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3-4-6-12 tiling#Dual tiling

{{Short description|Uniform Tiling}}

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! style="background-color:#e7dcc3;" colspan=2|3-4-6-12 tiling

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style="background-color:#e7dcc3;width:105px;" |Type2-uniform tiling
style="background-color:#e7dcc3;" |Vertex configuration60px60px
3.4.6.4 and 4.6.12
style="background-color:#e7dcc3;" |Symmetryp6m, [6,3], (*632)
style="background-color:#e7dcc3;" |Rotation symmetryp6, [6,3]+, (632)
style="background-color:#e7dcc3;" |Properties2-uniform, 4-isohedral, 4-isotoxal

In geometry of the Euclidean plane, the 3-4-6-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, hexagons and dodecagons, arranged in two vertex configuration: 3.4.6.4 and 4.6.12.Critchlow, pp. 62–67Grünbaum and Shephard 1986, pp. 65–67[https://web.archive.org/web/20150710230158/http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf In Search of Demiregular Tilings] #4Chavey (1989)

It has hexagonal symmetry, p6m, [6,3], (*632). It is also called a demiregular tiling by some authors.

Geometry

Its two vertex configurations are shared with two 1-uniform tilings:

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!rhombitrihexagonal tiling

!truncated trihexagonal tiling

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3.4.6.4

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4.6.12

It can be seen as a type of diminished rhombitrihexagonal tiling, with dodecagons replacing periodic sets of hexagons and surrounding squares and triangles. This is similar to the Johnson solid, a diminished rhombicosidodecahedron, which is a rhombicosidodecahedron with faces removed, leading to new decagonal faces. The dual of this variant is shown to the right (deltoidal hexagonal insets).

: 160pxFile:Dual of Planar Tiling (Uniform Two 5) 4.6.12; 3.4.6.4 Corrected Variant O.png

Related ''k''-uniform tilings of regular polygons

The hexagons can be dissected into 6 triangles, and the dodecagons can be dissected into triangles, hexagons and squares.

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|+ Dissected polygons

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Hexagon

!colspan=2|Dodecagon
(each has 2 orientations)

File:Insetting_Polygon_for_Uniform_Tilings_1.png

|File:Insetting_Polygon_for_Uniform_Tilings_1.png

|File:Dissection_Polygon_3.png

colspan="3" |Dual Processes (Dual 'Insets')

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!colspan=3|3-uniform tilings

48

!26

!18 (2-uniform)

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[36; 32.4.3.4; 32.4.12]

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[3.42.6; (3.4.6.4)2]

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[36; 32.4.3.4]

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|File:Dual of Planar Tiling (Uniform Two 5) 4.6.12; 3.4.6.4 Corrected Variant I.png
V[36; 32.4.3.4; 32.4.12]

|File:Dual of Planar Tiling (Uniform Two 5) 4.6.12; 3.4.6.4 Corrected Variant II.png
V[3.42.6; (3.4.6.4)2]

|File:Dual of Planar Tiling (Uniform Two 5) 4.6.12; 3.4.6.4 Corrected Variant III.png
V[36; 32.4.3.4]

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! colspan="3" |3-uniform duals

Circle Packing

This 2-uniform tiling can be used as a circle packing. Cyan circles are in contact with 3 other circles (2 cyan, 1 pink), corresponding to the V4.6.12 planigon, and pink circles are in contact with 4 other circles (1 cyan, 2 pink), corresponding to the V3.4.6.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.

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|+

!C[3.4.6.12]

!a[3.4.6.12]

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= Dual tiling =

The dual tiling has right triangle and kite faces, defined by face configurations: V3.4.6.4 and V4.6.12, and can be seen combining the deltoidal trihexagonal tiling and kisrhombille tilings.

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Dual tiling

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V3.4.6.4
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V4.6.12

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Deltoidal trihexagonal tiling

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Kisrhombille tiling

Notes

{{reflist}}

References

  • Keith Critchlow, Order in Space: A design source book, 1970, pp. 62–67
  • Ghyka, M. The Geometry of Art and Life, (1946), 2nd edition, New York: Dover, 1977. Demiregular tiling #15
  • {{The Geometrical Foundation of Natural Structure (book)}} pp. 35–43
  • {{cite book | author-link=Branko Grünbaum | last1=Grünbaum | first1=Branko | last2=Shephard | first2=G. C. | title=Tilings and Patterns | publisher=W. H. Freeman | date=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} p. 65
  • Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37 [https://www.amazon.com/exec/obidos/ASIN/0965640582]

External links

  • {{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons—II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147–165 | doi=10.1016/0898-1221(89)90156-9| doi-access= }}
  • {{cite web | author = Dutch, Steve | title = Uniform Tilings | url = http://www.uwgb.edu/dutchs/symmetry/uniftil.htm | access-date = 2006-09-09 | archive-url = https://web.archive.org/web/20060909053826/http://www.uwgb.edu/dutchs/SYMMETRY/uniftil.htm | archive-date = 2006-09-09 | url-status = dead }}
  • {{MathWorld | urlname=DemiregularTessellation | title=Demiregular tessellation}}
  • [https://web.archive.org/web/20160507010400/http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf In Search of Demiregular Tilings], Helmer Aslaksen
  • [http://probabilitysports.com/tilings.html n-uniform tilings] Brian Galebach, [http://probabilitysports.com/tilings.html?u=0&n=2&t=1 2-Uniform Tiling 1 of 20]

Category:Euclidean plane geometry

Category:Tessellation