33344-33434 tiling

In geometry of the Euclidean plane, a 33344-33434 tiling is one of two of 20 2-uniform tilings of the Euclidean plane by regular polygons. They contains regular triangle and square faces, arranged in two vertex configuration: 3.3.3.4.4 and 3.3.4.3.4.[2]

33344-33434 tilings

Faced colored by their symmetry positions
Type2-uniform tiling
Designation[1][33.42; 32.4.3.4]1[33.42; 32.4.3.4]2
Vertex configurations3.3.4.3.4 and 3.3.3.4.4
Symmetryp4g, [4,4+], (4*2)pgg, [4+,4+], (22×)
Rotation symmetryp4, [4,4]+, (442)p2, [4+,4+]+, (2222)
Properties4-isohedral, 5-isotoxal3-isohedral, 6-isotoxal

The first has triangles in groups of 3 and square in groups of 1 and 2. It has 4 types of faces and 5 types of edges.

The second has triangles in groups of 4, and squares in groups of 2. It has 3 types of face and 6 types of edges.

Geometry

Its two vertex configurations are shared with two 1-uniform tilings:


3.3.4.3.4
3.3.3.4.4

elongated triangular tiling
snub square tiling

Circle Packings

These 2-uniform tilings can be used as a circle packings.

In the first 2-uniform tiling (whose dual resembles a key-lock pattern): cyan circles are in contact with 5 other circles (3 cyan, 2 pink), corresponding to the V33.42 planigon, and pink circles are also in contact with 5 other circles (4 cyan, 1 pink), corresponding to the V32.4.3.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.

In the second 2-uniform tiling (whose dual resembles jagged streams of water): cyan circles are in contact with 5 other circles (2 cyan, 3 pink), corresponding to the V33.42 planigon, and pink circles are also in contact with 5 other circles (3 cyan, 2 pink), corresponding to the V32.4.3.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.

Circle Packings of and Ambo Operations on Two Pentagonal Isoperimetric 2-dual-uniform tilings.
C[33.42; 32.4.3.4]1 a33.42; 32.4.3.4]1 C[33.42; 32.4.3.4]2 a[33.42; 32.4.3.4]2

Dual tilings

The dual tilings have right triangle and kite faces, defined by face configurations: V3.3.3.4.4 and V3.3.4.3.4, and can be seen combining the prismatic pentagonal tiling and Cairo pentagonal tilings.

Faces1-uniform2-uniform
V3.3.3.4.4V3.3.4.3.4V3.3.3.4.4 and V3.3.4.3.4

V3.3.3.4.4

V3.3.4.3.4
prismatic pentagonal tiling
Cairo pentagonal tiling
Dual tiling I
Dual tiling II

Notes
  1. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1. p. 65-67
  2. Chavey (1989)

References
  • Keith Critchlow, Order in Space: A design source book, 1970, pp. 62–67
  • Ghyka, M. The Geometry of Art and Life, (1946), 2nd edition, New York: Dover, 1977. Demiregular tiling #15
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. pp. 35–43
  • Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37
  • Introduction to Tessellations, Dale Seymour, Jill Britton, (1989), p.57, Fig 3-24 Tessellations of regular polygons that contain more than one type of vertex point
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