SQUARE TILING

In geometry, the 'Square tiling' is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}.
The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling.

Contents

Uniform colorings

Wythoff constructions from square tiling

See also

References

External links

Uniform colorings

There are 9 distinct uniform colorings of a square tiling. (Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry.)

Wythoff constructions from square tiling

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular square tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms: 'square tiling', truncated square tiling, snub square tiling.
{| class="prettytable"
!Operation
!Schläfli
symbol
!Wythoff
Symbol
!Vertex figure
!Image
|-
!Parent
!t₀{4,4}
! 4 | 2 4
!'4⁴'
|
|-
!Truncation
!t_0,1{4,4}
! 2 4 | 4
!4.8.8
|
|-
!Rectification
!t₁{4,4}
! 2 | 4 4
!'(4.4)²'
|
|-
!Bitruncation
!t_1,2{4,4}
! 2 4 | 4
!4.8.8
|
|-
!Dual
!t₂{4,4}
! 4 | 2 4
!'4⁴'
|
|-
!Cantellation
!t_0,2{4,4}
! 4 4 | 2
!'4.4.4.4'
|
|-
!Omnitruncation
!t_0,1,2{4,4}
! 2 4 4 |
!4.8.8
|
|-
!Snubbing
!s{4,4}
! | 2 4 4
!3.3.4.3.4
|
|}

See also

★ List of uniform tilings

★ List of regular polytopes

★ Tilings of regular polygons

★ Square lattice

★ Checkerboard

References

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

★ Williams, Robert ''The Geometrical Foundation of Natural Structure: A Source Book of Design'' New York: Dover, 1979. p36

★ 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)

External links

★