All Topics  
Cubic honeycomb

 

 
   Email Print
   Bookmark   Link
 

 

Cubic honeycomb
 
 
  Topics Cubic honeycomb Topic Home

The cubic honeycomb is the only 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 honeycomb
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 Euclidean 3-space, made up of cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
s.






Discussion
Ask a question about 'Cubic honeycomb'
Start a new discussion about 'Cubic honeycomb'
Answer questions from other users
Full Discussion Forum



Encyclopedia


Partial Cubic Honeycomb
TypeRegular 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....
FamilyHypercube honeycomb
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....
t0,3
x
x x
t0
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....




Cell type
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
Face type
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/edge4
Faces/edge44
Cells/vertex8
Faces/vertex412
Edges/vertex6
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
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....
s		[4,3,4]
[4,31,1]
Dualself-dual
Propertiesvertex-transitive
Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
Cubic Honeycomb
The cubic honeycomb is the only 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 honeycomb
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 Euclidean 3-space, made up of cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
s. It is an analog of the square tiling
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
 of the plane, and part of a dimensional family called hypercube honeycombs.

It is one of 28 uniform honeycombs
Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
 using regular and semiregular polyhedral cells.

Four cubes exist on each edge, and 8 cubes around each vertex. It is a self-dual tessellation.

It is related to the regular tesseract
Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
 which exists in 4-space with 3 cubes on each edge.

Uniform colorings


There is a large number of uniform coloring
Uniform coloring

In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive. Different Symmetry can be expressed on the same geometric figure with the Face following different uniform color patterns....
s, derived from different symmetries. Some of the reflective symmetries include:
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....
Partial
honeycomb		Colors by letters
Partial Cubic Honeycomb
 1: aaaa/aaaa
2: aaaa/bbbb
2: abba/abba
2: abba/baab
4: abcd/abcd
Runcinated Cubic Honeycomb
 4: abbc/bccd


 8: abcd/efgh


See also

  • List of regular polytopes
    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....