Bilunabirotunda Explained

Type:	Johnson
Faces:	8 triangles 2 squares 4 pentagons
Edges:	26
Vertices:	14
Symmetry:	D _{2h}
Net:	Johnson solid 91 net.png

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 - a convex polyhedron in which all of the faces are regular polygon - 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:$$\backslash left(2\; +\; 2\backslash sqrt\; +\; \backslash sqrt\backslash right)a^2\; \backslash approx\; 12.346a^2,$$and the volume of a bilunabirotunda is:$$\backslash fraca^3\; \backslash approx\; 3.0937a^3.$$

Cartesian coordinates

One way to construct a bilunabirotunda with edge length

\sqrt{ 5 } - 1 is by union of the orbits of the coordinates$$(0,\; 0,\; 1),\; \backslash left(\backslash frac,\; 1,\; \backslash frac\; \backslash right),\; \backslash left(\backslash frac,\; \backslash frac\; \backslash right).$$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).^[1] Such clusters combine with regular dodecahedra to form a space-filling honeycomb.

External links

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

Notes and References

1. [Bonnie Stewart|B. M. Stewart]