Triangular tiling: Difference between revisions

Revision as of 21:29, 2 August 2015

Latest revision as of 10:12, 29 August 2025

(42 intermediate revisions by 22 users not shown)

Line 1:

{{short description|Regular tiling of the plane}}

{{Uniform tiles db|Reg tiling stat table|Ut}}

In [[geometry]], the '''triangular tiling''' or '''triangular tessellation''' is one of the three regular [[tessellation|tiling]]s of the [[Euclidean plane]]. Because the internal angle of the equilateral [[triangle]] is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has [[Schläfli symbol]] of {3,6}.

In [[geometry]], the '''triangular tiling''' or '''triangular tessellation''' is one of the three regular [[tessellationEuclidean tilings by convex regular polygons#Regular tilings|tilingregular tilings]]s of the [[Euclidean plane]], and is the only such tiling where the constituent shapes are not [[parallelogon]]s. Because the internal angle of the equilateral [[equilateral triangle]] is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has [[Schläfli symbol]] of {{math|{3,6}.}}

[[John Horton Conway|Conway]] calls it a '''deltille''', named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a '''kishextille''' by a [[Conway kis operator|kis]] operation that adds a center point and triangles to replace the faces of a [[hextille]].

English mathematician [[John Horton Conway|John Conway]] callscalled it a '''deltille''', named from the triangular shape of the Greek letter [[Delta (letter)|delta]] (Δ). The triangular tiling can also be called a '''kishextille''' by a [[Conway kis operator|kis]] operation that adds a center point and triangles to replace the faces of a [[hextille]].

It is one of [[List_of_regular_polytopes#Euclidean_tilings|three regular tilings of the plane]]. The other two are the [[square tiling]] and the [[hexagonal tiling]].

It is one of [[List_of_regular_polytopesList of regular polytopes#Euclidean_tilingsEuclidean tilings|three regular tilings of the plane]]. The other two are the [[square tiling]] and the [[hexagonal tiling]].

== Uniform colorings ==

== Uniform colorings ==

There are 9 distinct [[uniform coloring]]s of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314.<ref>Tilings and Patterns, p.102-107</ref>

[[File:Triangular_tiling_4-color.svg|thumb|A 2-uniform triangular tiling, 4 colored triangles, related to the [[geodesic polyhedron]] as {3,6+}_2,0.]]

There are 9 distinct [[uniform coloring]]s of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314.<ref>''[[Tilings and Patternspatterns]]'', p.102-107</ref>

There is one class of [[Archimedean coloring]]s, 111112, (marked with a *) which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example givenshown is 2-uniform, but there are infinitely many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows.

{| class="wikitable"

Line 20 ⟶ 22:

|BGCOLOR="#c0c0ff"|111112(*)

|- align=center

|[[File:Uniform triangular tiling 111111.pngsvg|75px]]

|[[File:Uniform triangular tiling 121212.pngsvg|75px]]

|[[File:Uniform triangular tiling 111222.pngsvg|75px]]

|[[File:Uniform triangular tiling 112122.pngsvg|75px]]

|[[File:2-uniform_triangular_tiling_111112.pngsvg|75px]]

|- align=center

|p6m (*632)

Line 52 ⟶ 54:

== A2 lattice and circle packings ==

{{distinguish|Strukturbericht designation#A-compounds{{!}}the A2 crystal lattice structure in the Strukturbericht classification system}}

[[File:Compound 3 triangular tilings.pngsvg|thumb|The A{{sup sub|*|2}} lattice as three triangular tilings: {{CDD|node_1|split1|branch}} + {{CDD|node|split1|branch_10lu}} + {{CDD|node|split1|branch_01ld}}]]

The [[vertex arrangement]] of the triangular tiling is called an [[A2Root latticesystem#Explicit construction of the irreducible root systems|A₂ lattice]].<ref>{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html|title = The Lattice A2}}</ref> It is the 2-dimensional case of a [[simplectic honeycomb]].

The A{{sup sub|*|2}} lattice (also called A{{sup sub|3|2}}) can be constructed by the union of all three A₂ lattices, and equivalent to the A₂ lattice.

:{{CDD|node_1|split1|branch}} + {{CDD|node|split1|branch_10lu}} + {{CDD|node|split1|branch_01ld}} = dual of {{CDD|node_1|split1|branch_11}} = {{CDD|node_1|split1|branch}}

The vertices of the triangular tiling are the centers of the densest possible [[circle packing]].<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, p.74-75, pattern 1</ref> Every circle is in contact with 6 other circles in the packing ([[kissing number]]). The packing density is <math>\{{frac|{{\pi}}|{{\sqrt{|12}}</math>}} or 90.69%. Since the union of 3 A₂ lattices is also an A₂ lattice, the circle packing can be given with 3 colors of circles.

The [[voronoi cell]] of a triangular tiling is a [[hexagon]], and so the [[voronoi tessellation]], the hexagonal tiling, has a direct correspondence to the circle packings.

:[[File:1-uniform-11-circlepack.svg|200px]]

The [[voronoi cell]] of a triangular tiling is a [[hexagon]], and so the [[voronoi tessellation]], the hexagonal tiling has a direct correspondence to the circle packings.

