Euclidean tilings by convex regular polygons: Difference between revisions

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|[[File:Distorted truncated square tiling.svg|150px]]<br/>A [[#Tilings_that_are_not_edge-to-edge|non-edge-to-edge tiling]] can have different-sized regular faces.

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Euclidean [[Plane (mathematics)|plane]] '''[[Tessellation|tilings]] by convex [[regular polygon]]s''' have been widely used since antiquity. The first systematic mathematical treatment was that of [[Johannes Kepler|Kepler]] in his ''{{lang|la|[[HarmonicesHarmonice Mundi]]''}} ([[Latin language|Latin]]: ''The Harmony of the World'', 1619).

== Notation of Euclidean tilings ==

Euclidean tilings are usually named after Cundy & Rollett’s notation.<ref>{{cite book |last1=Cundy |first1=H.M.|last2=Rollett |first2=A.P. |title=Mathematical Models; |date=1981 |publisher=Tarquin Publications |___location=Stradbroke (UK)}}</ref> This notation represents (i) the number of vertices, (ii) the number of polygons around each vertex (arranged clockwise) and (iii) the number of sides to each of those polygons. For example: 3⁶; 3⁶; 3⁴.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3"3-uniform (2-vertex types)’" tiling. Broken down, 3⁶; 3⁶ (both of different transitivity class), or (3⁶)², tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided polygons (triangles). With a final vertex 3⁴.6, 4 more contiguous equilateral triangles and a single regular hexagon.

However, this notation has two main problems related to ambiguous conformation and uniqueness <ref name="Gomez-Jauregui 2012">{{cite journal |last1=Gomez-Jauregui |first1=Valentin al.|last2=Otero |first2=Cesar |display-authors=etal |title=Generation and Nomenclature of Tessellations and Double-Layer Grids |journal=Journal of Structural Engineering |date=2012 |volume=138 |issue=7 |doi=10.1061/(ASCE)ST.1943-541X.0000532 |url=https://ascelibrary.org/doi/10.1061/%28ASCE%29ST.1943-541X.0000532}}</ref> First, when it comes to k-uniform tilings, the notation does not explain the relationships between the vertices. This makes it impossible to generate a covered plane given the notation alone. And second, some tessellations have the same nomenclature, they are very similar but it can be noticed that the relative positions of the hexagons are different. Therefore, the second problem is that this nomenclature is not unique for each tessellation.

In order to solve those problems, GomJau-Hogg’s notation <ref>{{cite journal |last1=Gomez-Jauregui |first1=Valentin |last2=Hogg |first2=Harrison|display-authors=etal |title=GomJau-Hogg’s Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0 |journal=Symmetry |date=2021 |volume=13 |issue=12 |doi=10.3390/sym13122376 |url=https://doi.org/10.3390/sym13122376}}</ref> is a slightly modified version of the research and notation presented in 2012,<ref name="Gomez-Jauregui 2012" /> about the generation and nomenclature of tessellations and double-layer grids. Antwerp v3.0,<ref>{{cite web |last1=Hogg |first1=Harrison |last2=Gomez-Jauregui |first2=Valentin |title=Antwerp 3.0 |url=https://antwerp.hogg.io/<}}</ref> a free online application, allows for the infinite generation of regular polygon tilings through a set of shape placement stages and iterative rotation and reflection operations, obtained directly from the GomJau-Hogg’s notation.

However, this notation has two main problems related to ambiguous conformation and uniqueness <ref name="Gomez-Jauregui 2012">{{cite journal |last1=Gomez-Jauregui |first1=Valentin al.|last2=Otero |first2=Cesar |display-authors=etal |title=Generation and Nomenclature of Tessellations and Double-Layer Grids |journal=Journal of Structural Engineering |date=2012 |volume=138 |issue=7 |pages=843–852 |doi=10.1061/(ASCE)ST.1943-541X.0000532 |hdl=10902/5869 |url=https://ascelibrary.org/doi/10.1061/%28ASCE%29ST.1943-541X.0000532|hdl-access=free }}</ref> First, when it comes to k-uniform tilings, the notation does not explain the relationships between the vertices. This makes it impossible to generate a covered plane given the notation alone. And second, some tessellations have the same nomenclature, they are very similar but it can be noticed that the relative positions of the hexagons are different. Therefore, the second problem is that this nomenclature is not unique for each tessellation.

