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Euclidean tilings by convex regular polygons
Euclidean tilings by convex regular polygons
Euclidean tilings by convex regular polygons
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Example periodic tilings

A regular tiling has one type of regular face.
A semiregular or uniform tiling has one type of vertex, but two or more types of faces.

A k-uniform tiling has k types of vertices, and two or more types of regular faces.
A non-edge-to-edge tiling can have different-sized regular faces.

Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonice Mundi (Latin: The Harmony of the World, 1619).

Notation of Euclidean tilings

[edit]

Euclidean tilings are usually named after Cundy & Rollett’s notation.[1] 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-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 [2] 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 [3] is a slightly modified version of the research and notation presented in 2012,[2] about the generation and nomenclature of tessellations and double-layer grids. Antwerp v3.0,[4] 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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Following Grünbaum and Shephard (section 1.3), a tiling is said to be regular if the symmetry group of the tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons. There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three regular tessellations.

Regular tilings (3)
p6m, *632 p4m, *442

C&R: 36
GJ-H: 3/m30/r(h2)
(t = 1, e = 1)
C&R: 63
GJ-H: 6/m30/r(h1)
(t = 1, e = 1)
C&R: 44
GJ-H: 4/m45/r(h1)
(t = 1, e = 1)

C&R: Cundy & Rollet's notation
GJ-H: Notation of GomJau-Hogg

Archimedean, uniform or semiregular tilings

[edit]
Further information: List of Euclidean uniform tilings

Vertex-transitivity means that for every pair of vertices there is a symmetry operation mapping the first vertex to the second.[5]

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 or semiregular tilings. Note that there are two mirror image (enantiomorphic or 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.

Uniform tilings (8)
p6m, *632



C&R: 3.122
GJ-H: 12-3/m30/r(h3)
(t = 2, e = 2)
t{6,3}


C&R: 3.4.6.4
GJ-H: 6-4-3/m30/r(c2)
(t = 3, e = 2)
rr{3,6}


C&R: 4.6.12
GJ-H: 12-6,4/m30/r(c2)
(t = 3, e = 3)
tr{3,6}


C&R: (3.6)2
GJ-H: 6-3-6/m30/r(v4)
(t = 2, e = 1)
r{6,3}



C&R: 4.82
GJ-H: 8-4/m90/r(h4)
(t = 2, e = 2)
t{4,4}


C&R: 32.4.3.4
GJ-H: 4-3-3,4/r90/r(h2)
(t = 2, e = 2)
s{4,4}


C&R: 33.42
GJ-H: 4-3/m90/r(h2)
(t = 2, e = 3)
{3,6}:e


C&R: 34.6
GJ-H: 6-3-3/r60/r(h5)
(t = 3, e = 3)
sr{3,6}

C&R: Cundy & Rollet's notation
GJ-H: Notation of GomJau-Hogg

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.

Plane-vertex tilings

[edit]

There are 17 combinations of regular convex polygons that form 21 types of plane-vertex tilings.[6][7] Polygons in these meet at a point with no gap or overlap. Listing by their vertex figures, one has 6 polygons, three have 5 polygons, seven have 4 polygons, and ten have 3 polygons.[8]

Three of them can make regular tilings (63, 44, 36), and eight more can make 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 (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).[9]

Plane-vertex tilings
6
36
5
3.3.4.3.4
3.3.3.4.4
3.3.3.3.6
4
3.3.4.12
3.4.3.12
3.3.6.6
(3.6)2
3.4.4.6
3.4.6.4
44
3
3.7.42
3.8.24
3.9.18
3.10.15
3.12.12
4.5.20
4.6.12
4.8.8
5.5.10
63

k-uniform tilings

[edit]

Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of vertices, a tiling is known as k-uniform or k-isogonal; if there are t orbits of tiles, as t-isohedral; if there are e orbits of edges, as e-isotoxal.

k-uniform tilings with the same vertex figures can be further identified by their wallpaper group symmetry.

