henri Poincaré, the great French scientist), this interpretation of Lobachevsky geometry is also called the Poincaré disk.
1. Point in the Lobachevsky plane. Period - it is a point in Africa too.
2. A segment on the Lobachevsky plane. We connect two points with a line along the shortest distance in the sense of the Lobachevsky plane.
The shortest distance is constructed as follows:
It is necessary to draw a circle orthogonal to the Poincaré disk through the given two points (Z and V in the figure). The center of this circle will always be outside the disk. The arc connecting the original two points will be the shortest distance in the sense of the Lobachevsky plane.
3. Removing the auxiliary arcs, we obtain the straight line E1 - H1 in the Lobachevsky plane.
Points E1, H1 “lie” on the infinity of the Lobachevsky plane; in general, the edge of the Poincaré disk is all infinitely distant points of the Lobachevsky plane.
4. And finally, what is a triangle in the Lobachevsky plane? We take three points and connect them with segments.
By analogy with a triangle, you can draw an arbitrary polygon on the Lobachevsky plane. For us, the property of a hyperbolic triangle is fundamentally important, namely that the sum of the angles of such a triangle is always less than 180 degrees, which we are accustomed to in the Euclidean triangle. Moreover, the sum of the angles of two different hyperbolic triangles is different. The larger the triangle is in area, the LESS the sum of its angles.
Accordingly, the 4th sign of equality of hyperbolic triangles also takes place here - by three angles: two hyperbolic triangles are equal to each other if their corresponding angles are equal.
For simplicity, the Poincaré disk itself can sometimes not be drawn, then the triangle will look a little “shrunken”, “deflated”:
The Lobachevsky plane (and in general the Lobachevsky space of any dimension) is also called the space of constant NEGATIVE curvature. The curvature of space precisely leads to the fact that the shortest distance is an arc, and not the straight line segment we are used to. The segment seems to be bent.
Regular partitions of a two-dimensional Sphere and regular three-dimensional polyhedra
Everything said about the sphere and the Lobachevsky plane refers to two-dimensionality, i.e. The surface of a sphere is two-dimensional. What does this have to do with the three-dimensionality indicated in the title of the article? It turns out that each three-dimensional regular Euclidean polyhedron has a one-to-one correspondence with its own partition of the two-dimensional sphere. This is best seen in the figure:
To obtain a partition of a sphere from a regular polyhedron, you need to describe a sphere around the polyhedron. The vertices of the polyhedron will appear on the surface of the sphere, connecting these points with segments on the sphere (arcs), we obtain a partition of the two-dimensional sphere into regular spherical polygons. As an example, a video demonstration was made of how the icosahedron corresponds to the division of a sphere into spherical triangles and vice versa, and how the division of a sphere into spherical triangles converging in fives at the apex corresponds to the icosahedron.
In order to construct a polyhedron from a partition of a sphere, the vertices of the partition corresponding to the arcs must be connected by ordinary, rectilinear, Euclidean segments.
Accordingly, the Schläfli symbol of the icosahedron (3, 5)—triangles converging five at a vertex—specifies not only the structure of this polyhedron, but also the structure of the partition of a two-dimensional sphere. Similarly with other polytopes, their Schläfli symbols also determine the structure of the corresponding partitions. Moreover, partitions of the Euclidean plane and the Lobachevsky plane into regular polygons can also be specified by the Schläfli symbol. For example, (4, 4) - quadrilaterals converging in fours - this is the squared notebook we are all familiar with, i.e. This is a division of the Euclidean plane into squares. Are there other divisions of the Euclidean plane? We'll see further.
Construction of partitions of a two-dimensional sphere, the Euclidean plane and the Lobachevsky plane
To construct partitions of two-dimensional spaces of constant curvature (this is the general name of these three spaces), we need elementary school geometry and knowledge that the sum of the angles of a spherical triangle is greater than 180 degrees (greater than Pi), that the sum of the angles of a hyperbolic triangle is less than 180 degrees (less than Pi) and What is the Schläfli symbol? All this has already been said above.So, let’s take an arbitrary Schläfli symbol (p1, p2), it specifies a partition of one of three spaces of constant curvature (for a plane this is true, for spaces of higher dimensions the situation is more complicated, but nothing prevents us from exploring all combinations of the symbol).
Let's consider a regular p1 square and draw segments connecting its center and vertices. We get p1 pieces of isosceles triangles (only one such triangle is shown in the figure). We denote the sum of the angles of each of these triangles as t and express t in terms of pi and the lambda coefficient.
Then if lambda = 1, then the Euclidean triangle, i.e. is in the Euclidean plane, if lambda is in the interval (1, 3), then this means that the sum of the angles is greater than pi and this means that this triangle is spherical (it is not difficult to imagine that when increasing a spherical triangle in the limit, a circle with three points on it is obtained, in at each point the angle of the triangle is equal to pi, and the total is 3*pi. This explains the upper limit of the interval = 3). If lambda is in the interval (0, 1), then the triangle is hyperbolic, since the sum of its angles is less than pi (i.e. less than 180 degrees). Briefly it can be written like this:
On the other hand, for convergence at the vertex of p2 pieces (i.e., an integer number) of the same polygons, it is necessary that
Equating the expressions for 2*betta found from the convergence condition and from the polygon:
We have obtained an equation that shows which of the three spaces is divided by the figure given by its Schläfli symbol (p1, p2). To solve this equation, we must also remember that p1, p2 are integers greater than or equal to 3. This, so to speak, follows from their physical meaning, since these are p1 angles (at least 3 angles) converging along p2 pieces at the vertex (also no less than 3, otherwise it will not be a vertex).
