# Bilunabirotunda (EntityTopic, 14)

(Redirected from Bilbiro)

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 = 16 · (4φ2 + 5φ) · l3

## Incidence matrix

Dual: J91 dual

 # TXID Type Name Va Vb Vc Ea Eb Ec Ed Ee 3a 3b 4a 5a 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

 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