Bilunabirotunda

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.^[1] It is one of the Johnson solids—a convex polyhedron in which all of the faces are regular polygon—enumerated as 91st Johnson solid ${\displaystyle J_{91}}$.^[2]

The surface area of a bilunabirotunda with edge length ${\displaystyle a}$ is:^[1] $${\displaystyle \left(2+2{\sqrt {3}}+{\sqrt {5(5+2{\sqrt {5}})}}\right)a^{2}\approx 12.346a^{2},}$$ and the volume of a bilunabirotunda is:^[1] $${\displaystyle {\frac {17+9{\sqrt {5}}}{12}}a^{3}\approx 3.0937a^{3}.}$$

The bilunabirotunda is an elementary polyhedron: it cannot be separated by a plane into two small regular-faced polyhedra.^[3] One way to construct a bilunabirotunda is by attaching two wedges and two tridiminished icosahedrons.^[4]

For edge length ${\displaystyle {\sqrt {5}}-1}$, the coordinates of the bilunabirotunda may be given as^[5]$${\displaystyle {\begin{aligned}&(0,0,\pm 1),\\&\left(\pm {\frac {{\sqrt {5}}-1}{2}},\pm 1,\pm {\frac {{\sqrt {5}}-1}{2}}\right),\\&\left(\pm {\frac {{\sqrt {5}}+1}{2}},\pm {\frac {{\sqrt {5}}-1}{2}},0\right).\end{aligned}}}$$

Reynolds (2004) discusses the bilunabirotunda as a shape that could be used in architecture.^[6]

Six bilunabirotundae can be augmented around a cube with pyritohedral symmetry. B. M. Stewart labeled this six-bilunabirotunda model as 6J₉₁(P₄).^[7] Such clusters combine with regular dodecahedra to form a space-filling honeycomb.