In order to solve those problems, GomJau-Hogg's notation <ref>{{cite journal |last1=Gomez-Jauregui |first1=Valentin |last2=Hogg |first2=Harrison|display-authors=etal |title=GomJau-Hogg's Notation for Automatic Generation of k-Uniform Tessellations with ANTWERP v3.0 |journal=Symmetry |date=2021 |volume=13 |issue=12 |page=2376 |doi=10.3390/sym13122376 |bibcode=2021Symm...13.2376G |doi-access=free |hdl=10902/23907 |hdl-access=free }}</ref> is a slightly modified version of the research and notation presented in 2012,<ref name="Gomez-Jauregui 2012" /> about the generation and nomenclature of tessellations and double-layer grids. Antwerp v3.0,<ref>{{cite web |last1=Hogg |first1=Harrison |last2=Gomez-Jauregui |first2=Valentin |title=Antwerp 3.0 |url=https://antwerp.hogg.io/<}}</ref> a free online application, allows for the infinite generation of regular polygon tilings through a set of shape placement stages and iterative rotation and reflection operations, obtained directly from the GomJau-Hogg's notation.

== Regular tilings ==

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== Archimedean, uniform or semiregular tilings ==<!-- This section is linked from [[Archimedean tiling]] -->

{{Further|List of convexEuclidean uniform tilings}}

[[Vertex-transitive|Vertex-transitivity]] means that for every pair of vertices there is a [[symmetry operation]] mapping the first vertex to the second.<ref name="Critchlow 1969">{{cite book |last1=Critchlow |first1=K. |title=Order in Space: A Design Source Book, |date=1969 |publisher=Thames and Hudson |___location=London |pages=60-6160–61}}</ref>

If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as ''Archimedean'', ''[[Uniform tiling|uniform]]'' or ''demiregularsemiregular'' tilings. Note that there are two [[mirror image]] (enantiomorphic or [[Chirality (mathematics)|chiral]]) forms of 3⁴.6 (snub hexagonal) tiling, only one of which is shown in the following table. All other regular and semiregular tilings are achiral.

{| class="wikitable"

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Grünbaum and Shephard distinguish the description of these tilings as ''Archimedean'' as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as ''uniform'' as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform.

== ''k''Plane-uniformvertex tilings ==

There are 17 combinations of regular convex polygons that form 21 types of [[Vertex (geometry)#Of a plane tiling|plane-vertex tilings]].<ref>{{citation|title=The Elements of Plane Practical Geometry, Etc|first=Elmslie William|last=Dallas|publisher=John W. Parker & Son|year=1855|page=134|url=https://books.google.com/books?id=y4BaAAAAcAAJ&pg=PA134}}</ref><ref>[[Tilings and patterns]], Figure 2.1.1, p.60</ref> Polygons in these meet at a point with no gap or overlap. Listing by their [[vertex figure]]s, one has 6 polygons, three have 5 polygons, seven have 4 polygons, and ten have 3 polygons.<ref>[[Tilings and patterns]], p.58-69</ref>

{| class=wikitable align=right width=300

|+ 3-uniform tiling #57 of 61 colored

Three of them can make [[#Regular_tilings|regular tilings]] (6³, 4⁴, 3⁶), and eight more can make [[#Archimedean,_uniform_or_semiregular_tilings|semiregular or archimedean tilings]], (3.12.12, 4.6.12, 4.8.8, (3.6)², 3.4.6.4, 3.3.4.3.4, 3.3.3.4.4, 3.3.3.3.6). Four of them can exist in higher [[#k-uniform_tilings|''k''-uniform tilings]] (3.3.4.12, 3.4.3.12, 3.3.6.6, 3.4.4.6), while six can not be used to completely tile the plane by regular polygons with no gaps or overlaps - they only tessellate space entirely when irregular polygons are included (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, 5.5.10).<ref>{{Cite web|url=https://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/|title=Pentagon-Decagon Packing|website=American Mathematical Society|publisher=AMS|access-date=2022-03-07}}</ref>

|- align=center valign=top

|[[File:3-uniform 57.svg|150px]]<br/>by sides, yellow triangles, red squares (by polygons)

