Bilunabirotunda (EntityTopic, 14)
From Hi.gher. Space
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== Coordinates == | == Coordinates == | ||
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The following coordinates give an origin-centered bilunabirotunda with edge length 2: | The following coordinates give an origin-centered bilunabirotunda with edge length 2: | ||
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surface area = (2 + 2√3 + √(25+10√5)) {{DotHV}}<br> | surface area = (2 + 2√3 + √(25+10√5)) {{DotHV}}<br> | ||
volume = {{Over|1|6}} · (4φ<sup>2</sup> + 5φ) {{DotHV|3}}</blockquote> | volume = {{Over|1|6}} · (4φ<sup>2</sup> + 5φ) {{DotHV|3}}</blockquote> | ||
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| + | <[#polytope [id 66]]> | ||
{{Trishapes}} | {{Trishapes}} | ||
Revision as of 00:45, 23 February 2014
The bilunabirotunda is the 91st Johnson solid, J91. It suddenly became important when the castellated rhodoperihedral prism was discovered.
Coordinates
The following coordinates give an origin-centered bilunabirotunda with edge length 2:
- <±1, 0, ±φ2>
- <±φ, ±1, ±1>
- <0, ±φ, 0>
where φ=(1+√5)/2 is the Golden Ratio.
Equations
- The hypervolumes of a bilunabirotunda with side length l are given by:
total edge length = 26l
surface area = (2 + 2√3 + √(25+10√5)) · l2
volume = 1∕6 · (4φ2 + 5φ) · l3
Incidence matrix
Dual: J91 dual
| # | TXID | Va | Vb | Vc | Ea | Eb | Ec | Ed | Ee | 3a | 3b | 4a | 5a | Type | Name |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | Va | = point | ; cuboid corners | ||||||||||||
| 1 | Vb | = point | ; crosses | ||||||||||||
| 2 | Vc | = point | ; ends | ||||||||||||
| 3 | Ea | 2 | 0 | 0 | = digon | ; vertical cuboid | |||||||||
| 4 | Eb | 2 | 0 | 0 | = digon | ; horizontal cuboid | |||||||||
| 5 | Ec | 1 | 1 | 0 | = digon | ; cross-to-cuboid | |||||||||
| 6 | Ed | 1 | 0 | 1 | = digon | ; end-to-cuboid | |||||||||
| 7 | Ee | 0 | 0 | 2 | = digon | ; ends | |||||||||
| 8 | 3a | 2 | 1 | 0 | 1 | 0 | 2 | 0 | 0 | = triangle | ; cross | ||||
| 9 | 3b | 2 | 0 | 1 | 0 | 1 | 0 | 2 | 0 | = triangle | ; end | ||||
| 10 | 4a | 4 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | = square | ; | ||||
| 11 | 5a | 2 | 1 | 2 | 0 | 0 | 2 | 2 | 1 | = pentagon | ; | ||||
| 12 | C1a | 8 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | 4 | 4 | 2 | 4 | = bilunabirotunda | ; |
Usage as facets
- 30× 1-facets of a castellated rhodoperihedral prism
- 24× 1-facets of a D4.11
- 1× 1-facets of a bilunabirotunda pseudopyramid
- 30× 1-facets of a castellated rhodopantohedral prism
| 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 |
