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
- Variables:
l ⇒ length of edges of the octahedron
- The hypervolumes of a octahedron are given by:
total edge length = 12l
surface area = 2√3 · l2
volume = √3⁄3 · l3
- The planar cross-sections (n) of an octahedron are:
[!x, !y, !z] ⇒ square of side (√2⁄2 l − |n|) rotated by 45°
Dissection
The octahedron of side √2 may be dissected into 8× irregular tetrahedron with sides 3×1, 3×√2.
| Cross polytopes |
| diamond • octahedron • aerochoron • aeroteron • aeropeton |
| Notable Trishapes | |
| Regular: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
| Strict truncates: | tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |
| Lax truncates: | cuboctahedron • icosidodecahedron • cuboctahedral rectate • icosidodecahedral rectate • cuboctahedral truncate • icosidodecahedral truncate |
| Snubs: | cubic snub • dodecahedral snub |
| Curved: | sphere • torus • cylinder • cone • frustum • crind |
| 4. [III] Cube | 5. <III> Octahedron | 6. (III) Sphere |
| List of bracketopes | ||
