# Reading Notes on Tilings and Patterns

By Xah Lee. Date: . Last updated: .

This is some notes from reading [Tiling and Patterns By Branko Grunbaüm, G C Shaphard.]

### Some definitions

* A Plane Tiling PT is a countable family of closed sets PT = {T1, T2,…} which cover the plane without gaps or overlaps.

* to narrow down our study, we require that each tile is a (closed) topological disk. By topological disk, we mean any set whose boundary is a single simple closed curve. (this way, we eliminates some possibly useful but weird situations where "tiles" that are disconnected, has a hole, connected by just a point, connected by a line, or has lines, or unbounded.)

* A vertice of a tiling is a point in the plane that are intersection of two tiles. A edge of a tiling is a curve that is the intersection of two tiles. The vertices, edges, tiles of a tiling are called its elements.

* A vertice is connected to a number of edges. This number is called the valence of the vertice.

in the context of tilings by polygons, there is the concept of edge-to-edge tilings. That is, if every side of a polygon is also a edge of the tiling, than the tiling is edge-to-edge (as opposed to a edge a tiling consisting several sides of a tile). More to the point, all edges in the tiling are straight lines.

### 1.2 Tilings with tiles of a few shapes

A tiling is call monohedral if every tile in the tiling PT is congruent (directly or reflectively) to one fixed set T. The set T is called the prototile of PT.

### 1.3 Symmetry, transitivity, and regularity

An isometry or congruence transformation is any mapping of the Euclidean plane onto itself which preserves all distances.

If a tiling admits any symmetry in addition to the identity symmetry then it will be called symmetric. If its symmetry group contains at least two translations in nonparallel directions then the tiling will be called periodic. The linear combinations of any two nonparallel translations generates a lattice. Thus with every periodic tiling PT is associated a lattice, and the points of the lattice can be regarded (in many ways) as the vertices of a parallelogram tiling PT; the tiles of PT are known as period parallelograms.

Two tiles T1 and T2 of a tiling PT are said to be equivalent if the symmetry group S(PT) contains a transformation that maps T1 onto T2; the collection of all tiles of PT that are equivalent to T1 is called the transitivity class of T1. If all tiles of PT form one transitive class we say that PT is tile-transitive or isohedral.

If PT is a tiling with precisely k transitivity classes then PT is called k-isohedral.

If the symmetry group S(PT) of PT contains operations that map every vertex of PT onto any other vertex, then we say that the vertices form one transitivity class, or that the tiling is isogonal. A tiling is k-isogonal if its vertices form k transitivity classes, where k ≥ 1 is any integer.

1-gonal is called monogonal. It's the number of vertex types.
1-hedral is called monohedral. It's the number of tiles types.
1-toxal is called monotoxal. It's the number of edge types.

With symmetry of the tiling considered, we have k-isogonal, k-isohedral, and k-isotoxal corresponding to the number of classes with vertexes, tiles, and edges respectively.

By a flag in a tiling we mean a triple (V,E,T) consisting of a vertex V, an edge E and a tile T which are mutually incident.

A tiling PT is called regular if its symmetry group S(PT) is transitive on the flags of PT.

### 1.5 Monomorphism and k-morphism

A tile is called k-morphic if it can be used to tile the plane in exactly n ways. is the prototile of a unique monohedral tiling of the plane. Similarly, a tile is n-morphic if it is the prototile of exactly n tilings of the plane.

##### open questions

For every positive integer r, is it possible to find a tile which is r-morphic?

Is there a tile that admits a countable infinity of distinct tilings?
Xah: I thought so, but don't know. For example, figure 1.5.8, just change the neghboring row placements, one should be able to come up with infinity distinct tilings, where most of them are not symmetric. Form p.59, the discussion seems to indicate that my method above allows an “uncountable infinity”, so is not a solution.

## 2. Tilings by regular polygons and star polygons

### Edge-to-edge regular polygon tilings

#### k-hedral, k-gonal, k-toxal and k-isohedral, k-isogonal, k-isotoxal

• * if the tiling has k types of tiles (that is, uses k prototiles), then it is called k-hedral.
• * if the tiling has k types of vertexes, then it is called k-gonal.
• * if the tiling has k types edges, then it is called k-toxal.

If the tiles in a tiling forms h transitive classes with respect to the tiling's symmetry, then the tiling is called h-isohedral (or, having isohedral index of h). (two tiles are of the same transitive class if one can be mapped to the other by the symmetries of the tiling. Obviously, the isoheral index is always less or equal to the hedral index.) A tiling's isogonal and isotoxal indexes are similarly defined with respect to vertexes and edges.

