Triangular tiling: Difference between revisions

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|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)

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== 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 [[Root system#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]].

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<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>

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|[[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:Heptakis heptagonal tiling.svg|60px]]<BR/>[[Order-7 truncated triangular tiling|V7.6.6]]

|}

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{| class=wikitable

|-

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

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

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

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== References ==

{{Reflist}}

== Sources ==

* [[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

* {{cite book | author=Grünbaum, Branko | author-link=Branko Grünbaum | author2= 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.&nbsp;102–107)