World Library  
Flag as Inappropriate
Email this Article

Elongated triangular tiling

Article Id: WHEBN0002864821
Reproduction Date:

Title: Elongated triangular tiling  
Author: World Heritage Encyclopedia
Language: English
Subject: Demiregular tiling, Wythoff construction, Circle packing, Uniform tiling, Gilbert tessellation
Collection: Euclidean Tilings, Isogonal Tilings, Semiregular Tilings
Publisher: World Heritage Encyclopedia

Elongated triangular tiling

Elongated triangular tiling
Elongated triangular tiling
Type Semiregular tiling
Vertex configuration
Schläfli symbol {3,6}:e
Wythoff symbol 2 | 2 (2 2)
Coxeter diagram
Symmetry cmm, [∞,2+,∞], (2*22)
Rotation symmetry p2, [∞,2,∞]+, (2222)
Bowers acronym Etrat
Dual Prismatic pentagonal tiling
Properties Vertex-transitive

In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.

Conway calls it a isosnub quadrille.[1]

There are 3 regular and 8 semiregular tilings in the plane. This tiling is similar to the snub square tiling which also has 3 triangles and two squares on a vertex, but in a different order.


  • Construction 1
  • Uniform colorings 2
  • Circle packing 3
  • Related tilings 4
    • Prismatic pentagonal tiling 4.1
      • Geometric variations 4.1.1
      • Related 2-uniform dual tilings 4.1.2
  • See also 5
  • Notes 6
  • References 7
  • External links 8


It is also the only uniform tiling that can't be created as a Wythoff construction. It can be constructed as alternate layers of apeirogonal prisms and apeirogonal antiprisms.

Uniform colorings

There is one uniform colorings of an elongated triangular tiling. Two 2-uniform colorings have a single vertex figure, 11123, with two colors of squares, but are not 1-uniform, repeated either by reflection or glide reflection, or in general each row of squares can be shifted around independently. The 2-uniform tilings are also called Archimedean colorings. There are infinite variations of these Archimedean colorings by arbitrary shifts in the square row colorings.

11122 (1-uniform) 11123 (2-uniform or 1-Archimedean)
cmm (2*22) pmg (22*) pgg (22×)

Circle packing

The elongated triangular tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).[2]

Related tilings

It is first in a series of symmetry mutations[3] with hyperbolic uniform tilings with 2*n2 orbifold notation symmetry, vertex figure 4.n., and Coxeter diagram . Their duals have hexagonal faces in the hyperbolic plane, with face configuration V4.n.

Symmetry mutation 2*n2 of uniform tilings: 4.n.
2*22 2*32 2*42
or or

There are four related 2-uniform tilings, mixing 2 or 3 rows of triangles or squares.[4][5]

Double elongated Triple elongated Half elongated One third elongated

Prismatic pentagonal tiling

Prismatic pentagonal tiling
Type Dual uniform tiling
Coxeter diagram
Faces irregular pentagons
Face configuration V3.
Symmetry group cmm, [∞,2+,∞], (2*22)
Rotation group p2, [∞,2,∞]+, (2222)
Dual Elongated triangular tiling
Properties face-transitive

The prismatic pentagonal tiling is a dual uniform tiling in the Euclidean plane. It is one of 15 known isohedral pentagon tilings. It can be seen as a stretched hexagonal tiling with a set of parallel bisecting lines through the hexagons.

Conway calls it a iso(4-)pentille.[1] Each of its pentagonal faces has three 120° and two 90° angles.

It is related to the Cairo pentagonal tiling with face configuration V3.

Geometric variations

Monohedral pentagonal tiling type 6 has the same topology, but two edge lengths and a lower p2 (2222) wallpaper group symmetry:

a=d=e, b=c
B+D=180°, 2B=E

Related 2-uniform dual tilings

There are four related 2-uniform dual tilings, mixing in rows of squares or hexagons.

See also


  1. ^ a b Conway, 2008, p.288 table
  2. ^ Order in Space: A design source book, Keith Critchlow, p.74-75, circle pattern F
  3. ^ by Daniel HusonTwo Dimensional symmetry Mutations
  4. ^ Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications 17: 147–165.  
  5. ^


  • , p. 58-65) Regular and uniform tilings (Chapter 2.1:  
  • p37  
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [2]
  • Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern Q2, Dual p. 77-76, pattern 6
  • Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56

External links

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.