1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.[10]

Finally, if the number of types of vertices is the same as the uniformity (m = k below), then the tiling is said to be 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:

Colored 3-uniform tiling #57 of 61

by sides, yellow triangles, red squares (by polygons)
by 4-isohedral positions, 3 shaded colors of triangles (by orbits)
k-uniform, m-Archimedean tiling counts [11][12][13]
m-Archimedean
1 2 3 4 5 6 7 8 9 10 11 12 13 14 ≥ 15 Total
k-uniform 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11
2 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 20
3 0 22 39 0 0 0 0 0 0 0 0 0 0 0 0 61
4 0 33 85 33 0 0 0 0 0 0 0 0 0 0 0 151
5 0 74 149 94 15 0 0 0 0 0 0 0 0 0 0 332
6 0 100 284 187 92 10 0 0 0 0 0 0 0 0 0 673
7 0 175 572 426 218 74 7 0 0 0 0 0 0 0 0 1472
8 0 298 1037 795 537 203 20 0 0 0 0 0 0 0 0 2850
9 0 424 1992 1608 1278 570 80 8 0 0 0 0 0 0 0 5960
10 0 663 3772 2979 2745 1468 212 27 0 0 0 0 0 0 0 11866
11 0 1086 7171 5798 5993 3711 647 52 1 0 0 0 0 0 0 24459
12 0 1607 13762 11006 12309 9230 1736 129 15 0 0 0 0 0 0 49794
13 0 ? ? ? ? ? ? ? ? ? 0 0 0 0 0 103082
14 0 ? ? ? ? ? ? ? ? ? 0 0 0 0 0 ?
≥ 15 0 ? ? ? ? ? ? ? ? ? ? ? ? ? 0 ?
Total 11 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 0 ∞

2-uniform tilings

[edit]

There are twenty (20) 2-uniform tilings of the Euclidean plane. (also called 2-isogonal tilings or demiregular tilings) [5]: 62-67  [14][15] Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.

2-uniform tilings (20)
p6m, *632 p4m, *442

[36; 32.4.3.4]
3-4-3/m30/r(c3)
(t = 3, e = 3)
[3.4.6.4; 32.4.3.4]
6-4-3,3/m30/r(h1)
(t = 4, e = 4)
[3.4.6.4; 33.42]
6-4-3-3/m30/r(h5)
(t = 4, e = 4)
[3.4.6.4; 3.42.6]
6-4-3,4-6/m30/r(c4)
(t = 5, e = 5)
[4.6.12; 3.4.6.4]
12-4,6-3/m30/r(c3)
(t = 4, e = 4)
[36; 32.4.12]
12-3,4-3/m30/r(c3)
(t = 4, e = 4)
[3.12.12; 3.4.3.12]
12-0,3,3-0,4/m45/m(h1)
(t = 3, e = 3)
p6m, *632 p6, 632 p6, 632 cmm, 2*22 pmm, *2222 cmm, 2*22 pmm, *2222

[36; 32.62]
3-6/m30/r(c2)
(t = 2, e = 3)
[36; 34.6]1
6-3,3-3/m30/r(h1)
(t = 3, e = 3)
[36; 34.6]2
6-3-3,3-3/r60/r(h8)
(t = 5, e = 7)
[32.62; 34.6]
6-3/m90/r(h1)
(t = 2, e = 4)
[3.6.3.6; 32.62]
6-3,6/m90/r(h3)
(t = 2, e = 3)
[3.42.6; 3.6.3.6]2
6-3,4-6-3,4-6,4/m90/r(c6)
(t = 3, e = 4)
[3.42.6; 3.6.3.6]1
6-3,4/m90/r(h4)
(t = 4, e = 4)
p4g, 4*2 pgg, 22× cmm, 2*22 cmm, 2*22 pmm, *2222 cmm, 2*22

[33.42; 32.4.3.4]1
4-3,3-4,3/r90/m(h3)
(t = 4, e = 5)
[33.42; 32.4.3.4]2
4-3,3,3-4,3/r(c2)/r(h13)/r(h45)
(t = 3, e = 6)
[44; 33.42]1
4-3/m(h4)/m(h3)/r(h2)
(t = 2, e = 4)
[44; 33.42]2
4-4-3-3/m90/r(h3)
(t = 3, e = 5)
[36; 33.42]1
4-3,4-3,3/m90/r(h3)
(t = 3, e = 4)
[36; 33.42]2
4-3-3-3/m90/r(h7)/r(h5)
(t = 4, e = 5)

Higher k-uniform tilings

[edit]

k-uniform tilings have been enumerated up to 6. There are 673 6-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 10 6-uniform tilings with 6 distinct vertex types, as well as finding 92 of them with 5 vertex types, 187 of them with 4 vertex types, 284 of them with 3 vertex types, and 100 with 2 vertex types.

