Bilunabirotunda

In geometry, the bilunabirotunda is a Johnson solid with faces of 8 equilateral triangles, 2 squares, and 4 regular pentagons.

Properties

The bilunabirotunda is named from the prefix lune, meaning a figure featuring two triangles adjacent to opposite sides of a square. 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:

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 is by union of the orbits of the coordinates, the bilunabirotunda is:
under the group action generated by reflections about coordinate planes.

discusses the bilunabirotunda as a shape that could be used in architecture.

Six bilunabirotundae can be augmented around a cube with pyritohedral symmetry. B. M. Stewart labeled this six-bilunabirotunda model as 6J₉₁. Such clusters combine with regular dodecahedra to form a space-filling honeycomb.