In geometry of the Euclidean plane, the 3-4-6-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, hexagons and dodecagons, arranged in two vertex configuration: 3.4.6.4 and 4.6.12. It has hexagonal symmetry, p6m, [6,3], (*632). It is also called a demiregular tiling by some authors.
Geometry Its two vertex configurations are shared with two 1-uniform tilings:
It can be seen as a type of diminished rhombitrihexagonal tiling, with dodecagons replacing periodic sets of hexagons and surrounding squares and triangles. This is similar to the Johnson solid, a diminished rhombicosidodecahedron, which is a rhombicosidodecahedron with faces removed, leading to new decagonal faces. The dual of this variant is shown to the right (deltoidal hexagonal insets).
Related k-uniform tilings of regular polygons The hexagons can be dissected into 6 triangles, and the dodecagons can be dissected into triangles, hexagons and squares.
Circle Packing This 2-uniform tiling can be used as a circle packing. Cyan circles are in contact with 3 other circles (2 cyan, 1 pink), corresponding to the V4.6.12 planigon, and pink circles are in contact with 4 other circles (1 cyan, 2 pink), corresponding to the V3.4.6.4 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.
Dual tiling The dual tiling has right triangle and kite faces, defined by face configurations: V3.4.6.4 and V4.6.12, and can be seen combining the deltoidal trihexagonal tiling and kisrhombille tilings.
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 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
Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1. p. 65 Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37 [1]
External links 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. Dutch, Steve. "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09. Weisstein, Eric W. "Demiregular tessellation". MathWorld. In Search of Demiregular Tilings, Helmer Aslaksen n-uniform tilings Brian Galebach, 2-Uniform Tiling 1 of 20