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).

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]

It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Either one of the two clusters of two pentagons and two triangles can be aligned with a congruent patch of faces on the icosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the icosidodecahedron, then two vertices from the bilunabirotundae will meet in the very center of the icosidodecahedron.

The other two clusters of faces of the bilunabirotunda, each with one square and two triangles, can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron.

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 six-bilunabirotunda 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)