Octahedron (EntityTopic, 14)

From Hi.gher. Space

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.


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

(±√2,0, 0)


  • 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°


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

Incidence matrix

Dual: cube

0 Va = point ;
1 Ea 2 = digon ;
2 3a 33 = triangle ;
3 C1a 6128 = octahedron ;

Usage as facets

Cross polytopes

Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind

4. [III]
5. <III>
6. (III)
List of bracketopes