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demiregular tiling

In geometry, the demiregular tilings are a set of Euclidean tessellations made from 2 or more regular polygon faces. Different authors have listed different sets of tilings. A more systematic approach looking at symmetry orbits are the 2-uniform tilings of which there are 20. Some of the demiregular ones are actually 3-uniform tilings.

20 2-uniform tilings

Grünbaum and Shephard enumerated the full list of 20 2-uniform tilings in Tilings and patterns, 1987:

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|+ 2-uniform tilings

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|cmm, 2*22
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(44; 33.42)1

|cmm, 2*22
120px
(44; 33.42)2

|pmm, *2222
120px
(36; 33.42)1

|cmm, 2*22
120px
(36; 33.42)2

|cmm, 2*22
120px
(3.42.6; (3.6)2)2

|pmm, *2222
120px
(3.42.6; (3.6)2)1

|pmm, *2222
120px
((3.6)2; 32.62)

align=center valign=top

|p4m, *442
120px
(3.12.12; 3.4.3.12)

|p4g, 4*2
120px
(33.42; 32.4.3.4)1

|pgg, 2×
120px
(33.42; 32.4.3.4)2

|p6m, *632
120px
(36; 32.62)

|p6m, *632
120px
(36; 34.6)1

|p6, 632
120px
(36; 34.6)2

|cmm, 2*22
120px
(32.62; 34.6)

align=center valign=top

|p6m, *632
120px
(36; 32.4.3.4)

|p6m, *632
120px
(3.4.6.4; 32.4.3.4)

|p6m, *632
120px
(3.4.6.4; 33.42)

|p6m, *632
120px
(3.4.6.4; 3.42.6)

|p6m, *632
120px
(4.6.12; 3.4.6.4)

|p6m, *632
120px
(36; 32.4.12)

Ghyka's list (1946)

Ghyka lists 10 of them with 2 or 3 vertex types, calling them semiregular polymorph partitions.Ghyka (1946) pp. 73-80

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|120px

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|Plate XXVII
No. 12
4.6.12
3.4.6.4

|No. 13
3.4.6.4
3.3.3.4.4

|No. 13 bis.
3.4.4.6
3.3.4.3.4

|No. 13 ter.
3.4.4.6
3.3.3.4.4

|Plate XXIV
No. 13 quatuor.
3.4.6.4
3.3.4.3.4

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|

|

|

|120px

|120px

align=center valign=top

|No. 14
33.42
36

|Plate XXVI
No. 14 bis.
3.3.4.3.4
3.3.3.4.4
36

|No. 14 ter.
33.42
36

|No. 15
3.3.4.12
36

|Plate XXV
No. 16
3.3.4.12
3.3.4.3.4
36

Steinhaus's list (1969)

Steinhaus gives 5 examples of non-homogeneous tessellations of regular polygons beyond the 11 regular and semiregular ones.Steinhaus, 1969, p.79-82. (All of them have 2 types of vertices, while one is 3-uniform.)

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!colspan=4|2-uniform

!3-uniform

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|120px

|120px

|120px

|120px

align=center

|Image 85
33.42
3.4.6.4

|Image 86
32.4.3.4
3.4.6.4

|Image 87
3.3.4.12
36

|Image 89
33.42
32.4.3.4

|Image 88
3.12.12
3.3.4.12

Critchlow's list (1970)

Critchlow identifies 14 demi-regular tessellations, with 7 being 2-uniform, and 7 being 3-uniform.

He codes letter names for the vertex types, with superscripts to distinguish face orders. He recognizes A, B, C, D, F, and J can't be a part of continuous coverings of the whole plane.

