TOPICS
Octahedron
A (general) octahedron is apolyhedron 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, andtruncated tetrahedron.
There are 257 convex octahedra, corresponding to the duals of theoctahedral 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 thepolyhedral 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 |
Theregular octahedron, often simply called "the" octahedron, is thePlatonic solid
with sixpolyhedron vertices, 12polyhedron edges, and eight equivalentequilateral triangular faces, denoted
. It is alsouniform polyhedron
and Wenninger model
. It is given by theSchläfli symbol
andWythoff symbol
. The octahedron of unit side length is theantiprism of
sides with height
. The octahedron is also asquare dipyramid with equal edge lengths.
See also
Dürer's Solid,Escher's Solid,Icosahedron,Jumping Octahedron,Octahedral Graph,Octahedral Group,Regular Octahedron,Stella Octangula,Tritetrahedron,Truncated OctahedronExplore this topic in the MathWorld classroomExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Octahedron." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/Octahedron.html