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3-4-3-12 tiling

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

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style="background-color:#e7dcc3;width:105px;" |Type2-uniform tiling
style="background-color:#e7dcc3;" |Vertex configuration60px 60px
3.4.3.12 and 3.12.12
style="background-color:#e7dcc3;" |Symmetryp4m, [4,4], (*442)
style="background-color:#e7dcc3;" |Rotation symmetryp4, [4,4]+, (442)
style="background-color:#e7dcc3;" |Properties2-uniform, 3-isohedral, 3-isotoxal

In geometry of the Euclidean plane, the 3-4-3-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, and dodecagons, arranged in two vertex configuration: 3.4.3.12 and 3.12.12.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] #1Chavey (1989)

The 3.12.12 vertex figure alone generates a truncated hexagonal tiling, while the 3.4.3.12 only exists in this 2-uniform tiling. There are 2 3-uniform tilings that contain both of these vertex figures among one more.

It has square symmetry, p4m, [4,4], (*442). It is also called a demiregular tiling by some authors.

Circle Packing

This 2-uniform tiling can be used as a circle packing. Cyan circles are in contact with 3 other circles (1 cyan, 2 pink), corresponding to the V3.122 planigon, and pink circles are in contact with 4 other circles (2 cyan, 2 pink), corresponding to the V3.4.3.12 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 (one dimensional duals to the respective planigons). Both images coincide.

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!Circle Packing

!Ambo

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Dual tiling

The dual tiling has kite ('ties') and isosceles triangle faces, defined by face configurations: V3.4.3.12 and V3.12.12. The kites meet in sets of 4 around a center vertex, and the triangles are in pairs making planigon rhombi. Every four kites and four isosceles triangles make a square of side length 2+\sqrt{3}.

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|File:Small Star Uniform 2 Tiling-2.svg
Dual tiling

|File:Semiplanigon_V.3.4.3.12_(Desmos_Generated).png
V3.4.3.12
Semiplanigon
120px
V3.12.12
Planigon

This is one of the only dual uniform tilings which only uses planigons (and semiplanigons) containing a 30° angle. Conversely, 3.4.3.12; 3.122 is one of the only uniform tilings in which every vertex is contained on a dodecagon.

Related tilings

It has 2 related 3-uniform tilings that include both 3.4.3.12 and 3.12.12 vertex figures:

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3.4.3.12, 3.12.12, 3.4.6.4

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3.4.3.12, 3.12.12, 3.3.4.12

File:Alternate Jewel Uniform 3 Tiling-2.png
V3.4.3.12, V3.12.12, V3.4.6.4

|File:Alternating Petals Uniform 3 Tiling-2.png
V3.4.3.12, V3.12.12, V3.3.4.12

This tiling can be seen in a series as a lattice of 4n-gons starting from the square tiling. For 16-gons (n=4), the gaps can be filled with isogonal octagons and isosceles triangles.

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!4

!8

!12

!16

!20

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Square tiling
Q

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Truncated square tiling
tQ

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3-4-3-12 tiling

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Twice-truncated square tiling
ttQ

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20-gons, squares
trapezoids, triangles

Notes

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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
  • {{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=2 2-Uniform Tiling 2 of 20]

Category:Euclidean plane geometry

Category:Tessellation