Snub square tiling

In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is s{4,4}.

Conway calls it a snub quadrille, constructed by a snub operation applied to a square tiling (quadrille).

There are 3 regular and 8 semiregular tilings in the plane.

Uniform colorings

There are two distinct uniform colorings of a snub square tiling. (Naming the colors by indices around a vertex (3.3.4.3.4): 11212, 11213.)

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!Coloring

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11212

|120px
11213

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!Symmetry

|4*2, [4⁺,4], (p4g)

|442, [4,4]⁺, (p4)

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!Schläfli symbol

|s{4,4}

|sr{4,4}

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!Wythoff symbol

| {{pipe}} 4 4 2

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!Coxeter diagram

|{{CDD|node_h|4|node_h|4|node}}

|{{CDD|node_h|4|node_h|4|node_h}}

Circle packing

The snub square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern C

:File:1-uniform-9-circlepack.svg

Wythoff construction

The snub square tiling can be constructed as a snub operation from the square tiling, or as an alternate truncation from the truncated square tiling.

An alternate truncation deletes every other vertex, creating a new triangular faces at the removed vertices, and reduces the original faces to half as many sides. In this case starting with a truncated square tiling with 2 octagons and 1 square per vertex, the octagon faces into squares, and the square faces degenerate into edges and 2 new triangles appear at the truncated vertices around the original square.

If the original tiling is made of regular faces the new triangles will be isosceles. Starting with octagons which alternate long and short edge lengths, derived from a regular dodecagon, will produce a snub tiling with perfect equilateral triangle faces.

Example:

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Regular octagons alternately truncated

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→

(Alternate
truncation)

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Isosceles triangles (Nonuniform tiling)

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Nonregular octagons alternately truncated

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→

(Alternate
truncation)

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Equilateral triangles

Related tilings

File:Snub snub square tiling.svg|A snub operator applied twice to the square tiling, while it doesn't have regular faces, is made of square with irregular triangles and pentagons.

File:Isogonal snub square tiling-8x8.svg|A related isogonal tiling that combines pairs of triangles into rhombi

File:Triangular heptagonal tiling.svg|A 2-isogonal tiling can be made by combining 2 squares and 3 triangles into heptagons.

File:P2_dual.png|The Cairo pentagonal tiling is dual to the snub square tiling.

= Related k-uniform tilings=

This tiling is related to the elongated triangular tiling which also has 3 triangles and two squares on a vertex, but in a different order, 3.3.3.4.4. The two vertex figures can be mixed in many k-uniform tilings.{{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons—II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147–165 | doi=10.1016/0898-1221(89)90156-9| doi-access= }}{{cite web |url=http://www.uwgb.edu/dutchs/symmetry/uniftil.htm |title=Uniform Tilings |access-date=2006-09-09 |url-status=dead |archive-url=https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm |archive-date=2006-09-09 }}

class="wikitable collapsible" !colspan=6\|Related tilings of triangles and squares
snub square !elongated triangular !colspan=2\| 2-uniform !colspan=3\| 3-uniform
align=center !p4g, (42) !p2, (2222) !p2, (2222) !cmm, (222) !colspan=2\|p2, (2222)
align=center \|100px [3²434] \|100px [3³4²] \|100px [3³4²; 3²434] \|100px [3³4²; 3²434] \|100px [2: 3³4²; 3²434] \|100px [3³4²; 2: 3²434]
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!colspan=6|Related tilings of triangles and squares

snub square

!elongated triangular

!colspan=2| 2-uniform

!colspan=3| 3-uniform

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!p4g, (4*2)

!p2, (2222)

!cmm, (2*22)

!colspan=2|p2, (2222)

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|100px
[3²434]

|100px
[3³4²]

|100px
[3³4²; 3²434]

|100px
[2: 3³4²; 3²434]

|100px
[3³4²; 2: 3²434]

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|30px

|30px 30px

|30px 30px 30px

= Related topological series of polyhedra and tiling=

The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.

References

John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]
{{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|s4s4s - snasquat - O10}}
{{cite book | author1=Grünbaum, Branko | author-link=Branko Grünbaum | author2=Shephard, G. C. | title=Tilings and Patterns | location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} (Chapter 2.1: Regular and uniform tilings, p. 58-65)
{{The Geometrical Foundation of Natural Structure (book)}} p38
Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, {{isbn|978-0866514613}}, pp. 50–56, dual p. 115