# Bilunabirotunda (EntityTopic, 14)

### From Hi.gher. Space

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, ±φ
^{2}> - <±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)) ·l^{2}

volume =^{1}∕_{6}· (4φ^{2}+ 5φ) ·l^{3}

## 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 |