Bilunabirotunda (EntityTopic, 14)
From Hi.gher. Space
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where φ=(1+√5)/2 is the Golden Ratio. | where φ=(1+√5)/2 is the Golden Ratio. | ||
+ | |||
+ | == Construction from icosahedron == | ||
+ | |||
+ | The bilunabirotunda can be constructed from an icosahedron, as follows: | ||
+ | |||
+ | Firstly, write the icosahedron in [2,2,2]-symmetry: | ||
+ | |||
+ | x2o2f | ||
+ | f2x2o | ||
+ | o2f2x | ||
+ | |||
+ | Then, apply a caleido-faceting to the last node: | ||
+ | |||
+ | x2o2f -> x2o2f | ||
+ | f2x2o -> f2x2o | ||
+ | o2f2x -> o2f2(-x) | ||
+ | |||
+ | Finally, apply a Stott-expansion to the last node: | ||
+ | |||
+ | x2o2f -> x2o2F | ||
+ | f2x2o -> f2x2x | ||
+ | o2f2(-x) -> o2f2o | ||
+ | |||
+ | The result is J91, written in [2,2,2]-symmetry: xfo2oxf2Fxo&#zx. | ||
+ | |||
+ | You can compare this representation with the coordinates as given above. | ||
== Equations == | == Equations == |
Latest revision as of 10:38, 2 January 2018
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.
Construction from icosahedron
The bilunabirotunda can be constructed from an icosahedron, as follows:
Firstly, write the icosahedron in [2,2,2]-symmetry:
x2o2f f2x2o o2f2x
Then, apply a caleido-faceting to the last node:
x2o2f -> x2o2f f2x2o -> f2x2o o2f2x -> o2f2(-x)
Finally, apply a Stott-expansion to the last node:
x2o2f -> x2o2F f2x2o -> f2x2x o2f2(-x) -> o2f2o
The result is J91, written in [2,2,2]-symmetry: xfo2oxf2Fxo&#zx.
You can compare this representation with the coordinates as given above.
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 |