The solution to this equation is to enumerate all possible values for p1, p2 greater than or equal to 3 and calculate the lambda value. If it turns out to be equal to 1, then (p1, p2) partitions the Euclidean plane, if it is greater than 1 but less than 3, then this is a partition of the Sphere, if from 0 to 1, then this is a partition of the Lobachevsky plane. It is convenient to summarize all these calculations in a table.
From where it can be seen that:
1. The sphere corresponds to only 5 solutions; when lamda is greater than 1 and less than 3, they are highlighted in green in the table. These are: (3, 3) - tetrahedron, (3, 4) - octahedron, (3, 5) - icosahedron, (4, 3) - cube, (5, 3) - dodecahedron. Their pictures were presented at the beginning of the article.
2. Partitions of the Euclidean plane correspond to only three solutions, when lambda = 1, they are highlighted in blue in the table. This is what these splits look like.
3. And finally, all other combinations (p1, p2) correspond to partitions of the Lobachevsky plane; accordingly, there is an infinite (countable) number of such partitions. It remains only to illustrate some of them, for example.
Results
Thus, there are only 5 regular polyhedra, they correspond to five partitions of the two-dimensional sphere, there are only 3 partitions of the Euclidean plane, and there are a countable number of partitions of the Lobachevsky plane.What is the application of this knowledge?
There are people who are directly interested in partitions of a sphere.
Regular polyhedron A polyhedron is called such that all its faces are equal and are equal regular polygons, all edges and all vertices are also equal to each other. While there are any number of regular polygons, there are a limited number of regular polyhedra.
Just as regular polygons begin with a triangle, so regular polyhedra begin with its analogue - tetrahedron (i.e., in Greek, tetrahedron). It has the minimum possible number of vertices and faces - four of each, and six edges (three vertices always lie in the same plane; for a volumetric body, therefore, at least four vertices are needed; a finite volume in space cannot be limited by three flat faces). At each vertex three triangular faces and, accordingly, three edges converge. A tetrahedron is a pyramid, and the simplest one is trihedral (any pyramid consists of a base and side faces; a pyramid is called n-faceted if it has n side faces; it is easy to see that for an n-sided pyramid the base must inevitably have the shape of an n-gon ). Everything we have said so far about the tetrahedron applies to any tetrahedron, not necessarily the regular one; the faces of a regular tetrahedron are regular triangles.
You are very familiar with the following regular polyhedron - this is cube. If a tetrahedron is in a certain sense similar to a triangle, then a cube is similar to a square. A cube is a rectangular parallelepiped with all its faces being squares. Try, without looking at the picture, to figure out how many faces a cube (and, in fact, any rectangular parallelepiped) has, how many vertices, how many edges, and how many faces and edges converge at each vertex.
Another regular polyhedron has octahedron (i.e. octahedron) - there are no analogues in the flat world, because it looks a little like a triangle, and a little like a square. An octahedron can be made from two tetrahedral pyramids by gluing their bases. The faces of a regular octahedron are regular triangles. At each of its vertices, not three, like a tetrahedron and a cube, meet, but four faces. For example, natural diamond crystals have an octahedral shape.
The octahedron is closely related to the so-called cube property of reciprocity : the centers of the faces of a cube are the vertices of a regular octahedron, and the centers of the faces of a regular octahedron are the vertices of a cube. If you connect the centers of adjacent faces of a cube with segments, then these segments will become the edges of the octahedron; if you do the same operation with an octahedron, you get a cube. By the way, based on this, it is clear that the number of vertices of the octahedron is equal to the number of faces of the cube, and vice versa; Moreover, their numbers of edges coincide.
The tetrahedron is related to itself by the property of reciprocity
Is it possible to formulate some analogue of the reciprocity property for regular polygons?
By the way, the tetrahedron is also related to the cube. Namely, if you choose four vertices of a cube, of which no two are adjacent, and connect them with segments, then these segments form a tetrahedron!
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Rice. 3. Cube and tetrahedron |
The most important property of regular polyhedra that immediately attracts attention is their high degree of symmetry. A certain number of reflections around different planes, as well as a number of rotations around different axes, transform each of the polyhedra into itself. Each of them has a center through which all these planes of symmetry and axes pass; the vertices are equidistant from this center, the same is true for faces and edges. Therefore, a sphere can be inscribed in every regular polyhedron, and a sphere can be described around each of them. (In this regard, however, they are quite similar to regular polygons, into each of which a circle can be inscribed and around each of which a circle can also be described).
How many planes of symmetry does a cube, tetrahedron, or octahedron have? How many rotation axes does each of them have that transform the polyhedron into itself?