{| class=wikitable

|[[File:3-uniform n57.svg|150px]]<br/>by 4-isohedral positions, 3 shaded colors of triangles (by orbits)

|+ Plane-vertex tilings

|- align=center

!6

|valign=bottom|[[File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|40px]]<BR>3⁶

|- align=center

!5

|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 4 3 4.svg|40px]]<BR>3.3.4.3.4

|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 3 4 4.svg|40px]]<BR>3.3.3.4.4

|valign=bottom|[[File:Regular polygons meeting at vertex 5 3 3 3 3 6.svg|40px]]<BR>3.3.3.3.6

|- align=center

!4

|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 3 4 12.svg|40px]]<BR>3.3.4.12

|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 3 12.svg|40px]]<BR>3.4.3.12

|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 3 6 6.svg|40px]]<BR>3.3.6.6

|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 6 3 6.svg|40px]]<BR>(3.6)²

|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 4 6.svg|40px]]<BR>3.4.4.6

|valign=bottom|[[File:Regular polygons meeting at vertex 4 3 4 6 4.svg|40px]]<BR>3.4.6.4

|valign=bottom|[[File:Regular polygons meeting at vertex 4 4 4 4 4.svg|40px]]<BR>4⁴

|- align=center

!3

|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 7 42.svg|50px]]<BR>3.7.42

|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 8 24.svg|40px]]<BR>3.8.24

|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 9 18.svg|40px]]<BR>3.9.18

|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 10 15.svg|40px]]<BR>3.10.15

|valign=bottom|[[File:Regular polygons meeting at vertex 3 3 12 12.svg|40px]]<BR>3.12.12

|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 5 20.svg|40px]]<BR>4.5.20

|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 6 12.svg|40px]]<BR>4.6.12

|valign=bottom|[[File:Regular polygons meeting at vertex 3 4 8 8.svg|50px]]<BR>4.8.8

|valign=bottom|[[File:Regular polygons meeting at vertex 3 5 5 10.svg|40px]]<BR>5.5.10

|valign=bottom|[[File:Regular polygons meeting at vertex 3 6 6 6.svg|50px]]<BR>6³

|}

== ''k''-uniform tilings ==

Such periodic tilings may be classified by the number of [[Group action (mathematics)#Orbits and stabilizers|orbits]] of vertices, edges and tiles. If there are {{mvar|k}} orbits of vertices, a tiling is known as {{mvar|k}}-uniform or {{mvar|k}}-isogonal; if there are {{mvar|t}} orbits of tiles, as {{mvar|t}}-isohedral; if there are {{mvar|e}} orbits of edges, as {{mvar|e}}-isotoxal.

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Finally, if the number of types of vertices is the same as the uniformity (''m'' = ''k'' below), then the tiling is said to be ''[[oeis:A068600|Krotenheerdt]]''. In general, the uniformity is greater than or equal to the number of types of vertices (''m'' ≥ ''k''), as different types of vertices necessarily have different orbits, but not vice versa. Setting ''m'' = ''n'' = ''k'', there are 11 such tilings for ''n'' = 1; 20 such tilings for ''n'' = 2; 39 such tilings for ''n'' = 3; 33 such tilings for ''n'' = 4; 15 such tilings for ''n'' = 5; 10 such tilings for ''n'' = 6; and 7 such tilings for ''n'' = 7.

Below is an example of a 3-unifom tiling:

{| class=wikitable align=left width=300

|+ Colored 3-uniform tiling #57 of 61

|- align=center valign=top

|[[File:3-uniform 57.svg|150px]]<br/>by sides, yellow triangles, red squares (by polygons)

|[[File:3-uniform n57.svg|150px]]<br/>by 4-isohedral positions, 3 shaded colors of triangles (by orbits)

|}

{{Clr}}

{| class="wikitable" style="margin: auto; text-align:center;"

{| class="wikitable" align=left style="margin: auto; text-align:center;"

|+ ''k''-uniform, ''m''-Archimedean tiling counts<ref>{{Cite web|url=http://probabilitysports.com/tilings.html|title=n-Uniform Tilings|website=probabilitysports.com|access-date=2019-06-21}}</ref>

|+ ''k''-uniform, ''m''-Archimedean tiling counts <ref>{{Cite web|url=http://probabilitysports.com/tilings.html|title=n-Uniform Tilings|website=probabilitysports.com|access-date=2019-06-21}}</ref><ref>{{Cite OEIS |A068599 |Number of n-uniform tilings. |access-date=2023-01-07 }}</ref><ref>{{Cite web|url=https://zenorogue.github.io/tes-catalog/?c=k-uniform%2F|title=Enumeration of n-uniform k-Archimedean tilings|website=zenorogue.github.io/tes-catalog/?c=|access-date=2024-08-24}}</ref>

!colspan=2 rowspan=2 | !! colspan="16" |''m''-Archimedean

|-

Line 165 ⟶ 207:

|-

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| 0 || {{Dunno}}'''175''' || {{Dunno}}'''572''' || {{Dunno}}'''426''' || {{Dunno}}'''218''' || {{Dunno}}'''74''' ||'''7'''||0

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Line 204 ⟶ 246:

