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where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, and hence 6 faces.
The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces.
The number of vertices and edges has remained the same, but the number of faces has been reduced by 1. Therefore, proving Euler's formula for the polyhedron reduces to proving V − E + F = 1 {\displaystyle \ V-E+F=1\ } for this deformed, planar object.
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.
Regular star, great stellated dodecahedron, with regular pentagram faces Degenerate, 12 vertices in the center The concave equilateral dodecahedron, called an endo-dodecahedron. [clarification needed] A cube can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions.
where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, since a cube has 12 edges and 6 faces, the formula implies that it has eight vertices.
The relations can be made apparent by examining the vertex figures obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example, the cube has vertex figure 4.4.4, which is to say, three adjacent square faces. The possible faces are 3 - equilateral ...
The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the null polytope or empty set. These abstract elements can be mapped into ordinary space or realised as geometrical figures. Some abstract polyhedra have well-formed or faithful realisations, others do not.