A edge-to-edge regular polygon tiling has the following type of properties:

• * isohedral index (k-isohedral) and number of tile types (k-hedral). (if the two are equal, the tiling is called equitransitive.)
• * isogonal index (k-isogonal) and number of vertex types (k-gonal).
• * isotoxal index (k-isotoxal) and number of edge types (k-toxal). (the latter is always 1.)
• * its symmetry group.

k-uniform is the name for edge-to-edge regular polygon tilings that's k-isohedral. Archimedean tiling are 1-gonal edge-to-edge regular polygon tilings (by definition). Archimedean tilings are also 1-isogonal(i.e. happens to be), therefore also uniform.

## 4. The topology of tilings

### 4.1 Homeomorphisms and topologicalequivalence

A mapping φ:E²→E² of the plane onto itself is called a homeomorphism or a topological transformation if it is one-to-one and bicontinuous. (bicontinuous means both pre-image and image are continuous.)

Two tilings are said to be of the same topological type (or topologically equivalent) if there is a homeomorphism which maps one onto the other. The fact that the composition of two homeomorphisms, and the inverse of a homeomorphism, are also homeomorphisms shows that topological equivalence is an equivalence relation (inthat it is reflexive, symmetric, and transitive). It therefore partitions the set of all tilings into “topological types”.

A normal tiling after a homeomorphism may no longer be normal. (see text.)

Two tilings are said to be combinatorial equivalent if one is the image of an inclusion-preserving map of the other. (inclusion meaning subset.) Basically meaning that elements (vertexes, edges, tiles) remain neighbors. For normal tilings the concepts of topological equivalence and combinatorial equivalence coincide.
Two tilings are said to be isoptic if there is a series of continuous mappings that transforms one to the other. i.e. Their topological equivalence can be demonstrated visually. (the book notes that this is not easy to prove that isotopy and topoligical equivalence are equivalent.)

### 4.2 Duality

Two tilings PT and PT* are said to be dual to each other if there exists a one-to-one inclusion-reversing map phi; from a set E[PT] onto the set E[PT*]. (we recall that E[PT] is the set of elements -- vertices, edges, and tiles -- of the tiling PT.) By inclusion-reversing we mean that whenever e1 and e2 are elements of T, then phi[e1] includes phi[e2] iff e2 includes e1.

Two dual tilings are dually-situated if one can be superimposed on the other to make their dual relation visually apparant.

### 4.3 Homeohedral Tilings

A tiling PT is called homeohedral or topologically tiletransitive if it is a normal tiling and is such that for any two tiles T1, T2 of PT there exists a homeomorphism of the plane that maps PT onto PT and T1 onto T2.

If T is a tile of the tiling PT then we say that T is of valence-type {j1,j2,…,jk} provided T has k vertices which, if considered in a suitable cyclic order, have valences {j1,j2,…,jk}. If all tiles of a normal tiling PT have the same valence-type {j1,j2,…,jk}, we say that PT is homogeneous of type {j1,j2,…,jk}. It is clear that veery homeohedral tiling is homogeneous. We do not distinguish between types if their symbols can be obtained from each other by a cyclic permutation or reversal of order. We will choose the standard order to be the one ordered lexicographically.

Theorem: If PT is a homogeneous tiling then it is of one of the element types corresponding to the vertice types of the 11 Lave tilings. Each homogeneous tiling is homeohedral, all homogeneous tiling of the same type are topologically equivalent to each other.

## Personal notes

### To do

* Investigate tilings by regular square and triangle.
* Find m-isogonal tiling with n vertice types, where m > n.

### possible errors in the book

Suggestion:
p.37, section 1.4.
Considering introducting the orbifold notation for symmetry groups.

Suggestion:
p.44, Table 1.4.2.
Is there a reason why the transitive class are considered within the type of symmetry?

Error?
p.56, upper right.
According to Symmetries of Islamic Geometrical Patterns, by Syed Jan Abas, Amer Shaker Salman (World Scientific. 1995), all types of wallpaper groups are found in Islamic area. They also provide photographs.

Suggestion:
p.61, Table 2.1.1
The middle section of the table is annoying. The info is explicit in the tilings' symbol.

Suggestion:
p.95, section 2.7
While writing computer program to generate Archimedean tilings and their duals, I found that the centroids of Archimedean tiles are the vertexes of the Lave tilings, but the centroid of Lave tiles are not vertexes of Archimedean tilings. This is suprising, because I think intuitive we'd think that the centroids mutually forms the other's vertexes. Perhaps a good exercise.

Suggestion:
p.221, lower left
“We find it convenient to consider …”
Q: How exactly is group diagram defined? Does it contain the pattern themselves?
probably not a good comment.

#### Book wishlist

A chapter that intorduce Bill Thurston's idea of viewing wallpaper as an orbifold, and the orbifold notation for wallpaper groups.
A chapter on layered patterns and isonemal fabric.
Possibly a chapter on group theory, and Cayley diagram.