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|{{Dunno}}

|-

Line 255 ⟶ 297:

|}

{{Clr}}

=== Dissected regular polygons ===

Some of the ''k''-uniform tilings can be derived by symmetrically dissecting the tiling polygons with interior edges, for example (direct dissection):

{| class="wikitable" style="margin: auto;"

|+ Dissected polygons with original edges

|[[File:Vertex_type_3-3-3-3-3-3.svg|alt=|center|80x80px]]

|[[File:Hexagonal_cupola_flat.svg|alt=|center|80x80px]]

|[[File:Dissected_dodecagon.svg|alt=|center|80x80px]]

|-

![[Hexagon]]

!colspan=2|[[Dodecagon]]<br/>(each has 2 orientations)

|}

Some ''k''-uniform tilings can be derived by dissecting regular polygons with new vertices along the original edges, for example (indirect dissection):

{| class="wikitable" style="margin: auto;"

|+ Dissected with 1 or 2 middle vertex

|[[File:Face figure 3-333.svg|80px]]

|[[File:Dissected triangle-36.png|80px]]

|[[File:Dissected triangle-3b.png|80px]]

|[[File:Vertex type 4-4-4-4.svg|80px]]

|[[File:dissected square-3x3.png|80px]]

|[[File:Dissected hexagon 36a.png|80px]]

|[[File:Dissected hexagon 36b.png|80px]]

|[[File:Dissected hexagon 3b.png|80px]]

|-

!colspan=3|[[Triangle]]

!colspan=2|[[Square]]

!colspan=3|[[Hexagon]]

|}

Finally, to see all types of vertex configurations, see [[Planigon]].

=== 2-uniform tilings ===

There are twenty (20) '''2-uniform tilings''' of the Euclidean plane. (also called '''2-[[isogonal figure|isogonal]] tilings''' or '''[[demiregular tiling]]s''') {{r|"Critchlow 1969"|p=62-67}} <ref>[[Tilings and Patternspatterns]], Grünbaum and Shephard 1986, pp. 65-67</ref><ref>{{Cite web |url=http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf |title=In Search of Demiregular Tilings |access-date=2015-06-04 |archive-url=https://web.archive.org/web/20160507010400/http://www.math.nus.edu.sg/aslaksen/papers/Demiregular.pdf |archive-date=2016-05-07 |url-status=dead }}</ref> Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.

{| class="wikitable" style="text-align:center;"

Line 295 ⟶ 307:

! colspan=6|p6m, *632 !! p4m, *442

|- valign=top

| [[File:2-uniform n18.svg|120px]]<br/>[[Rhombitrihexagonal_tiling#Related_tilings|[3⁶; 3².4.3.4]]]]]]<br/>3-4-3/m30/r(c3) <br/>(''t'' = 3, ''e'' = 3)

| [[File:2-uniform n9.svg|120px]]<br/>[[Truncated_hexagonal_tiling#Related_2-uniform_tilings|[3.4.6.4; 3².4.3.4]]]<br/> 6-4-3,3/m30/r(h1)<br/>(''t'' = 4, ''e'' = 4)

| [[File:2-uniform n8.svg|120px]]<br/>[[Truncated_hexagonal_tiling#Related_2-uniform_tilings|[3.4.6.4; 3³.4²]]]<br/>6-4-3-3/m30/r(h5) <br/>(''t'' = 4, ''e'' = 4)

Line 316 ⟶ 328:

|- valign=top

| [[File:2-uniform n16.svg|120px]]<br/>[[33344-33434 tiling|[3³.4²; 3².4.3.4]₁]]<br/>4-3,3-4,3/r90/m(h3) <br/>(''t'' = 4, ''e'' = 5)

| [[File:2-uniform n17-1.pngsvg|120px]]<br/>[[33344-33434 tiling|[3³.4²; 3².4.3.4]₂]]<br/>4-3,3,3-4,3/r(c2)/r(h13)/r(h45)<br/>(''t'' = 3, ''e'' = 6)

| [[File:2-uniform n4.svg|120px]]<br/>[4⁴; 3³.4²]₁<br/>4-3/m(h4)/m(h3)/r(h2)<br/>(''t'' = 2, ''e'' = 4)

| [[File:2-uniform n3.svg|120px]]<br/>[4⁴; 3³.4²]₂<br/>4-4-3-3/m90/r(h3)<br/>(''t'' = 3, ''e'' = 5)

Line 336 ⟶ 348:

|- valign="center"

! valign="center" |Shape

|[[File:The Triangle.pngsvg|alt=|center|frameless|80px]]

|[[File:A Square Tile.pngsvg|alt=|center|frameless|80px]]

|[[File:A Hexagon Tile.pngsvg|center|frameless|120px]]

|[[File:A Dissected Dodecagon.pngsvg|center|frameless|120px]]

|-

!Fractalizing

|[[File:Truncated Hexagonal Fractal Triangle.pngsvg|center|frameless|80px]]

|[[File:Truncated Hexagonal Fractal Square.pngsvg|center|frameless|80px]]

|[[File:Truncated Hexagonal Fractal Hexagon.pngsvg|center|frameless|90px]]

|[[File:Truncated Hexagonal Fractal Dissected Dodecagon.pngsvg|center|frameless|100px]]

|}

The side lengths are dilated by a factor of <math>2+\sqrt{3}</math>.

