Bilunabirotunda

Bilunabirotunda
Type	Johnson J90 - J91 - J92
Faces	2x4 triangles 2 squares 4 pentagons
Edges	26
Vertices	14
Vertex configuration	4(3.5^2) 8(3.4.3.5) 2(3.5.3.5)
Symmetry group	D2h
Dual polyhedron	-
Properties	convex
Net

In geometry, the bilunabirotunda is one of the Johnson solids (J₉₁). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.^[1]

Cartesian coordinates

The following define the vertices of a bilunabirotunda centered at the origin with edge length 1:

$\left(0, 0, \pm\frac{\varphi}{2}\right)$

$\left(\pm\frac{(\varphi+1)}{2}, \pm\frac{1}{2}, 0\right)$

$\left(\pm\frac{1}{2},\pm\frac{\varphi}{2},\pm\frac{1}{2}\right)$

where $\varphi=\frac{1+\sqrt{5}}{2}$ is the golden ratio.

Related polyhedra and honeycombs

Six bilunabirotundae can be augmented around a cube with pyritohedral symmetry. B. M. Stewart labeled this 6 bilunabirotundae model as 6J₉₁(P4).^[2]

The bilunabirotunda can be used with the regular dodecahedron and cube as a space-filling honeycomb.

Spacefilling honeycomb

6 bilunabirotundae around a cube

External links

1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. B. M. Stewart, Adventures Among the Toroids: A Study of Quasi-Convex, Aplanar, Tunneled Orientable Polyhedra of Positive Genus Having Regular Faces With Disjoint Interiors (1980) ISBN 978-0686119364, (page 127, 2nd ed.) polyhedron 6J₉₁(P4).

• Eric W. Weisstein, Bilunabirotunda (Johnson solid) at MathWorld
• Miracle Spacefilling (Dodecahedron&Cube&Johnson solid No.91)