Login
Login

Uniform honeycomb

{{short description|Isogonal honeycomb of uniform polytope facets}}

In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex. Its dimension can be clarified as {{mvar|n}}-honeycomb for an {{mvar|n}}-dimensional honeycomb.

An {{mvar|n}}-dimensional uniform honeycomb can be constructed on the surface of {{mvar|n}}-spheres, in {{mvar|n}}-dimensional Euclidean space, and {{mvar|n}}-dimensional hyperbolic space. A 2-dimensional uniform honeycomb is more often called a uniform tiling or uniform tessellation.

Nearly all uniform tessellations can be generated by a Wythoff construction, and represented by a Coxeter–Dynkin diagram. The terminology for the convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb articles were coined by Norman Johnson.

Wythoffian tessellations can be defined by a vertex figure. For 2-dimensional tilings, they can be given by a vertex configuration listing the sequence of faces around every vertex. For example, {{math|4.4.4.4}} represents a regular tessellation, a square tiling, with 4 squares around each vertex. In general an {{mvar|n}}-dimensional uniform tessellation vertex figures are define by an {{math|(n−1)}}-polytope with edges labeled with integers, representing the number of sides of the polygonal face at each edge radiating from the vertex.

Examples of uniform honeycombs

class="wikitable"

!colspan=5|2-dimensional tessellations

 

!Spherical

!Euclidean

!colspan=2|Hyperbolic

 

|{{main|Uniform polyhedron}}

|{{main|List of uniform tilings}}

|colspan=2|{{main|Uniform tilings in hyperbolic plane}}

Coxeter diagram

|valign=top align=center|{{CDD|node_1|5|node_1|3|node_1}}

|valign=top align=center|{{CDD|node_1|6|node_1|3|node_1}}

|valign=top align=center|{{CDD|node_1|7|node_1|3|node_1}}

|valign=top align=center|{{CDD|node_1|infin|node_1|3|node_1}}

valign=top

!Picture

|width=160|150px
Truncated icosidodecahedron

|width=160|150px
Truncated trihexagonal tiling

|width=160|150px
Truncated triheptagonal tiling
(Poincaré disk model)

|width=160|150px
Truncated triapeirogonal tiling

Vertex figure

|120px

|120px

|120px

|

colspan=5|3-dimensional honeycombs
 

!3-spherical

!3-Euclidean

!colspan=2|3-hyperbolic

 

|{{main|Uniform polychoron}}

|{{main|Convex uniform honeycomb}}

|colspan=2|{{main|Uniform honeycombs in hyperbolic space}}and paracompact uniform honeycomb

valign=top align=center

!Coxeter diagram

|{{CDD|node_1|3|node|3|node|4|node}}

|{{CDD|node_1|4|node|3|node|4|node}}

|{{CDD|node_1|5|node|3|node|4|node}}

|{{CDD|node_1|6|node|3|node|4|node}}

valign=top

!Picture

|width=160|150px
(Stereographic projection)
16-cell

|width=160|150px
cubic honeycomb

|width=160|150px
order-4 dodecahedral honeycomb
(Beltrami–Klein model)

|150px
order-4 hexagonal tiling honeycomb
(Poincaré disk model)

valign=bottom align=center

!Vertex figure

|80px
(Octahedron)

|80px
(Octahedron)

|80px
(Octahedron)

|80px
(Octahedron)

See also

References

  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  • Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49–56.
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
  • {{The Geometrical Foundation of Natural Structure (book)}}
  • {{cite book | author=Grünbaum, Branko | author-link=Branko Grünbaum | author2=Shephard, G. C. | author2-link=G.C. Shephard | title=Tilings and Patterns | publisher=W. H. Freeman and Company | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }}
  • H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • {{cite book | first=Keith | last=Critchlow | author-link=Keith Critchlow | title=Order in Space: A design source book | publisher=Viking Press| year=1970 | isbn=0-500-34033-1 }}
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.

External links

  • {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
  • [https://www.orchidpalms.com/polyhedra/tessellations/tessel.htm Tessellations of the Plane]
  • {{KlitzingPolytopes|flat.htm|2D|Euclidean tesselations}}

{{Tessellation}}

{{DEFAULTSORT:Uniform Tessellation}}

Category:Uniform tilings

Category:Honeycombs (geometry)