Truncated hexagonal tiling

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Truncated hexagonal tiling
Truncated hexagonal tiling
Type Semiregular tiling
Vertex configuration Truncated hexagonal tiling vertfig.png
Schläfli symbol t{6,3}
Wythoff symbol 2 3 | 6
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Symmetry p6m, [6,3], (*632)
Rotation symmetry p6, [6,3]+, (632)
Bowers acronym Toxat
Dual Triakis triangular tiling
Properties Vertex-transitive

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.

As the name implies this tiling is constructed by a truncation operation applies to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}.

Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling (hextille).

There are 3 regular and 8 semiregular tilings in the plane.

Uniform colorings[edit]

There is only one uniform coloring of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.)

Uniform polyhedron-63-t01.png

Topologically identical tilings[edit]

The dodecagonal faces can be distorted into different geometries, like:

Truncated hexagonal tiling0.png Gyrated truncated hexagonal tiling.png
Gyrated truncated hexagonal tiling3.png Gyrated truncated hexagonal tiling2.png

Related polyhedra and tilings[edit]

Wythoff constructions from hexagonal and triangular tilings[edit]

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
{6,3} t{6,3} r{6,3} t{3,6} {3,6} rr{6,3} tr{6,3} sr{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Uniform tiling 63-t0.png Uniform tiling 63-t01.png Uniform tiling 63-t1.png Uniform tiling 63-t12.png Uniform tiling 63-t2.png Uniform tiling 63-t02.png Uniform tiling 63-t012.png Uniform tiling 63-snub.png
Config. 63 3.12.12 (6.3)2 6.6.6 36 4.6.12

Symmetry mutations[edit]

This tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: 3.2n.2n
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
[12i,3] [9i,3] [6i,3]
Spherical triangular prism.png Uniform tiling 332-t01-1-.png Uniform tiling 432-t01.png Uniform tiling 532-t01.png Uniform tiling 63-t01.png H2 tiling 237-3.png H2 tiling 238-3.png H2 tiling 23i-3.png H2 tiling 23j12-3.png H2 tiling 23j9-3.png H2 tiling 23j6-3.png
Config. 3.4.4 3.6.6 3.8.8 3.10.10 3.12.12 3.14.14 3.16.16 3.∞.∞ 3.24i.24i 3.18i.18i 3.12i.12i
Spherical trigonal bipyramid.png Spherical triakis tetrahedron.png Spherical triakis octahedron.png Spherical triakis icosahedron.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Ord7 triakis triang til.png Ord8 triakis triang til.png Ord-infin triakis triang til.png
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞

Related 2-uniform tilings[edit]

Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares.[1][2]

1-uniform Dissection 2-uniform dissections
1-uniform 4.png
Regular dodecagon.svg
Hexagonal cupola flat.png
2-uniform 8.png
( & (
2-uniform 9.png
( & (

Circle packing[edit]

The truncated hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point.[3] Every circle is in contact with 3 other circles in the packing (kissing number). This is the lowest density packing that can be created from a uniform tiling. The dodecagonal gaps can be filled perfectly with 7 circles, creating a denser 3-uniform packing.

Truncated hexagonal tiling circle packing.png Truncated hexagonal tiling circle packing2.png Truncated hexagonal tiling circle packing3.png

Triakis triangular tiling[edit]

Triakis triangular tiling
1-uniform 4 dual.svg
Type Dual semiregular tiling
Coxeter diagram CDel node.pngCDel 3.pngCDel node f1.pngCDel 6.pngCDel node f1.png
Faces triangle
Face configuration V3.12.12
Tiling face 3-12-12.svg
Symmetry group p6m, [6,3], (*632)
Rotation group p6, [6,3]+, (632)
Dual Truncated hexagonal tiling
Properties face-transitive
On painted porcelain, China

The triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by face configuration V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, and two with 12 triangles.

Conway calls it a kisdeltile,[4] constructed as a kis operation applied to a triangular tiling (deltille).

In Japan the pattern is called asanoha for hemp leaf, although the name also applies to other triakis shapes like the triakis icosahedron and triakis octahedron.[5]

It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex.[6]

P4 dual.png

Related duals to uniform tilings[edit]

It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular duals.

Dual uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+, (632)
Uniform tiling 63-t2.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Rhombic star tiling.png Uniform tiling 63-t0.png Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
V63 V3.122 V(3.6)2 V36 V3.4.12.4 V.4.6.12 V34.6

See also[edit]


  1. ^ 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. 
  2. ^
  3. ^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern G
  4. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
  5. ^
  6. ^ Weisstein, Eric W., "Dual tessellation", MathWorld.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [2]
  • Grünbaum, Branko ; and 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. p. 39. ISBN 0-486-23729-X. 
  • Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern E, Dual p. 77-76, pattern 1
  • Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual p. 117

External links[edit]