Line 359 ⟶ 371:

|- valign="center"

! valign="center" |Shape

|[[File:The Triangle.pngsvg|alt=|center|frameless|80px]]

|[[File:A Square Tile.pngsvg|alt=|center|frameless|80px]]

|[[File:A Hexagon Tile.pngsvg|center|frameless|80px]]

|[[File:A Dissected Dodecagon.pngsvg|center|frameless|100px]]

|-

!Fractalizing

|[[File:Truncated Trihexagonal Fractal Triangle.pngsvg|center|frameless|143px]]

|[[File:Truncated Trihexagonal Fractal Square.pngsvg|center|frameless|143px]]

|[[File:Truncated Trihexagonal Fractal Hexagon.pngsvg|center|frameless|120px]]

|[[File:Truncated Trihexagonal Fractal Dissected Dodecagon.pngsvg|center|frameless|120px]]

|}

Line 386 ⟶ 398:

Convex regular polygons can also form plane tilings that are not edge-to-edge. Such tilings can be considered edge-to-edge as nonregular polygons with adjacent colinear edges.

There are seven families of [[isogonal figure|isogonal]]s, each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles. Two of the families are generated from shifted square, either progressive or zig-zagging positions. Grünbaum and Shephard call these tilings ''uniform'' although it contradicts Coxeter's definition for uniformity which requires edge-to-edge regular polygons.<ref>[{{Cite web |url=http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |title=Tilings by regular polygons] |p.=236 |archive-url=https://web.archive.org/web/20160303235526/http://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf |archive-date=2016-03-03 |url-status=dead}}</ref> Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions.

{| class=wikitable width=600

Line 403 ⟶ 415:

|[[File:Distorted truncated square tiling.svg|120px]]<br/>[[Pythagorean tiling|A tiling by squares]]

|[[File:Gyrated truncated hexagonal tiling.png|120px]]<br/>Three hexagons surround each triangle

|[[File:Gyrated hexagonal tiling2.pngsvg|120px]]<br/>Six triangles surround every hexagon.

|[[File:Trihexagonal tiling unequal2.svg|120px]]<br/>Three size triangles

|-

Line 424 ⟶ 436:

{{colbegin|colwidth=30em}}

* [[Grid (spatial index)]]

* [[Hyperbolic geometry]]

* [[Uniform tilings in hyperbolic plane]]

* [[Lattice (group)]]

* [[List of uniform tilings]]

* [[WythoffPenrose symboltiling]]

* [[Tessellation]]

* [[Wallpaper group]]

* [[Regular polyhedron]] (the [[Platonic solid]]s)

* [[Semiregular polyhedron]] (including the [[Archimedean solid]]s)

* [[Hyperbolic geometryTessellation]]

* [[Penrose tiling]]

* [[Tiling with rectangles]]

* [[Uniform tilings in hyperbolic plane]]

* [[Lattice (group)]]

* [[Wallpaper group]]

* [[Wythoff symbol]]

{{colend}}

Line 452 ⟶ 464:

| title=The boundary characteristic and Pick's theorem in the Archimedean planar tilings

| journal=J. Comb. Theory A | year=1987 | volume=44 | number=1 | pages=110–119

|doi=10.1016/0097-3165(87)90063-X | doi-access=free }}

* {{Cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons&mdash;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&ndash;165 | doi=10.1016/0898-1221(89)90156-9| doi-access=free | url-access=subscription }}

* Order in Space: A design source book, Keith Critchlow, 1970 {{ISBN|978-0-670-52830-1}}

*{{Cite book| first1= Duncan MacLaren Young |last1=Sommerville| title=An Introduction to the Geometry of '''n''' Dimensions | publisher=Dover Publications| year=1958}} Chapter X: The Regular Polytopes

* {{Cite journal|first1=P. | last1=Préa | title=Distance sequences and percolation thresholds in Archimedean Tilings | journal=Mathl. Comput. Modelling | volume=26

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