Bilunabirotunda

Template:Short description Template:Infobox polyhedron File:J91 bilunabirotunda.stl

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

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

Cartesian coordinates

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

Applications

Template:Harvtxt discusses the bilunabirotunda as a shape that could be used in architecture.Template:R

Related polyhedra and honeycombs

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Six bilunabirotundae can be augmented around a cube with pyritohedral symmetry. B. M. Stewart labeled this six-bilunabirotunda model as 6J₉₁(P₄).^[1] Such clusters combine with regular dodecahedra to form a space-filling honeycomb.

	Spacefilling honeycomb	6 bilunabirotundae around a cube	File:3D space tessellation of cubes and dodecahedra.ogg 12 bilunabirotundae around a dodecahedron