Square tiling

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

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

Flooring and game board

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 ${\displaystyle 4.4.4.4}$ or ${\displaystyle 4^{4}}$. ^[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. In this case, the flag consists of a mutually incident vertex, edge, and tile of the tiling. Simply put, every pair of flags has a symmetry operation mapping the first flag to the second: they are vertex-transitive (mapping the vertex of a tile to another), edge-transitive (mapping the edge to another), and face-transitive (mapping square tile to another). By meeting these three properties, the square tiling is categorized as one of three regular tilings; the remaining being triangular tiling and hexagonal tiling with its prototiles are equilateral triangles and regular hexagons, respectively. ^[7] The symmetry group of a square tiling is p4m: there is an order-4 dihedral group of a tile and an order-2 dihedral group 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] Its densest packing is ${\textstyle {\frac {\pi }{4}}\approx 0.785}$. ^[10]

Topologically equivalent tilings

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. ^[11]

Twelve isohedral quadrilateral tilings, and six triangular tilings that do not tile edge-to-edge.

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. ^[12]

Self-dual	Duals
4{4}4 or	2{8}4 or	4{8}2 or

See also

• Fenestrane
• Langton's ant
• Polyomino

References

1. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. AK Peters. p.  288. ISBN   978-1-56881-220-5.
2. Adams, Colin (2022). The Tiling Book: An Introduction to the Mathematical Theory of Tilings. American Mathematical Society. pp.  23. ISBN   9781470468972.
3. Grünbaum & Shephard (1987), p.  59.
4. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. p.  21, 29.
5. Lorenzo, Sadun (2008). Topology of Tiling Spaces. American Mathematical Society. p. 1. ISBN   978-0-8218-4727-5.
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.
7. Grünbaum & Shephard (1987), p.  35.
8. Grünbaum & Shephard (1987), p.  42, see p. 38 for detail of symbols.
9. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications. p. 36. ISBN   0-486-23729-X.
10. O'Keeffe, M.; Hyde, B. G. (1980). "Plane nets in crystal chemistry". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 295 (1417): 553–618. Bibcode:1980RSPTA.295..553O. doi:10.1098/rsta.1980.0150. JSTOR   36648. S2CID   121456259.
11. Grünbaum & Shephard (1987), p.  473–481.
12. 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

• Weisstein, Eric W. "Square Grid". MathWorld .
• Weisstein, Eric W. "Regular tessellation". MathWorld .
• Weisstein, Eric W. "Uniform tessellation". MathWorld .

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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
En−1	Uniform (n−1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21