Octahedron (EntityTopic, 14)
From Hi.gher. Space
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| - | {{Shape | + | {{STS Shape |
| - | + | ||
| name=Octahedron | | name=Octahedron | ||
| image=http://img473.imageshack.us/img473/3851/octahedron3vn.png | | image=http://img473.imageshack.us/img473/3851/octahedron3vn.png | ||
| Line 6: | Line 5: | ||
| elements=8, 12, 6 | | elements=8, 12, 6 | ||
| genus=0 | | genus=0 | ||
| - | |||
| ssc=<xyz> or {G3<sup>4</sup>} | | ssc=<xyz> or {G3<sup>4</sup>} | ||
| - | |||
| - | |||
| pv_circle=<sup>1</sup>⁄<sub>π</sub> ≈ 0.3183 | | pv_circle=<sup>1</sup>⁄<sub>π</sub> ≈ 0.3183 | ||
| + | | extra={{STS Bracketope | ||
| + | | index=9 | ||
| + | | bracket=<xyz> | ||
| + | }}{{STS Uniform polytope | ||
| wythoff=<nowiki>4 | 2 3, 2 | 3 3, or | 2 2 3 |</nowiki> | | wythoff=<nowiki>4 | 2 3, 2 | 3 3, or | 2 2 3 |</nowiki> | ||
| schlaefli={[[Triangle|3,]]4}, r{3,3}, sr{2,3} or s{3}h{ } | | schlaefli={[[Triangle|3,]]4}, r{3,3}, sr{2,3} or s{3}h{ } | ||
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| kana=ｵｸ | | kana=ｵｸ | ||
| dual=[[Cube]] | | dual=[[Cube]] | ||
| - | }} | + | }}}} |
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== Equations == | == Equations == | ||
*Variables: | *Variables: | ||
Revision as of 14:30, 14 March 2008
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 | ||
