Search results
Results from the WOW.Com Content Network
The (n − 3)-faces of an n-polytope are called peaks. A peak contains a rotational axis of facets and ridges in a regular polytope or honeycomb. For example: The peaks of a 3D polyhedron or plane tiling are its 0-faces or vertices. The peaks of a 4D polytope or 3-honeycomb are its 1-faces or edges.
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 edges is 2 less than the sum of the numbers of vertices and faces. For example, a cube has 8 vertices and 6 faces, and hence 12 edges.
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.
In a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa. The regular polyhedra show this duality as follows: The tetrahedron is self-dual, i.e. it pairs with itself. The cube and octahedron are dual to each other. The icosahedron and dodecahedron are dual to each other.
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.
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 hemi-cube is a regular projective polyhedron with 3 square faces, 6 edges, and 4 vertices. The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of even dihedra and hosohedra: [4] Hemi-cube, {4,3}/2; Hemi-octahedron, {3,4}/2
Faces are reduced to half as many sides, and square faces degenerate into edges. For example, the tetrahedron is an alternated cube, h{4,3}. Diminishment is a more general term used in reference to Johnson solids for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices.