Truncated trihexagonal tiling: Difference between revisions

Kisrhombille tiling
Kisrhombille tiling
Type	Dual semiregular tiling
Faces	30-60-90 triangle
Coxeter diagram
Symmetry group	p6m, [6,3], (*632)
Rotation group	p6, [6,3]+, (632)
Dual polyhedron	truncated trihexagonal tiling
Face configuration	V4.6.12
Properties	face-transitive

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Truncated trihexagonal tiling

Type	Semiregular tiling
Vertex configuration	4.6.12
Schläfli symbol	tr{6,3} or $t{\begin{Bmatrix}6\\3\end{Bmatrix}}$
Wythoff symbol	2 6 3 \|
Coxeter diagram
Symmetry	p6m, [6,3], (*632)
Rotation symmetry	p6, [6,3]⁺, (632)
Bowers acronym	Othat
Dual	Kisrhombille tiling
Properties	Vertex-transitive

In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}.

Names

The name truncated trihexagonal tiling is analogous to truncated cuboctahedron and truncated icosidodecahedron, and misleading in the same way. An actual truncation of the trihexagonal tiling has rectangles instead of squares, and its hexagonal and dodecagonal faces can not both be regular.

Alternate interchangeable names are:

Great rhombitrihexagonal tiling
Rhombitruncated trihexagonal tiling
Omnitruncated hexagonal tiling, omnitruncated triangular tiling
Conway calls it a truncated hexadeltille.^[1]

Trihexagonal tiling and its truncation

Uniform colorings

There is only one uniform coloring of a truncated trihexagonal tiling, with faces colored by polygon sides. A 2-uniform coloring has two colors of hexagons. 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares.

	1-uniform	2-uniform	3-uniform
Coloring
Symmetry	p6m, [6,3], (*632)	p3m1, [3^[3]], (*333)

Related 2-uniform tilings

The truncated trihexagonal tiling has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling. The first dissects the hexagons into 6 triangles. The other two dissect the dodecagons into a central hexagon and surrounding triangles and square, in two different orientations.^[2]^[3]

Semiregular	Dissections	Semiregular	2-uniform		3-uniform

Dual	Insets

Circle packing

The Truncated trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing number).^[4]

Kisrhombille tiling

The kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 30-60-90 triangles with 4, 6, and 12 triangles meeting at each vertex.

Subdividing the faces of these tilings creates the kisrhombille tiling. (Compare the disdyakis hexa-, dodeca- and triacontahedron, three Catalan solids similar to this tiling.)

The kisrhombille tiling under its dual (left) and under the floret pentagonal tiling (right), from which it can be created as a partial truncation.

Construction from rhombille tiling

Conway calls it a kisrhombille^[1] for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings, like 3-7 kisrhombille.

It can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. (Alternately it can be seen as a bisected triangular tiling divided into 6 triangles, or as an infinite arrangement of lines in six parallel families.)

It is labeled V4.6.12 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 12 triangles.

Symmetry

The kisrhombille tiling triangles represent the fundamental domains of p6m, [6,3] (*632 orbifold notation) wallpaper group symmetry. There are a number of small index subgroups constructed from [6,3] by mirror removal and alternation. [1⁺,6,3] creates *333 symmetry, shown as red mirror lines. [6,3⁺] creates 3*3 symmetry. [6,3]⁺ is the rotational subgroup. The commutator subgroup is [1⁺,6,3⁺], which is 333 symmetry. A larger index 6 subgroup constructed as [6,3*], also becomes (*333), shown in blue mirror lines, and which has its own 333 rotational symmetry, index 12.

Small index subgroups [6,3] (*632)
Index	1	2		3	6
Diagram
Intl (orb.) Coxeter	p6m (*632) [6,3] = =	p3m1 (*333) [1⁺,6,3] = =	p31m (3*3) [6,3⁺] =	cmm (2*22)	pmm (*2222)	p3m1 (333) [6,3] = =
Direct subgroups
Index	2	4		6	12
Diagram
Intl (orb.) Coxeter	p6 (632) [6,3]⁺ = =	p3 (333) [1⁺,6,3⁺] = =		p2 (2222)	p2 (2222)	p3 (333) [1⁺,6,3*] = =

Related polyhedra and tilings

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 v t e
Symmetry: [6,3], (*632)							[6,3]⁺ (632)	[6,3⁺] (3*3)
{6,3}	t{6,3}	r{6,3}	t{3,6}	{3,6}	rr{6,3}	tr{6,3}	sr{6,3}	s{3,6}


