Bilunabirotunda
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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 <math> J_{91} </math>.Template:R
The surface area of a bilunabirotunda with edge length <math> a </math> is:Template:R <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:Template:R <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.Template:R One way to construct a bilunabirotunda is by attaching two wedges and two tridiminished icosahedrons.Template:R
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.Template:R
Applications
Template:Harvtxt discusses the bilunabirotunda as a shape that could be used in architecture.Template:R
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).<ref>B. M. Stewart, Adventures Among the Toroids: A Study of Quasi-Convex, Aplanar, Tunneled Orientable Polyhedra of Positive Genus Having Regular Faces With Disjoint Interiors (1980) Template:ISBN, (page 127, 2nd ed.) polyhedron 6J91(P4).</ref> Such clusters combine with regular dodecahedra to form a space-filling honeycomb.