Fractalizing k-uniform tilings

[edit]

There are many ways of generating new k-uniform tilings from old k-uniform tilings. For example, notice that the 2-uniform [3.12.12; 3.4.3.12] tiling has a square lattice, the 4(3-1)-uniform [343.12; (3.122)3] tiling has a snub square lattice, and the 5(3-1-1)-uniform [334.12; 343.12; (3.12.12)3] tiling has an elongated triangular lattice. These higher-order uniform tilings use the same lattice but possess greater complexity. The fractalizing basis for theses tilings is as follows:[16]

Triangle Square Hexagon Dissected
Dodecagon
Shape
Fractalizing

The side lengths are dilated by a factor of .

This can similarly be done with the truncated trihexagonal tiling as a basis, with corresponding dilation of .

Triangle Square Hexagon Dissected
Dodecagon
Shape
Fractalizing

Fractalizing examples

[edit]
Truncated Hexagonal Tiling Truncated Trihexagonal Tiling
Fractalizing

Tilings that are not edge-to-edge

[edit]

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 figures, 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.[17] Such isogonal tilings are actually topologically identical to the uniform tilings, with different geometric proportions.

Periodic isogonal tilings by non-edge-to-edge convex regular polygons
1 2 3 4 5 6 7

Rows of squares with horizontal offsets
Rows of triangles with horizontal offsets
A tiling by squares
Three hexagons surround each triangle
Six triangles surround every hexagon.
Three size triangles
cmm (2*22) p2 (2222) cmm (2*22) p4m (*442) p6 (632) p3 (333)
Hexagonal tiling Square tiling Truncated square tiling Truncated hexagonal tiling Hexagonal tiling Trihexagonal tiling

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Euclidean tilings by convex regular polygons

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Euclidean tilings by convex regular polygons are edge-to-edge tessellations of the infinite using one or more congruent copies of regular polygons, such that the tiles cover the plane without gaps or overlaps and meet vertex-to-vertex. These tilings are characterized by their high degree of , with the regular cases—using a single type—limited to three: the triangular tiling (with six equilateral triangles meeting at each vertex), the (four squares per vertex), and the (three regular hexagons per vertex). More broadly, the topic encompasses uniform tilings, known as Archimedean tilings, which use multiple types but maintain identical vertex configurations throughout; there are exactly 11 such convex uniform tilings of the plane. The study of these tilings dates back to antiquity, with practical uses in and , but the first systematic mathematical classification was provided by in his 1619 work Harmonices Mundi, where he identified the 11 Archimedean tilings. These configurations arise from the geometric constraint that the sum of angles at each vertex must equal exactly 360 degrees, a condition derived from the interior angle formula of a regular n-gon, (n-2)×180°/n, which limits viable combinations to polygons with 3 to 6 sides. Beyond the uniform cases, higher-order classifications include k- tilings (with k distinct vertex types) and m-Archimedean tilings (with m distinct vertex figures), leading to extensive families such as the 20 known 2-uniform tilings. Notable aspects include their symmetry groups, which belong to the 17 wallpaper groups of the plane, and their applications in crystallography, where they model atomic arrangements, as well as in computational geometry for generating periodic structures. While only three monohedral regular tilings exist, the inclusion of multiple polygon types expands the possibilities dramatically, with catalogs enumerating thousands of periodic examples up to high uniformity orders. These tilings exemplify the interplay between discrete geometry and symmetry, influencing fields from materials science to aesthetic design.