6³	3.12²	(3.6)²	6.6.6	3⁶	3.4.6.4	4.6.12	3.3.3.3.6	3.3.3.3.3.3
Uniform duals

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

Symmetry mutations

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedra), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

*n32 symmetry mutation of omnitruncated tilings: 4.6.2n v t e
Sym. *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
Sym. *n32 [n,3]	*232 [2,3]	*332 [3,3]	*432 [4,3]	*532 [5,3]	*632 [6,3]	*732 [7,3]	*832 [8,3]	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]	[3i,3]
Figures
Config.	4.6.4	4.6.6	4.6.8	4.6.10	4.6.12	4.6.14	4.6.16	4.6.∞	4.6.24i	4.6.18i	4.6.12i	4.6.6i
Duals
Config.	V4.6.4	V4.6.6	V4.6.8	V4.6.10	V4.6.12	V4.6.14	V4.6.16	V4.6.∞	V4.6.24i	V4.6.18i	V4.6.12i	V4.6.6i

Notes

^ ^a ^b Conway, 2008, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table
^ 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.
^ "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern D

References

Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 41. ISBN 0-486-23729-X.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern G, Dual p. 77-76, pattern 4
Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56

External links

Weisstein, Eric W. "Uniform tessellation". MathWorld.
Weisstein, Eric W. "Semiregular tessellation". MathWorld.
Klitzing, Richard. "2D Euclidean tilings x3x6x - othat - O9".

[conway2008-1] Conway, 2008, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table

[2] 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.

[3] "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.

[Critchlow-4] Order in Space: A design source book, Keith Critchlow, p.74-75, pattern D

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
 {{Uniform tiles db|Uniform tiling stat table|Ugrth}}
 In [[geometry]], the '''truncated trihexagonal tiling''' is one of eight [[List of uniform tilings|semiregular tilings]] of the Euclidean plane. There are one [[Square (geometry)|square]], one [[hexagon]], and one [[dodecagon]] on each [[vertex (geometry)|vertex]].  It has [[Schläfli symbol]] of ''tr''{3,6}.
+[[File:Rhombic_truncated_trihexagonal_tiling.svg|thumb|An equilateral variation with rhombi instead of squares, and [[isotoxal figure|isotoxal]] hexagons instead of regular]]
-== Other names ==
+== Names ==
+{|
+| <!-- Does someone know a better way to make the image float right, but not below the infobox? -->
+The name ''truncated trihexagonal tiling'' is analogous to ''[[truncated cuboctahedron]]'' and ''[[truncated icosidodecahedron]]'', and misleading in the same way. An actual [[truncation (geometry)|truncation]] of the [[trihexagonal tiling]] has rectangles instead of squares, and its hexagonal and dodecagonal faces can not both be regular.
+Alternate interchangeable names are:
 * Great rhombitrihexagonal tiling
 * Rhombitruncated trihexagonal tiling
 * Omnitruncated hexagonal tiling, omnitruncated triangular tiling
-* [[John Horton Conway|Conway]] calls it a '''truncated hexadeltille''', constructed as a [[truncation (geometry)|truncation]] operation applied to a [[trihexagonal tiling]] (hexadeltille).<ref name=conway2008>Conway, 2008, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table</ref>
+* [[John Horton Conway|Conway]] calls it a ''truncated hexadeltille''.<ref name=conway2008>Conway, 2008, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table</ref>
+|
+{{multiple image
+ | align = left  | width = 130
+ | image1 = Tiling 3-6 simple.svg
+ | image2 = Tiling 3-6, truncated.svg
+ | footer = Trihexagonal tiling and its truncation
+}}
+|}
 == Uniform colorings ==
@@ Line 29: / Line 43: @@
 == Related 2-uniform tilings==
-The ''truncated trihexagonal tiling'' has three related [[2-uniform tiling]]s, one being a 2-uniform coloring of the semiregular [[rhombitrihexagonal tiling]]. The first dissects the hexagons into 6 triangles. The other two dissect the [[dodecagon]]s into a central hexagon and surrounding triangles and square, in two different orientations.<ref>{{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons&mdash;II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147&ndash;165 | doi=10.1016/0898-1221(89)90156-9|ref=harv}}</ref><ref>http://www.uwgb.edu/dutchs/symmetry/uniftil.htm</ref>
+The ''truncated trihexagonal tiling'' has three related [[2-uniform tiling]]s, one being a 2-uniform coloring of the semiregular [[rhombitrihexagonal tiling]]. The first dissects the hexagons into 6 triangles. The other two dissect the [[dodecagon]]s into a central hexagon and surrounding triangles and square, in two different orientations.<ref>{{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons&mdash;II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147&ndash;165 | doi=10.1016/0898-1221(89)90156-9| doi-access=free }}</ref><ref>{{cite web |url=http://www.uwgb.edu/dutchs/symmetry/uniftil.htm |title=Uniform Tilings |access-date=2006-09-09 |url-status=dead |archive-url=https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm |archive-date=2006-09-09 }}</ref>
-{| class=wikitable
+{| class=wikitable valign = center
 !Semiregular
+!Dissections
-!Dissected
+!Semiregular
-!2-uniform
+! colspan="2" |2-uniform
 !3-uniform
+|- align = center
-|-
-|rowspan=3|[[File:1-uniform 3.png|150px]]
+|[[File:1-uniform n3.svg|200x200px]]
-![[File:Regular hexagon.svg|75px]]<BR>[[File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|75px]]
+|[[File:Regular hexagon.svg|75px]][[File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|75px]]<br>[[File:Hexagonal cupola flat.png|75px]][[File:Regular dodecagon.svg|75px]]
-|[[File:2-uniform 13b.png|150px]]
+|[[File:1-uniform_6b.png|201x201px]]
-|[[File:3-uniform 6b.png|150px]]
+|[[File:2-uniform_5b.png|201x201px]]
+|[[File:2-uniform 13b.png|202x202px]]
-|-
+|[[File:3-uniform 6b.png|202x202px]]
-!Dissected
+|- align = center
-!Semiregular
+!Dual
-!2-uniform
+! Insets
-|-
+! colspan="4" |
-|[[File:Hexagonal cupola flat.png|75px]]<BR>[[File:Regular dodecagon.svg|75px]]
+|- align = center
-|[[File:1-uniform 6b.png|150px]]
-|[[File:2-uniform 5b.png|150px]]
+|[[File:1-Uniform 3.png|frameless|201x201px]]
+|[[File:Inset Variations of Dual Uniform Tiling.svg|200x200px]]
-<!--|-
+|[[File:3 Inset to D.gif|frameless|201x201px]]
-!colspan=3|4-uniform
+|[[File:3 Inset to rD.gif|frameless|201x201px]]
-|-
-|[[File:4-uniform 25.svg|150px]]
+|[[File:3 Inset to SH.gif|frameless|201x201px]]
-|[[File:4-uniform 28.svg|150px]]
+|[[File:3 Inset to 3SH.gif|frameless|201x201px]]
-|[[File:4-uniform 29.svg|150px]]-->
 |}
 == Circle packing ==
-The Truncated trihexagonal tiling can be used as a [[circle packing]], placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing ([[kissing number]]).<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, pattern D</ref> <!--The packing density is ... % coverage.)--> Circles can be alternatedly colored in this packing with an even number of sides of all the regular polygons of this tiling.
+The Truncated trihexagonal tiling can be used as a [[circle packing]], placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing ([[kissing number]]).<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, pattern D</ref> <!--The packing density is ... % coverage.)-->
