Polyhedrons are the part of the space, enclosed by flat polygons and any two of the edges can be matched with a broken line, which intersect none of the polygons.
A polyhedron is called regular when its faces are regular polygons and the vertices are also regular. This means that each edge, angle and vertex is equal. Besides, its faces are regular and the number of the edges is the same at every vertex.
One of the most important theorems in connection with polyhedron is the Euler's formula. It is a simple relationship connecting the number v of vertices, e of edges and f of faces:
v-e+f=2
Proof: Every polyhedron can be drawn as a planar graph. These graphs can be constructed from an isolated point step by step. Firstly there is one vertex (the isolated point), there is one face and there are no edges. In every step a new edge is between a new vertex and an old one or it is between two old vertices. In the first case the number of the vertices and edges increase by one, but the number of the faces does not change. So v-e+f=2 will be true. The second case is the same. This means that the value of the v-e+f expression will not change. Q.E.D..
With this condition, it can be easily concluded that there are five regular polyhedrons.
Proof: Let us create a polyhedron with V vertices, F faces and E edges, in which at each vertex m n-gons meet. (This means, that the Schläfli symbol of the polyhedron is {m;n}.) The number of flat angles equals F*n , but also V*m , where
. Each edge has been counted twice, since each edge takes part in two flat angles.
So
F*n=V*m=2E
After combinating these relations with Euler's formula:
.
(n-2)(m-2)<4
The following cases are possible:
(n-2)(m-2)=1 => m=n=3;
(n-2)(m-2)=2 => m=3, n=4 or m=4, n=3;
(n-2)(m-2)=3 => m=3, n=5 or m=5, n=3;
The possible combinations of Schläfli symbol ({n,m}) are the following:
{3;3}{3;4}{3;5}{4;3}{5;3}. Q.E.D.
1. Tetrahedron (3;3)
The number of triangle-faces is 4
The number of vertices is 4
The number of edges is 6
2. Cube (hexahedron) (4;3)
The number square-faces is 6
The number of vertices is 8
The number of edges is 12
3. Octahedron (3;4)
The number triangle faces is 8
The number of vertices is 6
The number of edges is 12
4. Dodecahedron (5;3)
The number pentagonal-faces is 12
The number of vertices is 20
The number of edges is 30
5. Icosahedron (3;5)
The number triangle faces is 20
The number of vertices is 12
The number of edges is 30
The other interesting attributes of the polyhedrons are the following:
- If the diagonals in the faces of a cube are drawn, we can see that two tetrahedrons can be fitted into a cube:
- The midpoints of the faces of a cube are the vertices of an octahedron. (The cube and the octahedron are the duals of each other):
- An icosahedron can be fitted into a cube. If there is an icosahedron with an edge, which has a length of a. The length of the edge of that cube, in which the icosahedron can be fitted into is
:
- A dodecahedron can be fitted into a cube. The length of the edge of the cube and the distant of the dodecahedron's opposite edges are equals. If the dodecahedron has an a-length-edge, the length of the edge of the cube is
:
- But also a cube can be fitted into a dodecahedron, because planes can be laid to the edges of the cube and these 12 planes are the faces of a dodecahedron. If the dodecahedron with an edge length of a, the edge of the cube is
:
- . The icosahedron and the dodecahedron are the duals of each other. This means, that the midpoints of the faces of the icosahedron are the vertices of the dodecahedron and the midpoints of the faces of the dodecahedron are the vertices of the icosahedron:
The definition of the regular solids can be generalized in several ways. The first one is the definition of the semi-regular polyhedron, the second is the star polyhedron and the third is the definition of the politops.
Semi-regular polyhedron is a convex polyhedron whose
1. faces are regular polygons and the vertices are equal
2. polyhedral angles are regular and its faces are congruent.
The 13 Archimedean solids, the uniform prisms and the uniform antiprisms are the member of the first group. The Catalan-solids are in the second group. There are 13 Catalan-solids, too. These are the duals of the Archimedean solids.
Let us list the semi-regular polyhedrons. The names of these solids and the Schlälfli-symbols are listed. Schlälfli-symbol can show us how many (m) regular n-gons meet at a vertex. For example (3,6,6) means that in any vertex of this solid meet a regular triangle-faces and two regular, congruent hexagonal-faces. The duals of the Archimedean solids can be derived from a sphere, in which these solids can be ins.
1.Truncated tetrahedron (3,6,6)
The number of triangle faces is 4
The number of hexagonal-faces is 4
The number of vertices is 12
The number of edges is 18.
2. Cuboctahedron (3,4,3,4)
The number of triangle faces is 8
The number of square-faces is 4
The number of vertices is 12
The number of edges is 24
3. Truncated cube (3,8,8)
The number of triangle faces is 8
The number of square-faces is 6
The number of vertices is 24
The number of edges is 36
4. Truncated octahedron (4,6,6)
The number of triangle faces is 6
The number of square-faces is 8
The number of vertices is 24
The number of edges is 36
5. Small rhombicuboctahedron (3,4,4,4)
The number of triangle faces is 8
The number of square-faces is 18
The number of vertices is 24
The number of edges is 48
6. Icosidodecahedron (3,5,3,5)
The number of triangle faces is 20
The number of pentagonal-faces is 12
The number of vertices is 30
The number of edges is 60
7. Snub cube (3,3,3,3,4)
The number of triangle faces is 32
The number of square-faces is 6
The number of vertices is 24
The number of edges is 60
8. Truncated cuboctahedron (4,6,8)
The number of square faces is 12
The number of hexagonal-faces is 8
The number of octagonal-faces is 6
The number of vertices is 48
The number of edges is 72
9. Truncated icosahedron (5,6,6)
The number of pentagonal faces is 12
The number of hexagonal-faces is 20
The number of vertices is 60
The number of edges is 90
10. Truncated dodecahedron (3,10,10)
The number of triangle faces is 20
The number of decagonal-faces is 12
The number of vertices is 60
The number of edges is 90
11. Small rhombicosidodecahedron (3,4,5,4)
The number of triangle faces is 20
The number of square-faces is 30
The number of pentagonal-faces is 12
The number of vertices is 60
The number of edges is 120
12. Snub dodecahedron (3,3,3,3,5)
The number of triangle faces is 80
The number of pentagonal-faces is 12
The number of vertices is 60
The number of edges is 150
13. Truncated icosahedron (4,6,10)
The number of square faces is 30
The number of hexagonal-faces is 20
The number of decagonal-faces is 12
The number of vertices is 120
The number of edges is 180
Uniform prism:
These are polyhedrons, whose bottom face and top face are regular and congruent polygons and its sides are squares. (The cube is a special prism.) Let the top and bottom faces are a regular n-gons.
The number of the faces is n+2.
The number of vertices is 2n
The number of edges is 3n
Uniform antiprism:
These are solids whose bottom face and top face are regular polygons and its side faces are regular and congruent triangles. (The octahedron is a special antiprism, whose top face and bottom face are triangles.)
The number of the faces is 2n+2.
The number of vertices is 2n
The number of edges is 4n