Fundamentals and Notation

Notation for Euclidean tilings

In Euclidean tilings by convex regular polygons, the arrangement of polygons meeting at each vertex is described using a symbolic notation known as the vertex configuration or Cundy-Rollett symbol. This system, introduced by H. Martyn Cundy and A. P. Rollett in their 1961 book Mathematical Models, represents the cyclic sequence of polygons around a vertex as a series of s separated by dots, where each denotes the number of sides of the adjacent (e.g., 3 for a , 4 for ). For repeated identical polygons, a superscript indicates the count, such as 3^6 for six triangles. The sequence is arranged in counterclockwise order starting from a reference point, often chosen to begin with the smallest for standardization, ensuring the notation captures the full 360-degree cycle around the vertex. The notation adheres to specific rules to maintain consistency and uniqueness. Polygons are listed in the order they meet at the vertex, reflecting the tiling's local , and the arrangement must be cyclic, meaning rotations of represent the same configuration. To avoid redundancy, the sequence is normalized by rotating it to start with the lowest number and, if ties occur, selecting the lexicographically smallest variant among rotations and reflections. Critically, the sum of the interior angles of the polygons at the vertex must equal exactly 360 degrees to ensure a Euclidean (flat) tiling without gaps or overlaps, a condition derived from the interior angle formula for a regular n-gon: ((n-2)/n) × 180 degrees. This notation applies primarily to tilings, where all vertices are congruent, but can extend to describe vertex types in more general tilings. This symbolic system originated in the mid-20th century amid efforts to classify symmetric tilings, building on foundational work by H. S. M. Coxeter and others who explored regular polytopes and honeycombs in the and . Coxeter's analyses in works like Regular Polytopes (, third edition 1973) laid groundwork for systematic descriptions of vertex figures, influencing the adoption of sequences for plane tilings. Cundy and Rollett formalized and popularized the dotted notation specifically for educational models of tilings and polyhedra, making it a standard in geometric literature. The 11 uniform Euclidean tilings—comprising three regular and eight Archimedean—illustrate the notation's application. Each has a unique vertex configuration satisfying the angle sum condition.
Tiling TypeNotationDescription
Regular triangular3^6Six equilateral s meet at each vertex.
Regular square4^4Four squares meet at each vertex.
Regular hexagonal6^3Three regular s meet at each vertex.
Truncated square4.8.8One square and two octagons meet at each vertex.
Truncated hexagonal3.12.12One and two s meet at each vertex.
Rhombitrihexagonal3.4.6.4Alternating s, squares, and s meet at each vertex.
Snub hexagonal3.3.3.3.6Four s and one meet at each vertex.
Trihexagonal3.6.3.6Alternating s and s meet at each vertex.
Snub square3.3.4.3.4Three s and two squares meet at each vertex.
Elongated triangular3.3.3.4.4Three s and two squares meet at each vertex.
Truncated trihexagonal4.6.12One square, one , and one meet at each vertex.
These examples, drawn from Cundy and Rollett's classification, highlight how the notation concisely encodes the diversity of uniform tilings while emphasizing their shared requirement for congruent vertices.

Definitions and basic properties

A Euclidean tiling by convex regular polygons consists of a collection of congruent copies of one or more types of convex regular polygons that cover the entire without gaps or overlaps. Convexity of the polygons is a prerequisite, ensuring that the tiles have no self-intersections or indentations and are limited to those with at least three sides, as regular digons are degenerate in the . Such tilings are typically required to be edge-to-edge, meaning that adjacent polygons share entire edges, and the vertices of one polygon align precisely with those of neighboring polygons, preventing partial overlaps or interior vertex placements. A regular polygon with nn sides (n≥3n \geq 3) is both equilateral, with all sides of equal length, and equiangular, with all interior angles equal to (n−2)×180∘n\frac{(n-2) \times 180^\circ}{n}. This interior angle formula derives from the fact that the sum of interior angles in any nn-gon is (n−2)×180∘(n-2) \times 180^\circ, divided equally among the vertices in the regular case. In a tiling, the geometric constraint at each vertex requires that the sum of the interior angles from the meeting polygons equals exactly 360∘360^\circ, ensuring a flat Euclidean embedding without gaps or overlaps. For regular tilings, where a single type of regular pp-gon meets qq times around each vertex, the configuration is denoted by the {p,q}\{p, q\}. This symbol encapsulates the local , with pp specifying the number of sides per polygon and qq the number of polygons per vertex. The topological structure of these tilings adheres to the for the , which in the infinite limit yields V−E+F=0V - E + F = 0, where VV, EE, and FF are the numbers of vertices, edges, and faces, respectively. This relation imposes constraints on possible configurations; for instance, in a monohedral tiling by pp-gons, it implies 2E=pF2E = pF from edge counting and further relations that limit viable {p,q}\{p, q\} pairs when combined with the angle sum condition.

