Octahedron (EntityTopic, 14)
From Hi.gher. Space
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The '''octahedron''' is a [[regular polyhedron]] with four [[triangle]]s around each vertex. However, it can be alternatively constructed as the [[mesotruncate]]d [[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 '''octahedron''' is a [[regular polyhedron]] with four [[triangle]]s around each vertex. However, it can be alternatively constructed as the [[mesotruncate]]d [[tetrahedron]], so it is also in the sequence of mesotruncated [[simplices]]. In addition, it is the central vertex-first [[cross-section]] of the [[tesseract]]. | ||
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== 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]] with sides 3×1, 3×√2. | ||
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{{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}} | ||
Revision as of 22:08, 16 February 2014
The octahedron is a regular polyhedron with four triangles around each vertex. However, it can be alternatively constructed as the mesotruncated tetrahedron, so it is also in the sequence of mesotruncated simplices. In addition, it is the central vertex-first cross-section of the tesseract.
Equations
- The hypervolumes of a octahedron with side length l are given by:
total edge length = 12l
surface area = 2√3 · l2
volume = √3⁄3 · 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 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 | ||
