# Octahedron (EntityTopic, 14)

The octahedron is a regular polyhedron with four triangles around each vertex, having 8 triangles in all. However, it can be alternatively constructed as the mesotruncated (rectified) tetrahedron, so it is also in the sequence of mesotruncated simplices. In addition, it is the central vertex-first cross-section of the tesseract.

## Coordinates

The coordinates of an octahedron of edge length 2 are all permutations of:

(±√2,0, 0)

## Equations

• The hypervolumes of a octahedron with side length l are given by:
total edge length = 12l
surface area = 2√3 · l2
volume = √23 · l3
[!x, !y, !z] ⇒ square of side (√22 l − |n|) rotated by 45°

## Dissection

The octahedron of side √2 may be dissected into 8× irregular tetrahedron (triangular pyramid) with sides 3×1, 3×√2.

## Incidence matrix

Dual: cube

 # TXID Type Name Va Ea 3a 0 Va = point ; 1 Ea 2 = digon ; 2 3a 3 3 = triangle ; 3 C1a 6 12 8 = octahedron ;

## Usage as facets

 Cross polytopes diamond • octahedron • aerochoron • aeroteron • aeropeton

 Notable Trishapes Regular: tetrahedron • cube • octahedron • dodecahedron • icosahedron Direct truncates: tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate Mesotruncates: stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron Snubs: snub staurohedron • snub rhodohedron Curved: sphere • torus • cylinder • cone • frustum • crind

 4. [III]Cube 5. Octahedron 6. (III)Sphere List of bracketopes