33344-33434 tilingsFaced colored by their symmetry positionsType2-uniform tilingDesignation p. 65-67[33.42; 32.4.3.4]1[33.42; 32.4.3.4]2Vertex configurations3.3.4.3.4 and 3.3.3.4.4Symmetryp4g, [4,4+], (4*2)pgg, [4+,4+], (22×)Rotation symmetryp4, [4,4]+, (442)p2, [4+,4+]+, (2222)Properties4-isohedral, 5-isotoxal3-isohedral, 6-isotoxal 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.Chavey (1989)
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: 3.3.4.3.43.3.3.4.4snub square tilingelongated triangular 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]1a33.42; 32.4.3.4]1C[33.42; 32.4.3.4]2a[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-uniformV3.3.3.4.4V3.3.4.3.4V3.3.3.4.4 and V3.3.4.3.4V3.3.3.4.4V3.3.4.3.4prismatic pentagonal tilingCairo pentagonal tilingDual tiling IDual tiling II
Notes
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 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
External links
In Search of Demiregular Tilings, Helmer Aslaksen n-uniform tilings Brian Galebach, 2-Uniform Tiling 1 of 20