Hypercubic honeycomb

Square tiling uniform coloring 1.png
A regular square tiling.
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1 color
Partial cubic honeycomb.png
A cubic honeycomb in its regular form.
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1 color
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A checkboard square tiling
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2 colors
Bicolor cubic honeycomb.png
A cubic honeycomb checkerboard.
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2 colors
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Expanded square tiling
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3 colors
Runcinated cubic honeycomb.png
Expanded cubic honeycomb
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4 colors
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4 colors
Cubic 8-color honeycomb.png
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8 colors

In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter group Rn (or B~n-1) for n>=3.

The tessellation is constructed from 4 n-hypercubes per ridge. The vertex figure is a cross-polytope {3...3,4}.

The hypercubic honeycombs are self-dual.

Coxeter named this family as δn+1 for an n-dimensional honeycomb.

Wythoff construction classes by dimensionEdit

A Wythoff construction is a method for constructing a uniform polyhedron or plane tiling.

The two general forms of the hypercube honeycombs are the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard.

A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lower-dimensional elements. For example, an expanded cubic honeycomb has cubic cells centered on the original cubes, on the original faces, on the original edges, on the original vertices, creating 4 colors of cells around in vertex in 1:3:3:1 counts.

The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, in which each of the three axial directions may have different edge lengths. The facets are hyperrectangles, also called orthotopes; in 2 and 3 dimensions the orthotopes are rectangles and cuboids respectively.

δn Name Schläfli symbols Coxeter-Dynkin diagrams
Orthotopic
{∞}n
(2m colors, m<n)
Regular
(Expanded)
{4,3n-1,4}
(1 color, n colors)
Checkerboard
{4,3n-4,31,1}
(2 colors)
δ2 Apeirogon {∞}       
δ3 Square tiling {∞}2
{4,4}
           
     
     
δ4 Cubic honeycomb {∞}3
{4,3,4}
{4,31,1}
                
       
     
δ5 4-cube honeycomb {∞}4
{4,32,4}
{4,3,31,1}
                     
         
       
δ6 5-cube honeycomb {∞}5
{4,33,4}
{4,32,31,1}
                          
           
         
δ7 6-cube honeycomb {∞}6
{4,34,4}
{4,33,31,1}
                               
             
           
δ8 7-cube honeycomb {∞}7
{4,35,4}
{4,34,31,1}
                                    
               
             
δ9 8-cube honeycomb {∞}8
{4,36,4}
{4,35,31,1}
                                         
                 
               
δn n-hypercubic honeycomb {∞}n
{4,3n-3,4}
{4,3n-4,31,1}
...

See alsoEdit

ReferencesEdit

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
    1. pp. 122–123. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
    2. pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}
    3. p. 296, Table II: Regular honeycombs, δn+1
Fundamental convex regular and uniform honeycombs in dimensions 2-9
          /   /  
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21