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 trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is 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 polytope ...

dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...

with 6 rhombic and 6

trapezoidal A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eucli ...

faces. It has symmetry. A concave form can be constructed with an identical net, seen as excavating

trigonal trapezohedra In geometry, a trigonal trapezohedron is a rhombohedron (a polyhedron with six rhombus-shaped faces) in which, additionally, all six faces are congruent. Alternative names for the same shape are the ''trigonal deltohedron'' or ''isohedral rhomboh ...

from the top and bottom.

Construction

This polyhedron could be constructed by taking a tall uniform

hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.. Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used ...

, and making 3 angled cuts on the top and bottom. The trapezoids represent what remains of the original prism sides, and the 6 rhombi a result of the top and bottom cuts.

Space-filling tessellation

A space-filling tessellation, the

trapezo-rhombic dodecahedral honeycomb The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal ...

, can be made by translated copies of this cell. Each "layer" is a hexagonal tiling, or a

rhombille tiling In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape a ...

, and alternate layers are connected by shifting their centers and rotating each polyhedron so the rhombic faces match up. : Trapezo-rhombic_dodecahedron_honeycomb

Trapezo-rhombic_dodecahedron-concave_honeycomb

In the special case that the long sides of the trapezoids equals twice the length of the short sides, the solid now represents the 3D

Voronoi cell In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed t ...

of a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

in a Hexagonal Close Packing (HCP), next to Face-Centered-Cubic an optimal way to stack spheres in a lattice. It is therefore similar to the

rhombic dodecahedron In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahedro ...

, which can be represented by turning the lower half of the picture at right over an angle of 60 degrees. The rhombic dodecahedron is a Voronoi cell of the other optimal way to stack spheres. As the Voronoi cell of a regular space pattern, it is a

plesiohedron In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. Th ...

. It is the polyhedral dual of the

triangular orthobicupola In geometry, the triangular orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by attaching two triangular cupolas () along their bases. It has an equal number of squares and triangles at each vertex; howev ...

Variations

The ''trapezo-rhombic dodecahedron'' can be seen as an elongation of another dodecahedron, which can be called a rhombo-triangular dodecahedron, with 6 rhombi (or squares) and 6 triangles. It also has d_3h symmetry and is space-filling. It has 21 edges and 11 vertices. With square faces it can be seen as a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

split across the 3-fold axis, separated with the two halves rotated 180 degrees, and filling the gaps with triangles. When used as a space-filler, connecting dodecahedra on their triangles leaves two cubical step surfaces on the top and bottom which can connect with complementary steps.

References

* * ''Mathematical Recreations and Essays''

Walter William Rouse Ball Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...

Harold Scott Macdonald Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, p.151 * ''Structure in Nature Is a Strategy for Design'',

Peter Jon Pearce Peter Jon Pearce (October 8, 1936) is an American product designer, author, and inventor. He is the designer of the Cachet Chair, Manufactured by Steelcase, as well as the designer of the Curved Space Diamond Structure, a playground climbing ...

, p.48 Spacefilling systems based on rhombic dodecahedron

External links

* *

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphi ...

mode

{{Polyhedron-stub Space-filling polyhedra

Construction

Space-filling tessellation

Variations

See also

References

External links