Hyperbolic small dodecahedral honeycomb

	Email		Print
	Bookmark		Link

Hyperbolic small dodecahedral honeycomb

The order-4 dodecahedral honeycomb is one of four regular space-filling tessellation

Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
(or honeycombs

Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
) in hyperbolic 3-space

Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
.

Four dodecahedra

Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
exist on each edge, and 8 dodecahedra around each vertex.

Discussion

Ask a question about 'Hyperbolic small dodecahedral honeycomb'

Start a new discussion about 'Hyperbolic small dodecahedral honeycomb'

Answer questions from other users

Full Discussion Forum

Encyclopedia

Order-4 dodecahedral honeycomb
Perspective projection view within Beltrami-Klein model
Type	Hyperbolic regular honeycomb List of regular polytopes This page lists the regular polytopes in Euclidean geometry, spherical geometry and hyperbolic geometry spaces.The Schl?fli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each....
Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
Cells	dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
Faces	pentagon Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
Edge figure	square Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
Vertex figure	octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
Cells/edge	⁴
Cells/vertex	⁸
Euler characteristic Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....	0
Dual	Order-5 cubic honeycomb
Coxeter group Coxeter group In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....	[5,3,4] [5,3^1,1]
Properties	Regular

The order-4 dodecahedral honeycomb is one of four regular space-filling tessellation

Tessellation

Honeycomb (geometry)

Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
.

Four dodecahedra

Dodecahedron

Cubic honeycomb

The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs....
of Euclidean 3-space.

It can also be constructed from the bifurcating Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
[5,3^1,1] which can be represented by alternation of two colors of dodecahedral cells.

There is another regular honeycomb in hyperbolic 3-space called the order-5 dodecahedral honeycomb which has 5 dodecahedra per edge.

This honeycomb is also related to the 120-cell

120-cell

In geometry, the 120-cell is the convex regular 4-polytope with Schl?fli symbol .The boundary of the 120-cell is composed of 120 dodecahedral cell with 4 meeting at each vertex....
which has 120 dodecahedra in 4-dimensional space, with 3 dodecahedra on each edge.

The dihedral angle

Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
of a dodecahedron is ~116.6°, so it is impossible to fit 4 of them on an edge in Euclidean 3-space. However in hyperbolic space a properly scaled dodecahedron can be scaled so that its dihedral angles are reduced to 90 degrees, and then four fit exactly on every edge.

Hyperbolic small dodecahedral honeycomb

See also