Octahedron (EntityTopic, 14)
From Hi.gher. Space
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| - | {{Shape|Octahedron|http://img473.imageshack.us/img473/3851/octahedron3vn.png|3|8, 12, 6|0|{[[Triangle|3,]]4}, r{3,3}, sr{2,3}, or s{3}h{ }|<nowiki>4 | 2 3, 2 | 3 3, or | 2 2 3 |</nowiki>|[[Line (object)|E]][[Square|D]]D|N/A|[[Square]], edge 1|Oct|[[Cube]]|N/A|<xyz>| | + | {{Shape|Octahedron|http://img473.imageshack.us/img473/3851/octahedron3vn.png|3|8, 12, 6|0|{[[Triangle|3,]]4}, r{3,3}, sr{2,3}, or s{3}h{ }|<nowiki>4 | 2 3, 2 | 3 3, or | 2 2 3 |</nowiki>|[[Line (object)|E]][[Square|D]]D|N/A|[[Square]], edge 1|Oct|[[Cube]]|N/A|<xyz>|9}} |
== Geometry == | == Geometry == | ||
=== Equations === | === Equations === | ||
Revision as of 13:40, 18 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 |
