3-4-6-12 tiling

3-4-6-12 tiling
Type2-uniform tiling
Vertex configuration
3.4.6.4 and 4.6.12
Symmetryp6m, [6,3], (*632)
Rotation symmetryp6, [6,3]+, (632)
Properties2-uniform, 4-isohedral, 4-isotoxal

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.[1][2][3][4]

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:

rhombitrihexagonal tiling truncated trihexagonal tiling

3.4.6.4

4.6.12

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.

Related k-uniform tilings of regular polygons

The hexagons can be dissected into 6 triangles, and the dodecagaons can be dissected into triangles, hexagons and squares.

Dissected polygons
Hexagon Dodecagon
(each has 2 orientations)
3-uniform tilings
48 26 18

[36; 3.3.4.3.4; 3.3.4.12]

[3.4.4.6; (3.4.6.4)2]

[36; (3.3.4.3.4)2]

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.


Dual tiling

V3.4.6.4

V4.6.12

Deltoidal trihexagonal tiling

Kisrhombille tiling

Notes

  1. Critchlow, p.62-67
  2. Grünbaum and Shephard 1986, pp. 65-67
  3. In Search of Demiregular Tilings #4
  4. Chavey (1989)

References

External links

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