HEXAGONAL TILING

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In geometry, the 'hexagonal tiling' is a regular tiling of the Euclidean plane. It has Schläfli symbol of ''t₀{6,3}'' or ''t₂{3,6}''.
The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the square tiling and the triangular tiling.
This hexagonal pattern exists in nature in a beehive's honeycomb.

Contents

Nonregular forms

Uniform colorings

Wythoff constructions from hexagonal and triangular tilings

See also

External links

References

Nonregular forms

The ''hexagonal tiling'' can be stretched and adjusted to other geometric proportions and different symmetries. For example, the standard brick pattern can be considered a nonregular hexagonal tiling. Each rectangular brick has vertices inserted on the two long edges, dividing them into two colinear edges.

Uniform colorings

There are 3 distinct uniform colorings of a hexagonal tiling. (Naming the colors by indices on the 3 hexagons around a vertex: 111, 112, 123.)
The 3 colorings, named by their generating Wythoff symbols and symmetry are:

3 > 6 2 ★ p632 (p6m)

2 6 > 3 ★ p632 (p6m)

3 3 3 > ★ 333 (p3)

This tiling is topologically related as a part of sequence of regular polyhedra with vertex figure (n³), and continue into the hyperbolic plane.

(33)

(43)

(53)

(63) tiling

(73) tiling

It is also topologically related as a part of sequence of uniform truncated polyhedra with vertex figure (n.6.6).

(3.6.6)

(4.6.6)

(5.6.6)

(6.6.6) tiling

(7.6.6) tiling

Wythoff constructions from hexagonal and triangular tilings

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The ''truncated triangular tiling'' is topologically identical to the hexagonal tiling.)
{| class="prettytable"
!Tiling
!Schläfli
symbol
!Wythoff
symbol
!Vertex
figure
!Image
|-
!Hexagonal tiling
|t₀{6,3}
| 3 | 6 2
|6³
|
|-
!Truncated hexagonal tiling
|t_0,1{6,3}
| 2 3 | 6
|3.12.12
|
|-
!Rectified hexagonal tiling
(Trihexagonal tiling)
|t₁{6,3}
| 2 | 6 3
|(3.6)²
|
|-
!Bitruncated hexagonal tiling
(Truncated triangular tiling)
|t_1,2{6,3}
| 2 6 | 3
|6.6.6
|
|-
!Dual hexagonal tiling
(Triangular tiling)
|t₂{6,3}
| 6 | 3 2
|3⁶
|
|-
!Cantellated hexagonal tiling
(Small rhombitrihexagonal tiling)
|t_0,2{6,3}
| 6 3 | 2
|3.4.6.4
|
|-
!Omnitruncated hexagonal tiling
(Great rhombitrihexagonal tiling)
|t_0,1,2{6,3}
| 6 3 2 |
|4.6.12
|
|-
!Snub hexagonal tiling
|s{6,3}
|| 6 3 2
|3.3.3.3.6
|
|}

See also

★ Hexagonal lattice

★ Hexagonal prismatic honeycomb

★ Tilings of regular polygons

★ List of uniform tilings

★ List of regular polytopes

★ Example: Carbon nanotube

★ Example: Settlers of Catan

★ Example: Chicken wire

External links

★

References

★ Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table II: Regular honeycombs

★ Tilings and Patterns, Grünbaum, Branko ; and Shephard, G. C., , , W. H. Freeman, 1987, ISBN 0-716-71193-1 (Chapter 2.1: ''Regular and uniform tilings'', p.58-65)

★ Williams, Robert ''The Geometrical Foundation of Natural Structure: A Source Book of Design'' New York: Dover, 1979. p35