Octahedron (EntityTopic, 14)
From Hi.gher. Space
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**when oriented such that the vertices are points on the [[Cartesian axes]]: | **when oriented such that the vertices are points on the [[Cartesian axes]]: | ||
<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> | ||
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{{Polyhedra}} | {{Polyhedra}} | ||
| + | {{Bracketope|8|9|10|<[xy]z><br>Wide octahedron|<xyz><br>Octahedron|<(xy)z><br>Bicone}} | ||
Revision as of 16:50, 19 June 2007
Geometry
Equations
- Assumption: Octahedron is centered at the origin.
- 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:
- when oriented such that the vertices are points on the Cartesian axes:
[!x, !y, !z] ⇒ square of side (sqrt(2)-1l-abs(n)) rotated by 45°
| 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 |
