Square tiling: Difference between revisions

Square tiling
Square tiling
Type	regular tiling
Vertex configuration	4.4.4.4
Schläfli symbol	{4,4}
Wallpaper group	p4m
Properties	vertex-transitive, edge-transitive, face-transitive, self-dual

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In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille.^[1]

Structure and properties

The square tiling has a structure consisting of one type of congruent prototile, the square, sharing two vertices with other identical ones. This is an example of monohedral tiling.^[2] Each vertex at the tiling is surrounded by four squares, which denotes in a vertex configuration as $4.4.4.4$ or $4^{4}$ .^[3] The vertices of a square can be considered as the lattice, so the square tiling can be formed through the square lattice.^[4] This tiling is commonly familiar with the flooring and game boards.^[5] It is self-dual, meaning the center of each square connects to another of the adjacent tile, forming square tiling itself.^[6]

The square tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge, and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. Hence, the square tiling is one of three regular tilings, with the remaining being triangular tiling and hexagonal tiling with its prototiles are equilateral triangles and regular hexagons, respectively.^[7] The square tiling is p4m: there is an order-8 dihedral group of a tile and a two-fold rotation around the vertex surrounded by four squares lying on the line of reflection.^[8]

The square tiling is alternatively formed by the assemblage of infinitely many circles arranged vertically and horizontally, wherein their equal diameter at the center of every point contact with four other circles.^[9]

Topologically equivalent tilings

An isogonal variation with two types of faces, seen as a snub square tiling with triangle pairs combined into rhombi.

Topological square tilings can be made with concave faces and more than one edge shared between two faces. This variation has three edges shared. Other quadrilateral tilings can be made which are topologically equivalent to the square tiling (four quads around every vertex)

Isohedral tilings have identical faces (face-transitivity) and vertex-transitivity. There are eighteen variations, with six identified as triangles that do not connect edge-to-edge, or as quadrilateral with two collinear edges. Symmetry given assumes all faces are the same color.^[10]

Isohedral quadrilateral tilings
Square p4m, (*442)	Rectangle pmm, (*2222)	Parallelogram p2, (2222)	Parallelogram pmg, (22*)	Rhombus cmm, (2*22)


Trapezoid cmm, (2*22)	Quadrilateral pgg, (22×)	Kite pmg, (22*)	Quadrilateral pgg, (22×)	Quadrilateral p2, (2222)

Degenerate quadrilaterals or non-edge-to-edge triangles
Isosceles pmg, (22*)	Isosceles pgg, (22×)		Scalene pgg, (22×)		Scalene p2, (2222)

Related regular complex apeirogons

There are 3 regular complex apeirogons, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal.^[11]

Self-dual	Duals

4{4}4 or	2{8}4 or	4{8}2 or

References

^ Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. AK Peters. p. 288. ISBN 978-1-56881-220-5.
^ Adams, Colin (2022). The Tiling Book: An Introduction to the Mathematical Theory of Tilings. American Mathematical Society. pp. 23. ISBN 9781470468972.
^ Grünbaum & Shephard (1987), p. 59.
^ Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. p. 21, 29.
^ Lorenzo, Sadun (2008). Topology of Tiling Spaces. American Mathematical Society. p. 2.
^ Nelson, Roice; Segerman, Henry (2017). "Visualizing hyperbolic honeycombs". Journal of Mathematics and the Arts. 11 (1): 4–39. arXiv:1511.02851. doi:10.1080/17513472.2016.1263789.
^ Grünbaum & Shephard (1987), p. 35.
^ Grünbaum & Shephard (1987), p. 42, see p. 38 for detail of symbols.
^ Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications. p. 36. ISBN 0-486-23729-X.
^ Grünbaum & Shephard (1987), p. 473–481.
^ Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.

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

External links

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	0_[3]	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	0_[4]	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	0_[5]	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	0_[6]	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	0_[7]	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	0_[8]	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	0_[9]	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	0_[10]	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	0_[11]	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

[conway-1] Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. AK Peters. p. 288. ISBN 978-1-56881-220-5.

[adams-2] Adams, Colin (2022). The Tiling Book: An Introduction to the Mathematical Theory of Tilings. American Mathematical Society. pp. 23. ISBN 9781470468972.

[FOOTNOTEGrünbaumShephard1987[httpsbooksgooglecombooksid0x0vDAAAQBAJpgPA59_59]-3] Grünbaum & Shephard (1987), p. 59.

[gs-4] Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. p. 21, 29.

[sadun-5] Lorenzo, Sadun (2008). Topology of Tiling Spaces. American Mathematical Society. p. 2.

[ns-6] Nelson, Roice; Segerman, Henry (2017). "Visualizing hyperbolic honeycombs". Journal of Mathematics and the Arts. 11 (1): 4–39. arXiv:1511.02851. doi:10.1080/17513472.2016.1263789.

[FOOTNOTEGrünbaumShephard1987[httpsbooksgooglecombooksid0x0vDAAAQBAJpgPA35_35]-7] Grünbaum & Shephard (1987), p. 35.

[FOOTNOTEGrünbaumShephard1987[httpsbooksgooglecombooksid0x0vDAAAQBAJpgPA42_42]see_p._[httpsbooksgooglecombooksid0x0vDAAAQBAJpgPA38_38]_for_detail_of_symbols-8] Grünbaum & Shephard (1987), p. 42, see p. 38 for detail of symbols.

[williams-9] Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications. p. 36. ISBN 0-486-23729-X.

[FOOTNOTEGrünbaumShephard1987[httpsbooksgooglecombooksid0x0vDAAAQBAJpgPA473_473&ndash;481]-10] Grünbaum & Shephard (1987), p. 473–481.

[11] Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.

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 }}</ref>
-The square tiling [[Group action (mathematics)#Types of actions|acts transitively]] on the ''flags'' of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge, and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. Hence, the square tiling is one of three regular tilings, with the remaining being [[triangular tiling]] and [[hexagonal tiling]] with its prototiles are [[equilateral triangle]]s and [[regular hexagon]]s, respectively.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA35 35]}} The square tiling is p4m: there is a four-fold rotation around the center of a tile and a two-fold rotation around the vertex surrounded by four squares lying on the line of reflection.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA42 42]|loc=see p. [https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA38 38] for detail of symbols}}
+The square tiling [[Group action (mathematics)#Types of actions|acts transitively]] on the ''flags'' of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge, and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. Hence, the square tiling is one of three regular tilings, with the remaining being [[triangular tiling]] and [[hexagonal tiling]] with its prototiles are [[equilateral triangle]]s and [[regular hexagon]]s, respectively.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA35 35]}} The square tiling is p4m: there is an [[Dihedral group of order 8|order-8 dihedral group]] of a tile and a two-fold rotation around the vertex surrounded by four squares lying on the line of reflection.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA42 42]|loc=see p. [https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA38 38] for detail of symbols}}
 The square tiling is alternatively formed by the [[Circle packing|assemblage of infinitely many circles]] arranged vertically and horizontally, wherein their equal diameter at the center of every point contact with four other circles.<ref name=williams>{{cite book