</math>

| vertex_config =

| properties = convex, elementary

| net = Johnson solid 91 net.png

thumb|3D model of a 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 <math> J_{91} </math>.

The surface area of a bilunabirotunda with edge length <math> a </math> is:

<math display="block"> \left(2 + 2\sqrt{3} + \sqrt{5(5 + 2\sqrt{5})}\right)a^2 \approx 12.346a^2, </math>

and the volume of a bilunabirotunda is:

<math display="block"> \frac{17 + 9\sqrt{5{12}a^3 \approx 3.0937a^3. </math>

Construction

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 <math> \sqrt{5} - 1 </math> is by union of the orbits of the coordinates, the bilunabirotunda is:

<math display="block"> (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}, 0 \right). </math>

under the group action (of order 8) generated by reflections about coordinate planes.

Applications

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

200px|thumb|Six bilunabirotundae around a cube

Six bilunabirotundae can be augmented around a cube with pyritohedral symmetry. B. M. Stewart labeled this six-bilunabirotunda model as 6J<sub>91</sub>(P<sub>4</sub>). Such clusters combine with regular dodecahedra to form a space-filling honeycomb.

References

</references>

  • Miracle Spacefilling (Dodecahedron&Cube&Johnson solid No.91)