Bilunabirotunda
Type	Johnson J90 – J91 – J92
Faces	8 triangles 2 squares 4 pentagons
Edges	26
Vertices	14
Vertex configuration	4(3.52) 8(3.4.3.5) 2(3.5.3.5)
Symmetry group	${\displaystyle \mathrm {D} _{2\mathrm {h} }}$
Properties	convex, elementary
Net

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

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.^[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] It is known as the elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.^[3]

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] $${\displaystyle {\frac {17+9{\sqrt {5}}}{12}}a^{3}\approx 3.0937a^{3}.}$$

One way to construct a bilunabirotunda with edge length ${\displaystyle {\sqrt {5}}-1}$ is by union of the orbits of the coordinates $${\displaystyle (0,0,1),\left({\frac {{\sqrt {5}}-1}{2}},1,{\frac {{\sqrt {5}}-1}{2}}\right),\left({\frac {{\sqrt {5}}-1}{2}},{\frac {{\sqrt {5}}+1}{2}}\right).}$$ under the group's action (of order 8) generated by reflections about coordinate planes.^[4]

Reynolds (2004) discusses the bilunabirotunda as a shape that could be used in architecture.^[5]

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

Spacefilling honeycomb

6 bilunabirotundae around a cube

12 bilunabirotundae around a dodecahedron

• Weisstein, Eric W., "Bilunabirotunda" ("Johnson solid") at MathWorld.
• Miracle Spacefilling (Dodecahedron&Cube&Johnson solid No.91)