Octahedron (EntityTopic, 14)
From Hi.gher. Space
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<blockquote>[!x, !y, !z] ⇒ [[square]] of side (<sup>√2</sup>⁄<sub>2</sub> ''l'' − |''n''|) rotated by 45°</blockquote> | <blockquote>[!x, !y, !z] ⇒ [[square]] of side (<sup>√2</sup>⁄<sub>2</sub> ''l'' − |''n''|) rotated by 45°</blockquote> | ||
| - | == | + | == Dissection == |
| - | The octahedron of side √2 may be [[ | + | The octahedron of side √2 may be [[dissect]]ed into 8× irregular [[tetrahedron]] with sides 3×1, 3×√2. |
{{Cross polytopes|3}} | {{Cross polytopes|3}} | ||
{{Trishapes}} | {{Trishapes}} | ||
{{Bracketope Nav|4|5|6|[III]<br>Cube|<nowiki><III></nowiki><br>Octahedron|(III)<br>Sphere|hedra}} | {{Bracketope Nav|4|5|6|[III]<br>Cube|<nowiki><III></nowiki><br>Octahedron|(III)<br>Sphere|hedra}} | ||
Revision as of 15:08, 21 November 2011
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 |
| 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. <III> Octahedron | 6. (III) Sphere |
| List of bracketopes | ||
