Octahedron (EntityTopic, 14)

From Higher Dimensions Database

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

Equations

  • The hypervolumes of a octahedron with side length l are given by:
total edge length = 12l
surface area = 2√3 · l2
volume = √33 · 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 with sides 3×1, 3×√2.

Incidence matrix

Dual: cube

#TXIDVaEa3aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 3a 33 = triangle ;
3 C1a 6128 = octahedron ;

Usage as facets


Cross polytopes
diamondoctahedronaerochoronaeroteronaeropeton


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


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