Bilunabirotunda (EntityTopic, 14)

From Hi.gher. Space

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