In geometry, the bilunabirotunda is a Johnson solid with faces of 8 equilateral triangles, 2 squares, and 4 regular pentagons.
Properties The bilunabirotunda consists of two (bi‑) lunes, each consisting of a square between two triangles, and two (bi‑) partial rotundae, in which two pentagons and two triangles alternate around a single vertex as in the pentagonal rotunda. Therefore, the faces of a bilunabirotunda possess 8 equilateral triangles, 2 squares, and 4 regular pentagons as it faces. It is one of the Johnson solids—a convex polyhedron in which all of the faces are regular polygon—enumerated as 91st Johnson solid .
The surface area of a bilunabirotunda with edge length is:
and the volume of a bilunabirotunda is:
Construction The bilunabirotunda is an elementary polyhedron: it cannot be separated by a plane into two small regular-faced polyhedra. One way to construct a bilunabirotunda is by attaching two wedges and two tridiminished icosahedrons.
For edge length , the coordinates of the bilunabirotunda may be given as
Applications discusses the bilunabirotunda as a shape that could be used in architecture.
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).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) , (page 127, 2nd ed.) polyhedron 6J91(P4). Such clusters combine with regular dodecahedra to form a space-filling honeycomb.
References
External links
Miracle Spacefilling (Dodecahedron&Cube&Johnson solid No.91)