Snub square tiling
|Snub square tiling|
|Wythoff symbol||| 4 4 2|
|Symmetry||p4g, [4+,4], (4*2)|
|Rotation symmetry||p4, [4,4]+, (442)|
|Dual||Cairo pentagonal tiling|
There are two distinct uniform colorings of a snub square tiling. (Naming the colors by indices around a vertex (188.8.131.52.4): 11212, 11213.)
|Symmetry||4*2, [4+,4], (p4g)||442, [4,4]+, (p4)|
|Wythoff symbol||| 4 4 2|
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).
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 will produce a snub tiling with perfect equilateral triangle faces.
Regular octagons alternately truncated
Isosceles triangles (Nonuniform tiling)
Nonregular octagons alternately truncated
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.
The snub square tiling can be seen related to this 3-colored square tiling, with the yellow and red squares being twisted rigidly and the blue tiles being distorted into rhombi and then bisected into two triangles.
Fractalizing and Dissection
Every uniform planar tiling can be reduced to a tiling in equilateral triangles and squares by dissecting regular hexagons into six triangles and dissecting regular dodecagons into hexakis rhombitrihexagonal rotundas:
|Dissecting a Hexagon||Dissecting a Dodecagon|
|Insetting a Degree 6 Vertex||Completely Insetting a Degree 12 Vertex|
Nonetheless, the duals of the resulting tilings may also contain regular hexagons in addition to Cairo and prismatic pentagons.
For instance, starting from a truncated-hexagonal-fractalizing of the elongated square tiling, the dodecagons are progressively dissected in either orientation. The corresponding fractalizing dual is progressively inset. Below, we see that the kites (ties), skew quadrilaterals, and isosceles triangles are reduced to Cairo pentagons, prismatic pentagons, and hexagons; with most of the tiles Cairo pentagons:
Related polyhedra and tilings
The snub square tiling is similar to the elongated triangular tiling with vertex configuration 184.108.40.206.4, and two 2-uniform dual tilings and 2 3-uniform duals which mix the two types of pentagons (dual planigons to scale):
|Unit of the Double Gyro Square Tiling|
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 220.127.116.11.n.
|4n2 symmetry mutations of snub tilings: 18.104.22.168.n|
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.
|4n2 symmetry mutations of snub tilings: 3.3.n.3.n|
|Uniform tilings based on square tiling symmetry|
|Symmetry: [4,4], (*442)||[4,4]+, (442)||[4,4+], (4*2)|
|Wikimedia Commons has media related to Uniform tiling 3-3-4-3-4.|
- List of uniform planar tilings
- Snub square prismatic honeycomb
- Tilings of regular polygons
- Elongated triangular tiling
- Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern C
- Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.
- "Archived copy". Archived from the original on 2006-09-09. Retrieved 2006-09-09.CS1 maint: Archived copy as title (link)
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 
- Klitzing, Richard. "2D Euclidean tilings s4s4s - snasquat - O10".
- Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65)
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. p38
- Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual p. 115