Vertex-Transitive Tilings

Regular tilings

Regular tilings of the are edge-to-edge tilings composed entirely of congruent regular convex polygons, where the arrangement exhibits full at every vertex, meaning that the same number of polygons meet at each vertex with equal angles. These tilings are the simplest examples of vertex-transitive tilings, as all vertices are equivalent under the tiling's . There are precisely three such infinite regular tilings: the triangular tiling, denoted as {3,6}\{3,6\} or 363^6, in which six equilateral triangles meet at each vertex; the , denoted as {4,4}\{4,4\} or 444^4, in which four squares meet at each vertex; and the , denoted as {6,3}\{6,3\} or 636^3, in which three regular hexagons meet at each vertex. In all regular tilings, the polygons have equal side lengths, ensuring that edges match perfectly without gaps or overlaps, and the interior angles at each vertex sum exactly to 360∘360^\circ, satisfying the geometric constraint for a flat . The triangular and hexagonal tilings are dual to each other: the vertices of the triangular tiling correspond to the faces of the hexagonal tiling, and vice versa, reflecting their complementary structures in the plane. The square tiling is self-dual, with its vertices, edges, and faces aligning symmetrically in a dual configuration. These properties arise from the regularity of the polygons and the vertex figures, which impose strict conditions on the possible arrangements. Regular tilings have been employed in decorative mosaics and architectural patterns since antiquity, appearing in ancient Roman floors and Islamic geometric designs, where their repetitive provided aesthetic harmony. The first systematic mathematical classification of these tilings was provided by in his 1619 work Harmonices Mundi, where he explored their geometric and harmonic properties as part of a broader study of regular figures in and cosmology. Kepler illustrated the three tilings and noted their role in filling the plane without distortion, influencing later studies in and . These tilings are monohedral, using only one type of (the ), and possess both isohedral , where all faces are equivalent under the , and isogonal , where all vertices are equivalent. Only these three configurations satisfy the conditions for regular tilings in the , as determined by the requirement that the vertex angle sum must equal 360∘360^\circ for polygons with three or more sides, excluding cases like pentagons or higher that would either underfill or overfill the plane. This underscores their uniqueness among monohedral tilings by regular convex polygons.

Archimedean tilings

Archimedean tilings, also known as semiregular tilings, are edge-to-edge tilings of the using two or more types of convex regular such that the arrangement of around every vertex is the same up to and reflection, ensuring vertex-transitivity. These tilings generalize the regular tilings by incorporating multiple polygon types while preserving uniformity at the vertices. Together with the three regular tilings, they form the complete set of eleven convex uniform tilings by regular . The eight Archimedean tilings are identified by their vertex configuration notations, which list the number of sides of the polygons meeting at a vertex in cyclic order. These are: 3.3.3.4.4 (three triangles and two squares), 3.3.4.3.4 (snub square tiling with three triangles and two squares), 3.6.3.6 (trihexagonal tiling with alternating triangles and hexagons), 3.4.6.4 (rhombitrihexagonal tiling with one triangle, one square, one hexagon, and one square), 3.12.12 (truncated hexagonal tiling with one triangle and two dodecagons), 4.6.12 (truncated trihexagonal tiling with one square, one hexagon, and one dodecagon), 4.8.8 (truncated square tiling with one square and two octagons), and 3.3.3.3.3.6 (snub hexagonal tiling with five triangles and one hexagon). The snub hexagonal and snub square tilings are chiral, occurring in enantiomorphic pairs related by reflection. In all Archimedean tilings, edges are of equal length, and the interior angles at each vertex sum precisely to 360 degrees, satisfying the condition ∑(1−2ki)=2\sum (1 - \frac{2}{k_i}) = 2 where kik_i are the numbers of sides of the polygons at the vertex, ensuring a flat without gaps or overlaps. Their groups are among the seventeen groups of the plane, acting transitively on vertices; for example, the rhombitrihexagonal tiling (3.4.6.4) and truncated square tiling (4.8.8) possess p4g , while the (3.6.3.6) has p6mm . The of these tilings are isohedral tilings by irregular polygons, analogous to the duals of uniform polyhedra but realized in the plane. The term "Archimedean" draws an analogy to the Archimedean solids, though himself did not describe these plane tilings; the eleven uniform tilings were first enumerated by in his 1619 treatise Harmonices Mundi. Modern rigorous classification, confirming the eight Archimedean cases and addressing subtleties in and congruence, was provided by Branko Grünbaum and G. C. Shephard in their seminal 1987 work Tilings and Patterns.

