geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the bilunabirotunda is one of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s ().

Geometry

It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the

Platonic Plato's influence on Western culture was so profound that several different concepts are linked by being called Platonic or Platonist, for accepting some assumptions of Platonism, but which do not imply acceptance of that philosophy as a whole. It ...

and Archimedean solids. However, it does have a strong relationship to the

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 id ...

, an Archimedean solid. Either one of the two clusters of two pentagons and two triangles can be aligned with a congruent patch of faces on the icosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the icosidodecahedron, then two vertices of the bilunabirotundae meet in the very center of the icosidodecahedron. The other two clusters of faces of the bilunabirotunda, the ''lunes'' (each ''lune'' featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the

rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square (geometry), square face ...

. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron. Each of the two pairs of adjacent pentagons (each pair of pentagons sharing an edge) can be aligned with the pentagonal faces of a

metabidiminished icosahedron In geometry, the metabidiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing two pentagonal pyramids () from a regular icosahedron, replacing two sets of five triangular faces of the ...

as well. The bilunabirotunda has a weak relationship with the

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

, as it may be created by replacing four square faces of the cuboctahedron with pentagons. Bilunabirrotunda

Cartesian coordinates

The following define the vertices of a bilunabirotunda centered at the origin with edge length 1: :

\left(0, 0, \pm\frac\right)

\left(\pm\frac, \pm\frac, 0\right)

\left(\pm\frac,\pm\frac,\pm\frac\right)

where

\varphi=\frac

is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

Related polyhedra and honeycombs

Six bilunabirotundae can be augmented around a cube with

pyritohedral symmetry 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that combine a reflection a ...

. B. M. Stewart labeled this six-bilunabirotunda model as 6J₉₁(P₄). 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) , (page 127, 2nd ed.) polyhedron 6J₉₁(P₄). The bilunabirotunda can be used with the regular dodecahedron and cube as a space-filling honeycomb.

External links

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Miracle Spacefilling (Dodecahedron&Cube&Johnson solid No.91)
Johnson solids {{Johnson solids navigator