geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the bilunabirotunda is a

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

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 triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

s, 2

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s, and 4

regular pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

s as it faces. It is one of the

s—a

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

polyhedron in which all of the faces are

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

—enumerated as 91st Johnson solid

J_

. The surface area of a bilunabirotunda with edge length

a

is:

\left(2 + 2\sqrt + \sqrt\right)a^2 \approx 12.346a^2,

and the volume of a bilunabirotunda is:

\fraca^3 \approx 3.0937a^3.

Construction

The bilunabirotunda is an

elementary polyhedron In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedrons when sliced by a plane. Examples can be found in Johnson solids. Definition and examples A convex polyhedron is said to be ''composite'' if there exists ...

: it cannot be separated by a plane into two small regular-faced polyhedra. One way to construct a bilunabirotunda is by attaching two

wedge A wedge is a triangle, triangular shaped tool, a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by conver ...

s and two

tridiminished icosahedron In geometry, the tridiminished icosahedron is a Johnson solid that is constructed by removing three pentagonal pyramids from a regular icosahedron. Construction The tridiminished icosahedron can be constructed by removing three regular pentago ...

s. For edge length

\sqrt - 1

is by union of the orbits of the

coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...

, the bilunabirotunda is:

(0, 0, 1), \left( \frac, 1, \frac \right), \left( \frac, \frac, 0 \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 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). Such clusters combine with regular dodecahedra to form a space-filling honeycomb.

References

External links

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