Octahedron

A (general) octahedron is a polyhedron having eight faces. Examples include the 4-trapezohedron, augmented triangular prism (Johnson solid ), bislit cube, Dürer solid, elongated gyrobifastigium, gyrobifastigium (Johnson solid ), heptagonal pyramid, hexagonal prism, regular octahedron, square dipyramid, triangular cupola (Johnson solid ), tridiminished icosahedron (Johnson solid ), tritetrahedron, and truncated tetrahedron.

There are 257 convex octahedra, corresponding to the duals of the octahedral graphs. The convex octahedra consisting of regular polygonal faces of equal edge lengths are summarized in the following table. They all have , as required by the polyhedral formula.

polyhedron	degree sequence
truncated tetrahedron	3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3	12	18
heptagonal pyramid	3, 3, 3, 3, 3, 3, 3, 7	8	14
triangular cupola	3, 3, 3, 3, 3, 3, 4, 4, 4	9	15
tridiminished icosahedron	3, 3, 3, 3, 3, 3, 4, 4, 4	9	15
gyrobifastigium	3, 3, 3, 3, 4, 4, 4, 4	8	14
augmented triangular prism	3, 3, 4, 4, 4, 4, 4	7	13
octahedron	4, 4, 4, 4, 4, 4	6	12

The regular octahedron, often simply called "the" octahedron, is the Platonic solid with six polyhedron vertices, 12 polyhedron edges, and eight equivalent equilateral triangular faces, denoted . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and Wythoff symbol . The octahedron of unit side length is the antiprism of sides with height . The octahedron is also a square dipyramid with equal edge lengths.