Rhombitrihexagonal tiling

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Semiregular tiling of the Euclidean plane

title: "Rhombitrihexagonal tiling" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["euclidean-tilings", "isohedral-tilings", "semiregular-tilings"] description: "Semiregular tiling of the Euclidean plane" topic_path: "general/euclidean-tilings" source: "https://en.wikipedia.org/wiki/Rhombitrihexagonal_tiling" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Semiregular tiling of the Euclidean plane ::

::data[format=table title="Infobox face-uniform tiling |"]

FieldValue
nameDeltoidal trihexagonal tiling
Image_FileTiling small rhombi 3-6 dual simple.svg
TypeDual semiregular tiling
Cox
Face_Listkite
Symmetry_Groupp6m, [6,3], (*632)
Rotation_Groupp6, [6,3]+, (632)
Face_TypeV3.4.6.4[[File:Tiling small rhombi 3-6 dual face.svg
DualRhombitrihexagonal tiling
Property_Listface-transitive
::

In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.

John Conway calls it a rhombihexadeltille. It can be considered a cantellated by Norman Johnson's terminology or an expanded hexagonal tiling by Alicia Boole Stott's operational language.

There are three regular and eight semiregular tilings in the plane.

Uniform colorings

There is only one uniform coloring in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.)

With edge-colorings there is a half symmetry form (3*3) orbifold notation. The hexagons can be considered as truncated triangles, t{3} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{3,6}. The bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, constructed as a snub triangular tiling, .

::data[format=table]

Symmetry[6,3], (*632)[6,3+], (3*3)NameRhombitrihexagonalCantic snub triangularSnub triangularImageSchläflisymbolCoxeterdiagram
[[File:Rhombitrihexagonal tiling uniform coloring.svg120px]]Uniform face coloring[[File:Rhombitrihexagonal tiling snub edge coloring.svg120px]]Uniform edge coloring[[File:Rhombitrihexagonal tiling snub edge coloring nonuniform.svg120px]]Nonuniform geometry[[File:Snub triangular tiling with rhombitrihexagonal coloring.svg120px]]Limit
rr{3,6}s2{3,6}s{3,6}
::

Examples

::data[format=table] | [[File:Wallpaper group-p6m-4.jpg|166px]]From The Grammar of Ornament (1856) | [[File:Kensington board.svg|180px]]The game Kensington | [[File:Semi-regular-floor-3464.JPG|180px]]Floor tiling, Archeological Museum of Seville, Sevilla, Spain | [[File:Nîmes-Temple de Diane-6.jpg|180px]]The Temple of Diana in Nîmes, France | [[File:0 Mosaïque de Castel Guido - Pal. Massimo 1.JPG|180px]] Roman floor mosaic in Castel di Guido | |---|---|---|---|---| ::

Related tilings

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/fc/Circular_rhombitrihexagonal_tilng.svg" caption="access-date=2006-09-09}} The ''rhombitrihexagonal tiling'' is also related to the [[truncated trihexagonal tiling]] by replacing some of the hexagons and surrounding squares and triangles with dodecagons:"] ::

::data[format=table] | 1-uniform | Dissection | 2-uniform dissections | [[File:Regular hexagon.svg|100x100px]][[File:Triangular tiling vertfig.png|100x100px]] | Dual Tilings | [[File:Regular dodecagon.svg|100px]][[File:Hexagonal cupola flat.svg|100px]] | |---|---|---|---|---|---| | [[File:1-uniform n6.svg|200x200px]]3.4.6.4 | [[File:2-uniform 18.png|202x202px]]3.3.4.3.4 & 36 | [[File:D Inset to CH.gif|201x201px]]to CH | | | | | [[File:1-uniform 6b.png|alt=|201x201px]]3.4.6.4 | [[File:1-uniform 3.png|alt=|200x200px]]4.6.12 | [[File:D Outset to 3.gif|alt=]]to 3 | | | | ::

Circle packing

The rhombitrihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with four other circles in the packing (kissing number). The translational lattice domain (red rhombus) contains six distinct circles. :[[File:1-uniform-6-circlepack.svg|alt=|201x201px]]

Wythoff construction

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 eight forms, seven topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

Symmetry mutations

This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

Deltoidal trihexagonal tiling

name=Deltoidal trihexagonal tiling | Image_File = Tiling small rhombi 3-6 dual simple.svg | Type = Dual semiregular tiling | Cox= | Face_List = kite | Symmetry_Group = p6m, [6,3], (*632) | Rotation_Group = p6, [6,3]+, (632) | Face_Type = V3.4.6.4[[File:Tiling small rhombi 3-6 dual face.svg|80px|right]] | Dual = Rhombitrihexagonal tiling | Property_List = face-transitive | ::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/bf/Aperiodic_monotile_smith_2023.svg" caption="A 2023 discovered aperiodic monotile, solving the [[Einstein problem]], is composed by a collection of 8 kites from the deltoidal trihexagonal tiling"] ::

The deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. Conway called it a tetrille. The edges of this tiling can be formed by the intersection overlay of the regular triangular tiling and a hexagonal tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90°. It is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.{{citation | last1 = Kirby | first1 = Matthew | last2 = Umble | first2 = Ronald | arxiv = 0908.3257 | doi = 10.4169/math.mag.84.4.283 | issue = 4 | journal = Mathematics Magazine | mr = 2843659 | pages = 283–289 | title = Edge tessellations and stamp folding puzzles | volume = 84 | year = 2011}}.

The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling. Its faces are deltoids or kites. :[[File:P5 dual.svg|320px]]

Related polyhedra and tilings

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

This tiling has face transitive variations, that can distort the kites into bilateral trapezoids or more general quadrilaterals. Ignoring the face colors below, the fully symmetry is p6m, and the lower symmetry is p31m with three mirrors meeting at a point, and threefold rotation points. ::data[format=table title="Isohedral variations"]

Symmetryp6m, [6,3], (*632)p31m, [6,3+], (3*3)
Form[[File:Isohedral tiling p4-41.png200px]]
FacesKiteHalf regular hexagon
::

This tiling is related to the trihexagonal tiling by dividing the triangles and hexagons into central triangles and merging neighboring triangles into kites. :[[File:P3 hull.png|320px]]

The deltoidal trihexagonal tiling is a part of a set of uniform dual tilings, corresponding to the dual of the rhombitrihexagonal tiling.

Symmetry mutations

This tiling is topologically related as a part of sequence of tilings with face configurations V3.4.n.4, and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

Other deltoidal (kite) tiling

Other deltoidal tilings are possible.

Point symmetry allows the plane to be filled by growing kites, with the topology as a square tiling, V4.4.4.4, and can be created by crossing string of a dream catcher. Below is an example with dihedral hexagonal symmetry.

Another face transitive tiling with kite faces, also a topological variation of a square tiling and with face configuration V4.4.4.4. It is also vertex transitive, with every vertex containing all orientations of the kite face. ::data[format=table title=""]

SymmetryD6, [6], (*66)pmg, [∞,(2,∞)+], (22*)p6m, [6,3], (*632)TilingConfigurationV4.4.4.4V6.4.3.4
[[File:Inscribedstar.svg180px]][[File:Isohedral tiling p4-53.svg320px]][[File:Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg210px]]
::

Notes

References

References

  1. Conway, 2008, p288 table
  2. Chavey, D.. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications.
  3. "Uniform Tilings".
  4. Order in Space: A design source book, Keith Critchlow, p.74-75, pattern B
  5. "Dual tessellation".
  6. [[Tilings and patterns]]

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euclidean-tilingsisohedral-tilingssemiregular-tilings