Octahedron (EntityTopic, 14)
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| image=<[#embed [hash C7W1GZMK7XHGM2W8VHM5RHT3WR] [width 180]]> | | image=<[#embed [hash C7W1GZMK7XHGM2W8VHM5RHT3WR] [width 180]]> | ||
| dim=3 | | dim=3 | ||
| - | | elements=8, 12, 6 | + | | elements=8 [[triangle]]s, 12 [[digon]]s, 6 [[point]]s |
| + | | sym=[[Staurohedral symmetry|O<sub>h</sub>, BC<sub>3</sub>, [4,3], (*432)]] | ||
| genus=0 | | genus=0 | ||
| ssc=<xyz> or {G3<sup>4</sup>} | | ssc=<xyz> or {G3<sup>4</sup>} | ||
| Line 11: | Line 12: | ||
| extra={{STS Bracketope | | extra={{STS Bracketope | ||
| index=5 | | index=5 | ||
| - | | | + | | notation=<III> |
}}{{STS Polytope | }}{{STS Polytope | ||
| flayout={{FLD|a3|er|e4}} | | flayout={{FLD|a3|er|e4}} | ||
| petrie=6,0 | | petrie=6,0 | ||
| dual=[[Cube]] | | dual=[[Cube]] | ||
| + | | bowers=Oct | ||
}}{{STS Uniform polytope | }}{{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{ } | ||
| + | | dynkin=o4o3x, o3x3o | ||
| vlayout=[[Triangle|3]]<sup>4</sup> | | vlayout=[[Triangle|3]]<sup>4</sup> | ||
| vfigure=[[Square]], edge 1 | | vfigure=[[Square]], edge 1 | ||
| - | |||
}}}} | }}}} | ||
| + | The '''octahedron''' is a [[regular polyhedron]] with four [[triangle]]s around each vertex, having 8 triangles in all. However, it can be alternatively constructed as the [[mesotruncate]]d (rectified) [[tetrahedron]], so it is also in the sequence of mesotruncated [[simplices]]. In addition, it is the central vertex-first [[cross-section]] of the [[tesseract]]. | ||
| - | The | + | ==Coordinates== |
| - | + | The coordinates of an octahedron of edge length 2 are all permutations of: | |
| + | <blockquote>(±√2,0, 0)</blockquote> | ||
== Equations == | == Equations == | ||
*The [[hypervolume]]s of a octahedron with side length ''l'' are given by: | *The [[hypervolume]]s of a octahedron with side length ''l'' are given by: | ||
<blockquote>total edge length = 12''l''<br> | <blockquote>total edge length = 12''l''<br> | ||
surface area = 2√3 · ''l''<sup>2</sup><br> | surface area = 2√3 · ''l''<sup>2</sup><br> | ||
| - | volume = <sup> | + | volume = <sup>√2</sup>⁄<sub>3</sub> · ''l''<sup>3</sup></blockquote> |
*The [[planar]] [[cross-section]]s (''n'') of an octahedron with side length ''l'' are: | *The [[planar]] [[cross-section]]s (''n'') of an octahedron with side length ''l'' are: | ||
| Line 36: | Line 40: | ||
== Dissection == | == Dissection == | ||
| - | The octahedron of side √2 may be [[dissect]]ed into 8× irregular [[tetrahedron]] with sides 3×1, 3×√2. | + | The octahedron of side √2 may be [[dissect]]ed into 8× irregular [[tetrahedron]] (triangular pyramid) with sides 3×1, 3×√2. |
| + | |||
| + | <[#polytope [id 3]]> | ||
{{Cross polytopes|3}} | {{Cross polytopes|3}} | ||
{{Trishapes}} | {{Trishapes}} | ||
{{Bracketope Nav|4|5|6|[III]<br>Cube|<nowiki><III></nowiki><br>Octahedron|(III)<br>Sphere|hedra}} | {{Bracketope Nav|4|5|6|[III]<br>Cube|<nowiki><III></nowiki><br>Octahedron|(III)<br>Sphere|hedra}} | ||
Latest revision as of 14:14, 26 March 2017
The octahedron is a regular polyhedron with four triangles around each vertex, having 8 triangles in all. However, it can be alternatively constructed as the mesotruncated (rectified) tetrahedron, so it is also in the sequence of mesotruncated simplices. In addition, it is the central vertex-first cross-section of the tesseract.
Coordinates
The coordinates of an octahedron of edge length 2 are all permutations of:
(±√2,0, 0)
Equations
- The hypervolumes of a octahedron with side length l are given by:
total edge length = 12l
surface area = 2√3 · l2
volume = √2⁄3 · l3
- The planar cross-sections (n) of an octahedron with side length l are:
[!x, !y, !z] ⇒ square of side (√2⁄2 l − |n|) rotated by 45°
Dissection
The octahedron of side √2 may be dissected into 8× irregular tetrahedron (triangular pyramid) with sides 3×1, 3×√2.
Incidence matrix
Dual: cube
| # | TXID | Va | Ea | 3a | Type | Name |
|---|---|---|---|---|---|---|
| 0 | Va | = point | ; | |||
| 1 | Ea | 2 | = digon | ; | ||
| 2 | 3a | 3 | 3 | = triangle | ; | |
| 3 | C1a | 6 | 12 | 8 | = octahedron | ; |
Usage as facets
- 24× 1-facets of a xylochoron
- 5× 1-facets of a pyrorectichoron
- prism: 2× 1-facets of a octahedral prism
- pyramid: 1× 1-facets of a octahedral pyramid
- 144× 1-facets of a (dual of rectified snub demitesseract)
- 24× 1-facets of a rectified snub demitesseract (named tet sym)
- 96× 1-facets of a rectified snub demitesseract (named 3 ap sym)
- 32× 1-facets of a D4.11
- 8× 1-facets of a D4.11
- 48× 1-facets of a D4.11 dual
- 2× 1-facets of a D4.16
- 4× 1-facets of a (4D analog of J37)
- 1× 1-facets of a triangular hebesphenorotunda pseudopyramid (named top to roof)
- 4× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named top to roof)
- 12× 1-facets of a D4.7
- 6× 1-facets of a D4.7 dual
| 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 |
| 4. [III] Cube | 5. <III> Octahedron | 6. (III) Sphere |
| List of bracketopes | ||