+:[[File:1-uniform-3-circlepack.svg|200px]]
-The gap inside each [[hexagon]] allows for one circle, and each [[dodecagon]] allows for 7 circles, creating a dense 4-uniform packing.
-[[File:Truncated rhombitrihexagonal tiling circle packing.png|240px]]
-[[File:Truncated rhombitrihexagonal tiling circle packing2.png|240px]]
-[[File:Truncated rhombitrihexagonal tiling circle packing3.png|240px]]
 == Kisrhombille tiling ==
 {{Infobox face-uniform tiling  |
   name=Kisrhombille tiling|
-  Image_File=1-uniform 3 dual.svg|
+  Image_File=Tiling great rhombi 3-6 dual simple.svg|
   Type=[[List of uniform tilings|Dual semiregular tiling]]|
   Cox={{CDD|node_f1|3|node_f1|6|node_f1}} |
@@ Line 74: / Line 83: @@
   Symmetry_Group=p6m, [6,3], (*632)|
   Rotation_Group  = p6, [6,3]<sup>+</sup>, (632) |
-  Face_Type=V4.6.12[[File:Tiling face 4-6-12.svg|60px|right]]|
+  Face_Type=V4.6.12[[File:Tiling great rhombi 3-6 dual face.svg|60px]]|
   Dual=truncated trihexagonal tiling|
   Property_List=[[face-transitive]]|
 }}
-The '''kisrhombille tiling''' or '''3-6 kisrhombille tiling''' is a tiling of the Euclidean plane. It is constructed by congruent 30-60 degree [[right triangle]]s with 4, 6, and 12 triangles meeting at each vertex.
+The '''kisrhombille tiling''' or '''3-6 kisrhombille tiling''' is a tiling of the Euclidean plane. It is constructed by congruent [[30 60 90|30-60-90 triangle]]s with 4, 6, and 12 triangles meeting at each vertex.
+Subdividing the faces of these tilings creates the kisrhombille tiling. (Compare the disdyakis [[Tetrakis hexahedron#Symmetry|hexa-]], [[Disdyakis dodecahedron#Symmetry|dodeca-]] and [[Disdyakis triacontahedron#Symmetry|triacontahedron]], three [[Catalan solid]]s similar to this tiling.)
+<gallery>
+File:Kisrhombille in deltoidal.svg|[[Deltoidal trihexagonal tiling|3-6 deltoidal]]
+File:Kisrhombille in rhombille (blue).svg|[[Rhombille tiling|rhombille]]
+File:Kisrhombille in hexagonal (red).svg|[[Hexagonal tiling|hexagonal]]
+[File:Kisrhombille in triangular (yellow).svg|[[Triangular tiling|triangular]]
+</gallery>
+{{multiple image
+ | align = left  | width = 150
+ | image1 = Kisrhombille under truncated trihexagonal.svg
+ | image2 = Kisrhombille under floret pentagonal left (light faces).svg
+ | footer = The kisrhombille tiling under its dual (left) and under the [[floret pentagonal tiling]] (right), from which it can be created as a [[Truncation (geometry)#Alternation or partial truncation|partial truncation]].
+}}
 === Construction from rhombille tiling ===
 [[John Horton Conway|Conway]] calls it a '''kisrhombille'''<ref name=conway2008 /> for his [[Conway kis operator|kis]] vertex bisector operation applied to the [[rhombille tiling]]. More specifically it can be called a '''3-6 kisrhombille''', to distinguish it from other similar hyperbolic tilings, like [[3-7 kisrhombille]].
-[[File:Rhombic star tiling.png|150px|thumb|left|The related [[rhombille tiling]] becomes the kisrhombille by cutting each rhombic face along its diagonals into four triangular faces]]
 It can be seen as an equilateral [[hexagonal tiling]] with each hexagon divided into 12 triangles from the center point. (Alternately it can be seen as a bisected [[triangular tiling]] divided into 6 triangles, or as an infinite [[arrangement of lines]] in six parallel families.)
@@ Line 89: / Line 112: @@
 It is labeled V4.6.12 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 12 triangles.
-{{Clear}}
 === Symmetry===
 The ''kisrhombille tiling'' triangles represent the fundamental domains of p6m, [6,3] (*632 [[orbifold notation]]) [[wallpaper group]] symmetry. There are a number of [[Template:632_symmetry_table|small index subgroups constructed from [6,3]]] by mirror removal and alternation. [1<sup>+</sup>,6,3] creates *333 symmetry, shown as red mirror lines. [6,3<sup>+</sup>] creates 3*3 symmetry. [6,3]<sup>+</sup> is the rotational subgroup. The commutator subgroup is [1<sup>+</sup>,6,3<sup>+</sup>], which is 333 symmetry. A larger index 6 subgroup constructed as [6,3*], also becomes (*333), shown in blue mirror lines, and which has its own 333 rotational symmetry, index 12.
 {{632 symmetry table}}
-=== Practical uses ===
-The ''kisrhombille tiling'' is a useful starting point for making paper models of [[deltahedron|deltahedra]], as each of the equilateral triangles can serve as faces, the edges of which adjoin isosceles triangles that can serve as tabs for gluing the model together.{{citation needed|date=January 2013}}
-{{Clear}}
 == Related polyhedra and tilings ==
@@ Line 111: / Line 126: @@
 == See also ==
-{{Commons category|Uniform tiling 4-6-12}}
+{{Commons category|Uniform tiling 4-6-12 (truncated trihexagonal tiling)}}
 * [[Tilings of regular polygons]]
 * [[List of uniform tilings]]
@@ Line 120: / Line 135: @@
 == References ==
 * {{The Geometrical Foundation of Natural Structure (book)|page=41}}
-* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} [http://www.akpeters.com/product.asp?ProdCode=2205]
+* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]
 * Keith Critchlow, ''Order in Space: A design source book'', 1970, p.&nbsp;69-61, Pattern G, Dual p.&nbsp;77-76, pattern 4
-* Dale Seymour and Jill Britton, ''Introduction to Tessellations'', 1989, {{isbn|978-0866514613}}, pp.&nbsp;50–56
+* Dale Seymour and [[Jill Britton]], ''Introduction to Tessellations'', 1989, {{isbn|978-0866514613}}, pp.&nbsp;50–56
 == External links==