Multi-Type Uniform Tilings

Plane-vertex tilings

Plane-vertex configurations are the possible edge-to-edge arrangements of convex regular polygons meeting at a single vertex in the , such that the polygons cover 360 degrees without gaps or overlaps. These local configurations are characterized by their vertex figures, using notation like 3.12.12 to indicate the sequence of polygon sides around the vertex. There are exactly 21 such configurations, as enumerated in early 20th-century research on arrangements. Of these 21, 11 can be extended globally to form vertex-transitive uniform tilings (the three regular and eight Archimedean tilings), while 4 can appear in 2-uniform tilings, and 6 are forbidden in any edge-to-edge tiling of the plane. The remaining configurations support periodic global tilings when combined with others, but all are constrained by the angle sum condition derived from interior angles. Representative examples include the configuration 3.3.3.4.4 for the elongated triangular tiling (using triangles and squares), 3.12.12 (triangles and dodecagons), and 4.8.8 (squares and octagons), each of which extends to a uniform tiling. Prismatic and certain Laves mesh-inspired configurations also draw from these vertex figures. These configurations form the foundation for broader tiling classifications, with the complete enumeration resolving classical challenges in verifying compatibility for planarity.

2-uniform tilings

A 2-uniform tiling of the is an edge-to-edge tiling by convex regular polygons in which the acts on the vertices, partitioning them into exactly two (transitive on each orbit separately), meaning there are precisely two inequivalent vertex types under the tiling's symmetries. These tilings extend the concept of (1-uniform) tilings by allowing limited vertex diversity while maintaining full edge-to-edge congruence and equal edge lengths across all polygons. The complete enumeration of 2-uniform tilings was achieved by R. Krötenheerdt in 1968, who identified 20 distinct types, all periodic and belonging to the 17 symmetry groups. This classification built upon earlier work on tilings, including H.S.M. Coxeter's group-theoretic framework for analyzing in arrangements, though Coxeter's primary focus was on 1-uniform cases. The 20 tilings arise from compatible combinations of vertex figures drawn from the 11 Archimedean () tilings, ensuring the overall structure covers the plane without gaps or overlaps. Geometrically, these tilings are constructed by interweaving two distinct vertex configurations—such as (3.4.3.12) and (3.12.3.4), involving equilateral triangles, squares, and regular dodecagons—along a periodic lattice, where the angles at each vertex sum precisely to 360 degrees and edges match uniformly. Another representative example is the pairing of (3^2.4.3.4) and (4.3^2.4.3), utilizing triangles, squares, and hexagons to form a semi-regular pattern with bilateral . Such constructions rely on the compatibility of angles and edge alignments to achieve planarity. Key properties of 2-uniform tilings include faces that are transitive within their types (though not necessarily overall) and isogonal edges within each vertex class, distinguishing them from higher k-uniform tilings (k > 2) by their restricted vertex complexity and fuller symmetry. Unlike fully vertex-transitive uniform tilings, the two vertex orbits prevent complete vertex equivalence, yet the tilings exhibit greater regularity than general multi-type arrangements, often featuring translation lattices with 4 to 12 vertices per unit cell.