{| class=wikitable

!A₂ lattice circle packing

!A{{sup sub|*|2}} lattice circle packing

|-

|[[File:triangular tiling circle packing.png|240px]]

|[[File:Triangular tiling circle packing3.png|240px]]

|}

== Geometric variations ==

Triangular tilings can be made with the equivalent {3,6} topology as the regular tiling (6 triangles around every vertex). With identical faces ([[Face-transitive|face-transitivity]]) and [[vertex-transitive|vertex-transitivity]], there are 5 variations. Symmetry given assumes all faces are the same color.<ref>''[[Tilings and Patterns]]'', from list of 107 isohedral tilings, p.473-481</ref>

<gallery>

Isohedral_tiling_p3-11.pngsvg|[[Scalene triangle]]<BR/>p2 symmetry

Isohedral_tiling_p3-12.pngsvg|Scalene triangle<BR/>pmg symmetry

Isohedral_tiling_p3-13.pngsvg|[[Isosceles triangle]]<BR/>cmm symmetry

Isohedral_tiling_p3-11b.png|[[Right triangle]]<BR/>cmm symmetry

Isohedral_tiling_p3-14.pngsvg|[[Equilateral triangle]]<BR/>p6m symmetry

</gallery>

== Related polyhedra and tilings ==

The planar tilings are related to [[polyhedra]]. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a [[pyramid (geometry)|pyramid]]. These can be expanded to [[Platonic solid]]s: five, four and three triangles on a vertex define an [[icosahedron]], [[octahedron]], and [[tetrahedron]] respectively.

This tiling is topologically related as a part of sequence of regular polyhedra with [[Schläfli symbol]]s {3,n}, continuing into the [[Hyperbolic space|hyperbolic plane]].

Line 92 ⟶ 87:

{| class="wikitable"

|- align=center

|[[File:Triakistetrahedron.jpg|60px]]<BR/>[[Triakis tetrahedron|V3.6.6]]

|[[File:Tetrakishexahedron.jpg|60px]]<BR/>[[Tetrakis hexahedron|V4.6.6]]

|[[File:Pentakisdodecahedron.jpg|60px]]<BR/>[[Pentakis dodecahedron|V5.6.6]]

|[[File:Uniform polyhedron-63-t2.pngsvg|60px]]<BR/>V6.6.6

|[[File:Order3 heptakisHeptakis heptagonal tiltiling.pngsvg|60px]]<BR/>[[Order-7 truncated triangular tiling|V7.6.6]]

|}

Line 108 ⟶ 103:

{{Triangular tiling table}}

== Related regular complex apeirogons ==

There are 4 [[regular complex apeirogon]]s, sharing the vertices of the triangular tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons ''p''{''q''}''r'' are constrained by: 1/''p'' + 2/''q'' + 1/''r'' = 1. Edges have ''p'' vertices, and vertex figures are ''r''-gonal.<ref>Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.</ref>

The first is made of 2-edges, and next two are triangular edges, and the last has overlapping hexagonal edges.

{| class=wikitable

|-

|[[File:Complex apeirogon 2-6-6.svg|160px]]

|[[File:Complex apeirogon 3-4-6.png|160px]]

|[[File:Complex apeirogon 3-6-3.png|160px]]

|[[File:Complex apeirogon 6-3-6.png|160px]]

|-

!2{6}6 or {{CDD|node_1|6|6node}}

!3{4}6 or {{CDD|3node_1|4|6node}}

!3{6}3 or {{CDD|3node_1|6|3node}}

!6{3}6 or {{CDD|6node_1|3|6node}}

|}

=== Other triangular tilings===

Line 113 ⟶ 126:

{| class=wikitable

|- align=center valign=bottom

|[[File:1-uniform 3 dual.svg|240px]]<br/>[[Kisrhombille tiling|Kisrhombille]]<BR/>30°-60°-90° right triangles

|[[File:1-uniform 2 dual.svg|240px]]<br/>[[Tetrakis square tiling|Kisquadrille]]<BR/>45°-45°-90° right triangles

|[[File:1-uniform 4 dual.svg|240px]]<br/>[[Triakis triangular tiling|Kisdeltile]]<BR/>30°-30°-120° isosceles triangles

|}

==See also==

{{CommonscatCommons category|Order-6 triangular tiling}}

* [[Triangular tiling honeycomb]]

* [[Simplectic honeycomb]]

Line 127 ⟶ 140:

== References ==

{{reflistReflist}}

* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.&nbsp;296, Table II: Regular honeycombs

== Sources ==

* {{cite book|author=[[Branko Grünbaum|Grünbaum, Branko]] ; and Shephard, G. C.| title=Tilings and Patterns| ___location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1}} (Chapter 2.1: ''Regular and uniform tilings'', p.&nbsp;58-65, Chapter 2.9 Archimedean and Uniform colorings pp.102-107)

* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN {{isbn|0-486-61480-8}} p.&nbsp;296, Table II: Regular honeycombs

* {{cite book | author=[[Grünbaum, Branko Grünbaum|Grünbaum, author-link=Branko]] ;Grünbaum | andauthor2= Shephard, G. C. | name-list-style= amp | title=Tilings and Patterns | ___location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} (Chapter 2.1: ''Regular and uniform tilings'', p.&nbsp;58-65, Chapter 2.9 Archimedean and Uniform colorings pp.102-107&nbsp;102–107)

* {{The Geometrical Foundation of Natural Structure (book)}} p35

* John H. Conway, Heidi Burgiel, Chaim Goodman-StrassStrauss, ''The Symmetries of Things'' 2008, ISBN {{isbn|978-1-56881-220-5}} [httphttps://wwwweb.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]

== External links ==

Line 146 ⟶ 161:

[[Category:Isohedral tilings]]

[[Category:Regular tilings]]

[[Category:Triangular tilings| ]]

[[Category:Regular tessellations]]