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!A
(none)

B
(none)		C
(none)		D
(none)		E
(semi)		F
(none)		G
(semi)		H
(semi)		J
(none)		K (2)
(reg)
align=center valign=bottom

|BGCOLOR="#e0e0e0"|40px
3.7.42

|BGCOLOR="#e0e0e0"|40px
3.8.24

|BGCOLOR="#e0e0e0"|40px
3.9.18

|BGCOLOR="#e0e0e0"|40px
3.10.15

|BGCOLOR="#ffe0e0"|40px
3.12.12

|BGCOLOR="#e0e0e0"|40px
4.5.20

|BGCOLOR="#ffe0e0"|40px
4.6.12

|BGCOLOR="#ffe0e0"|40px
4.8.8

|BGCOLOR="#e0e0e0"|40px
5.5.10

|BGCOLOR="#d0ffd0"|40px
63

L1
(demi)||L2
(demi)||M1
(demi)||M2
(semi)||N1
(demi)||N2
(semi)||P (3)
(reg)||Q1
(semi)||Q2
(semi)||R
(semi)||S (1)
(reg)
align=center valign=bottom

|BGCOLOR="#e0e0ff"|40px
3.3.4.12

|BGCOLOR="#e0e0ff"|40px
3.4.3.12

|BGCOLOR="#e0e0ff"|40px
3.3.6.6

|BGCOLOR="#ffe0e0"|40px
3.6.3.6

|BGCOLOR="#e0e0ff"|40px
3.4.4.6

|BGCOLOR="#ffe0e0"|40px
3.4.6.4

|BGCOLOR="#d0ffd0"|40px
44

|BGCOLOR="#ffe0e0"|40px
3.3.4.3.4

|BGCOLOR="#ffe0e0"|40px
3.3.3.4.4

|BGCOLOR="#ffe0e0"|40px
3.3.3.3.6

|BGCOLOR="#d0ffd0"|40px
36

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|+ 2-uniforms

!1

!2

!4

!6

!7

!10

!14

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|100px
(3.12.12; 3.4.3.12)

|100px
(36; 32.4.12)

|100px
(4.6.12; 3.4.6.4)

|100px
((3.6)2; 32.62)

|100px
(3.4.6.4; 32.4.3.4)

|100px
(36; 32.4.3.4)

|100px
(3.4.6.4; 3.42.6)

E+L2||L1+(1)||N1+G||M1+M2||N2+Q1||Q1+(1)||N1+Q2

class=wikitable width=600

|+ 3-uniforms

!3

!5

!8

!9

!11

!12

!13

(3.3.4.3.4; 3.3.4.12, 3.4.3.12)

| (36; 3.3.4.12; 3.3.4.3.4)

|(3.3.4.3.4; 3.3.3.4.4, 4.3.4.6)

|(36, 3.3.4.3.4)

|(36; 3.3.4.3.4, 3.3.3.4.4)

|(36; 3.3.4.3.4; 3.3.3.4.4)

|(3.4.6.4; 3.42.6)

L1+L2+Q1||L1+Q1+(1)||N1+Q1+Q2||Q1+(1)||Q1+Q2+(1)||Q1+Q2+(1)||N1+N2
colspan="7" |[https://web.archive.org/web/20160507010400/http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf Claimed Tilings] and Duals
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References

{{Reflist}}

  • Ghyka, M. The Geometry of Art and Life, (1946), 2nd edition, New York: Dover, 1977.
  • Keith Critchlow, Order in Space: A design source book, 1970, pp. 62–67
  • {{The Geometrical Foundation of Natural Structure (book)}} pp. 35–43
  • Steinhaus, H. Mathematical Snapshots 3rd ed, (1969), Oxford University Press, and (1999) New York: Dover
  • {{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
  • {{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= }}
  • [https://web.archive.org/web/20160507010400/http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf In Search of Demiregular Tilings], Helmer Aslaksen

External links

  • {{MathWorld | urlname=DemiregularTessellation | title=Demiregular tessellation}}
  • [http://probabilitysports.com/tilings.html n-uniform tilings] Brian Galebach

Category:Tessellation

Category:Semiregular tilings