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, ±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

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