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Bilunabirotunda

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Bilunabirotunda
TypeJohnson
J90J91J92
Faces8 triangles
2 squares
4 pentagons
Edges26
Vertices14
Vertex configuration4(3.52)
8(3.4.3.5)
2(3.5.3.5)
Symmetry group
Propertiesconvex, elementary
Net
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

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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 .[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 is:[1] and the volume of a bilunabirotunda is:[1]

Cartesian coordinates

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One way to construct a bilunabirotunda with edge length is by union of the orbits of the coordinates under the group's action (of order 8) generated by reflections about coordinate planes.[4]

Applications

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

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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).[6] Such clusters combine with regular dodecahedra to form a space-filling honeycomb.


Spacefilling honeycomb

6 bilunabirotundae around a cube
Animation of tessellation of cubes, dodecahedra and bilunabirotunda

12 bilunabirotundae around a dodecahedron

References

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  1. ^ a b c Berman, M. (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
  2. ^ Francis, D. (August 2013). "Johnson solids & their acronyms". Word Ways. 46 (3): 177.
  3. ^ Cromwell, P. R. (1997). Polyhedra. Cambridge University Press. p. 86–87, 89. ISBN 978-0-521-66405-9.
  4. ^ Timofeenko, A. V. (2009). "The Non-Platonic and Non-Archimedean Noncomposite Polyhedra". Journal of Mathematical Sciences. 162 (5): 710–729. doi:10.1007/s10958-009-9655-0.
  5. ^ Reynolds, M. A. (2004). "The Bilunabirotunda". Nexus Network Journal. 6: 43–47. doi:10.1007/s00004-004-0005-8.
  6. ^ 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) ISBN 978-0686119364, (page 127, 2nd ed.) polyhedron 6J91(P4).
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