Octahedron (EntityTopic, 14)
From Hi.gher. Space
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*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 (sqrt(2)<sup>-1</sup>''l''-abs(''n'')) rotated by 45°</blockquote> | <blockquote>[!x, !y, !z] ⇒ [[square]] of side (sqrt(2)<sup>-1</sup>''l''-abs(''n'')) rotated by 45°</blockquote> | ||
| + | |||
| + | == Segmentation == | ||
| + | The octahedron of side √2 may be [[segment]]ed into 8× irregular [[tetrahedron]] with sides 3×1, 3×2<sup>2<sup>-1</sup></sup>. | ||
{{Trishapes}} | {{Trishapes}} | ||
Revision as of 17:32, 16 September 2007
Geometry
Equations
- Variables:
l ⇒ length of edges of the octahedron
- The hypervolumes of a octahedron are given by:
total edge length = 12l
surface area = 2sqrt(3)l2
volume = sqrt(3)-1l3
- The planar cross-sections (n) of an octahedron are:
[!x, !y, !z] ⇒ square of side (sqrt(2)-1l-abs(n)) rotated by 45°
Segmentation
The octahedron of side √2 may be segmented into 8× irregular tetrahedron with sides 3×1, 3×22-1.
| 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 | ||
