Octahedron (EntityTopic, 14)
From Hi.gher. Space
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*The [[hypervolume]]s of a octahedron are given by: | *The [[hypervolume]]s of a octahedron are given by: | ||
<blockquote>total edge length = 12''l''<br> | <blockquote>total edge length = 12''l''<br> | ||
| - | surface area = | + | surface area = 2√3 · ''l''<sup>2</sup><br> |
| - | volume = | + | volume = <sup>√3</sup>⁄<sub>3</sub> · ''l''<sup>3</sup></blockquote> |
*The [[planar]] [[cross-section]]s (''n'') of an octahedron are: | *The [[planar]] [[cross-section]]s (''n'') of an octahedron are: | ||
| - | <blockquote>[!x, !y, !z] ⇒ [[square]] of side ( | + | <blockquote>[!x, !y, !z] ⇒ [[square]] of side (<sup>√2</sup>⁄<sub>2</sub> ''l'' − |''n''|) rotated by 45°</blockquote> |
== Segmentation == | == Segmentation == | ||
| - | The octahedron of side √2 may be [[segment]]ed into 8× irregular [[tetrahedron]] with sides 3×1, | + | The octahedron of side √2 may be [[segment]]ed into 8× irregular [[tetrahedron]] with sides 3×1, 3×√2. |
{{Cross polytopes|3}} | {{Cross polytopes|3}} | ||
{{Trishapes}} | {{Trishapes}} | ||
{{Bracketope Nav|8|9|10|<[xy]z><br>Wide octahedron|<xyz><br>Octahedron|<(xy)z><br>Bicone|hedra}} | {{Bracketope Nav|8|9|10|<[xy]z><br>Wide octahedron|<xyz><br>Octahedron|<(xy)z><br>Bicone|hedra}} | ||
Revision as of 18:24, 12 March 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°
Segmentation
The octahedron of side √2 may be segmented 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 |
| 8. <[xy]z> Wide octahedron | 9. <xyz> Octahedron | 10. <(xy)z> Bicone |
| List of bracketopes | ||
