Truncated hexagonal tiling

Truncated hexagonal tiling
Truncated hexagonal tiling
Type Semiregular tiling
Vertex configuration Truncated hexagonal tiling vertfig.png
3.12.12
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 coloringsEdit

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

 

Topologically identical tilingsEdit

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

   
   

Related polyhedra and tilingsEdit

Wythoff constructions from hexagonal and triangular tilingsEdit

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.)

Symmetry mutationsEdit

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.

Related 2-uniform tilingsEdit

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
 
(3.122)
 
 
 
(3.4.6.4) & (33.42)
 
(3.4.6.4) & (32.4.3.4)
Dual Tilings

V3.122
 

 


V3.4.6.4 & V33.42

V3.4.6.4 & V32.4.3.4

Circle packingEdit

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.

 

Triakis triangular tilingEdit

Triakis triangular tiling
 
TypeDual semiregular tiling
Facestriangle
Coxeter diagram     
Symmetry groupp6m, [6,3], (*632)
Rotation groupp6, [6,3]+, (632)
Dual polyhedronTruncated hexagonal tiling
Face configurationV3.12.12
Propertiesface-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 kisdeltille,[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]

 

It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile.[7]

Related duals to uniform tilingsEdit

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)
             
V63 V3.122 V(3.6)2 V36 V3.4.6.4 V.4.6.12 V34.6

See alsoEdit

ReferencesEdit

  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. ^ "Archived copy". Archived from the original on 2006-09-09. Retrieved 2006-09-09.CS1 maint: archived copy as title (link)
  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 "Archived copy". Archived from the original on 2010-09-19. Retrieved 2012-01-20.CS1 maint: archived copy as title (link) (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
  5. ^ Inose, Mikio. "mikworks.com : Original Work : Asanoha". www.mikworks.com. Retrieved 20 April 2018.
  6. ^ Weisstein, Eric W. "Dual tessellation". MathWorld.
  7. ^ Kirby, Matthew; Umble, Ronald (2011), "Edge tessellations and stamp folding puzzles", Mathematics Magazine, 84 (4): 283–289, arXiv:0908.3257, doi:10.4169/math.mag.84.4.283, MR 2843659.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
  • Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1.CS1 maint: multiple names: authors list (link) (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 linksEdit