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&mdash;a convex polyhedron in which all of the faces are regular polygon&mdash;enumerated as 91st Johnson solid $$ J_{91} $$. It is known as elementary, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.

The surface area of a bilunabirotunda with edge length $$ a $$ is: $$ \left(2 + 2\sqrt{3} + \sqrt{5(5 + 2\sqrt{5})}\right)a^2 \approx 12.346a^2, $$ and the volume of a bilunabirotunda is: $$ \frac{17 + 9\sqrt{5}}{12}a^3 \approx 3.0937a^3. $$

Cartesian coordinates
One way to construct a bilunabirotunda with edge length $$ \sqrt{5} - 1 $$ is by union of the orbits of the coordinates $$ (0, 0, 1), \left( \frac{\sqrt{5} - 1}{2}, 1, \frac{\sqrt{5} - 1}{2} \right), \left( \frac{\sqrt{5} - 1}{2}, \frac{\sqrt{5} + 1}{2} \right). $$ under the group's action (of order 8) generated by reflections about coordinate planes.

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). Such clusters combine with regular dodecahedra to form a space-filling honeycomb.