Generalized k-Uniform Tilings

Higher k-uniform tilings

A k-uniform tiling for k≥3k \geq 3 is an edge-to-edge tiling of the Euclidean plane by congruent convex regular polygons in which the symmetry group of the tiling acts transitively on exactly kk orbits of vertices, meaning there are kk distinct vertex types up to symmetry. These tilings generalize the Archimedean (1-uniform) and semi-regular (2-uniform) cases by allowing multiple vertex configurations while maintaining overall uniformity through global symmetries, typically belonging to one of the 17 wallpaper groups. All such tilings are periodic, with translational symmetries, and use only regular polygons of at least three sides (triangles, squares, hexagons, etc.), ensuring equal edge lengths and proper angle fits at vertices. Known examples include 39 distinct 3-uniform tilings, such as those combining triangular, square, and hexagonal faces in configurations like (3.3.3.3.3.6; 3.6.3.6; 3.3.6.6), enumerated via systematic analysis of possible vertex figures. For higher kk, the count decreases initially before examples become sparser: 33 for k=4k=4, 15 for k=5k=5, and 10 for k=6k=6. Enumeration is complete up to k=6k=6 through computer-assisted methods, but remains incomplete for k≥7k \geq 7, with recent computational catalogs providing examples up to k=12k=12, including a 10-uniform tiling with vertex types [3^6, 3^3.4^2, (3^2.4.3.4)^3, 3^2.4.12, 3^2.6^2, 3.4^2.6, 3.4.6.4, 6^3]. As kk increases, the tilings exhibit greater complexity in vertex variety and polygon arrangements, though no aperiodic k-uniform tilings by convex regular polygons are known in the Euclidean plane. Construction methods for higher k-uniform tilings often involve iterative refinement from lower-kk examples, such as subdividing or fusing polygons while preserving edge-to-edge regularity and transitivity. Computer algorithms, like those using to propagate vertex figures across the plane under constraints, have been pivotal in generating complete sets up to k=6k=6 and sporadic higher examples. Additional approaches include recursive gluing patterns, where initial seed configurations are extended by solving for compatible adjacent faces, enabling the creation of high-kk tilings like the 10-uniform case through resolutions. Mathematical constraints ensure feasibility: at each vertex of degree dd (typically 3 to 6), the interior angles must sum to 2π2\pi, yielding ∑i=1d1ni=d−22\sum_{i=1}^d \frac{1}{n_i} = \frac{d-2}{2}, where nin_i are the sides of incident polygons. Globally, the average vertex degree is 4, implying ∑nnpn=4\sum_n n p_n = 4 for polygon densities pnp_n (normalized so ∑pn=1\sum p_n = 1), as derived from in the limit. These density formulas, combined with requirements, limit possible polygon combinations and explain the finite but growing scarcity of high-kk examples.

Fractalizing k-uniform tilings

Fractalizing k-uniform tilings involves a recursive substitution process applied to an existing k-uniform tiling of the , where each tile or vertex is replaced by a scaled-down version of the entire original tiling, thereby generating self-similar structures composed of convex regular polygons. This method extends traditional uniform tilings by introducing scale variations while maintaining the use of regular polygons at every level of the . The process relies on hierarchical substitution rules that inflate individual polygons and reconnect their edges to preserve the regularity and vertex configurations of the base tiling. For instance, in a base triangular tiling, each equilateral triangle may be subdivided according to a self-similar rule, such as replacing it with a cluster of smaller triangles scaled by a factor that aligns edge lengths, ensuring the overall pattern remains a valid tiling without gaps or overlaps. These rules are applied iteratively, with each iteration producing finer details while adhering to the geometric constraints of regular polygons. The resulting tilings exhibit and often possess non-periodic or quasiperiodic properties, as the recursive scaling introduces irrational ratios that prevent translational periodicity, yet the structure covers the plane completely using convex regular polygons at all scales, approaching infinite resolution in the limit. Such tilings maintain edge-to-edge adjacency only if the base k-uniform tiling does, as mismatches in scaling could otherwise lead to overlapping or gapped boundaries. This approach was introduced by Karl Scherer in the late 1980s and 1990s through his explorations of rep-tiles and self-similar dissections, building on John Conway's foundational ideas in substitution systems for generating complex tilings from simpler prototypes. Scherer's work emphasized irregular rep-tiles but extended principles to polygonal hierarchies, influencing later developments in self-similar constructions. In relation to k-uniform tilings, fractalizing begins with a finite k-uniform base—such as a 3-uniform Archimedean tiling—and through repeated substitutions, the effective number of distinct vertex types increases in the limit, yielding tilings that approximate higher uniformity across scales. A key limitation is that edge-to-edge properties are preserved solely when the base tiling supports compatible scaling factors; for example, the fractalized triangular tiling, derived from recursive subdivision of equilateral triangles, remains edge-to-edge but becomes quasiperiodic due to non-integer scaling ratios.

Non-Edge-to-Edge Variations

Examples of non-edge-to-edge tilings

Non-edge-to-edge tilings by convex regular are arrangements where the completely cover the without gaps or overlaps, but the edges do not necessarily coincide fully at shared boundaries; instead, vertices of one may lie strictly in the interior of an edge of another . This contrasts with the standard edge-to-edge condition, where all vertices meet precisely at common points along edges. Such tilings preserve overall coverage through careful offsetting or staggering of , allowing for symmetries that would be impossible under strict edge alignment. Classical examples of these tilings include the uniform configurations cataloged by Grünbaum and Shephard, which organize all such uniform non-edge-to-edge tilings by regular polygons into seven infinite families, each characterized by periodic structures with high symmetry. These families primarily involve equilateral triangles, squares, and regular hexagons, as higher-sided regular polygons like pentagons fail to tile the plane even with offsets due to incompatible angle measures. For instance, one family consists of staggered rows of equilateral triangles, where each subsequent row is shifted laterally so that the vertices of upper triangles embed into the midpoints or fractions of the base edges of the lower row, creating a vertex figure that effectively fills 360 degrees around nodes despite the misalignment. Similar staggered row patterns apply to squares and hexagons, adjusting the offset to ensure no overlaps occur while maintaining convexity. A notable specific configuration is the offset square tiling, where horizontal rows of identical squares are displaced by a rational fraction of the side length (e.g., one-third or one-half, tuned to avoid gaps), resulting in vertical edges intersecting horizontal edges at interior points rather than vertices. This produces isohedral tilings with , and analogous setups for hexagons yield 6.6.6 vertex figures with embedded vertices. These arrangements demonstrate how non-edge-to-edge placement enables infinite families for allowable polygons, often achieving isogonal or vertex-transitive . The properties of these tilings include full planar coverage without global gaps or overlaps, though local edge mismatches occur, and they frequently exhibit periodic groups like pmm or p6. They are particularly valued in decorative contexts for their visual complexity arising from the offsets, which create interlocking patterns without requiring irregular shapes. Historically, non-edge-to-edge tilings by regular polygons appear in ornamental designs from medieval periods, including intricate motifs in Islamic and European patterns, predating formal mathematical classifications and serving aesthetic rather than strictly geometric purposes. Examples are documented in early 20th-century analyses of historical ornamentation, highlighting their use in textiles and . Constraints on these tilings stem from the convexity and fixed interior angles of regular polygons, restricting viable configurations to those where partial edge occupations allow the effective angular sum around each node to equal 360 degrees. Only regular triangles (60° angles), squares (90°), and hexagons (120°) support such families, as their angles divide evenly into multiples approaching full circles via offsets; polygons with angles like 108° (pentagons) or larger cannot satisfy coverage without or overlaps, limiting the scope despite the flexibility of non-edge-to-edge placement.

Recent extensions and developments

In the early 2020s, computational methods have advanced the study of periodic tilings by regular polygons, including non-edge-to-edge variants, through systematic enumeration and representation techniques. A significant methodological advance came in 2021 with the development of an integer-based representation for periodic tilings by regular polygons, proposed by Soto-Sánchez et al. This framework encodes tilings using a compact (2 + n) × 4 matrix that specifies vectors and vertices in the , facilitating automated generation and verification of periodic structures. By representing polygon arrangements through lattice coordinates and , the method enables efficient enumeration of new tilings and supports extensions to non-uniform variants, streamlining computational exploration beyond manual enumeration. In 2024, Colin Adams introduced spandrelized tilings as a novel construction technique for non-edge-to-edge arrangements using convex regular polygons. The process begins with an existing edge-to-edge tiling, places tangent disks at each vertex, and glues additional regular polygons into the resulting regions—the curvilinear triangular gaps between disks and edges—yielding new tilings that incorporate both the original polygons and the inserted ones. This approach generates infinite families of tilings from base uniform patterns, such as transforming the into one featuring squares and equilateral triangles, while preserving overall planarity and convexity. Supporting these advances, computational tools have emerged to aid visualization and discovery, though primarily focused on edge-to-edge cases. Ongoing as of 2025 continues to probe aperiodic arrangements incorporating subsets of regular polygons, though full aperiodic tilings solely with convex regular polygons remain elusive due to angle constraints. Future directions emphasize Euclidean-focused extensions, including deeper connections to models via substitution rules on periodic bases, and explorations of hyperbolic limits as approximations for dense Euclidean packings. Challenges persist in fully enumerating non-edge-to-edge families beyond the classical seven, with no major controversies noted in the field.

References

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