Tilings of surfaces and packings of space have been of interest to artisans and manufacturers throughout history; they are a means of artistic expression and lend economy and strength to modular constructions. Today scientists and mathematicians study tilings because they pose interesting mathematical questions and provide mathematical models for such diverse structures as the molecular anatomy of crystals, cell packings of viruses, ndimensional algebraic codes, and “nearest neighbor” regions for a set of discrete points. The basic questions are: What bodies can tile space? In what ways do they tile? However, in this generality such questions are intractable. To study tiles and tilings, we must impose constraints.
Tilings of surfaces and packings of space have been of interest to artisans and manufacturers throughout history; they are a means of artistic expression and lend economy and strength to modular constructions. Today scientists and mathematicians study tilings because they pose interesting mathematical questions and provide mathematical models for such diverse structures as the molecular anatomy of crystals, cell packings of viruses, ndimensional algebraic codes, and “nearest neighbor” regions for a set of discrete points. The basic questions are: What bodies can tile space? In what ways do they tile? However, in this generality such questions are intractable. To study tiles and tilings, we must impose constraints.
Even with constraints the subject is unmanageably large. In this chapter we restrict ourselves, for the most part, to tilings of unbounded spaces. In the next section we present some general results that are fundamental to the subject as a whole. Section 3.2 addresses tilings with congruent tiles. In Section 3.3 we discuss the classical subject of periodic tilings, which continues to be enriched with new results. Next, we briefly describe the newer theory of nonperiodic and aperiodic tilings, both of which are discussed in more detail in Chapter 62. We conclude with a very brief description of some kinds of tilings not considered here.
In this section we define terms that will be used throughout the chapter and state some basic results. Taken together, these results state that although there is no algorithm for deciding which bodies are tiles, there are criteria for deciding the question in certain cases. We can obtain some quantitative information about the tiling in particularly wellbehaved cases.
Unless otherwise stated, we assume that S is an ndimensional space, either Euclidean $\left({\mathbb{E}}^{n}\right)$
or hyperbolic. We also assume that the tiles are bounded and the tilings are locally finite (see the Glossary below). Throughout this chapter, n is the dimension of the space in which we are working.:A bounded region (of S) that is the closure of its (nonempty) interior.
Tiling (of S):A decomposition of S into a countable number of ndimensional bodies whose interiors are pairwise disjoint. In this context, the bodies are also called ncells and are the tiles of the tiling (see below). Synonyms: tessellation , parquetry (when n = 2), honeycomb (for n ≥ 2).
Tile:A body that is an ncell of one or more tilings of S. To say that a body tiles a region R ⊆ S means that R can be covered exactly by copies of the body without gaps or overlaps.
Locally finite tiling:Every nball of finite radius in S meets only finitely many tiles of the tiling.
Prototile set (for a tiling $\mathcal{T}$ of S):A minimal subset of tiles in $\mathcal{T}$
such that each tile in the tiling $\mathcal{T}$ is the congruent image of one of those in the prototile set. The tiles in the set are called prototiles and the prototile set is said to admit $\mathcal{T}$ . kface (of a tiling):An intersection of at least n − k + 1 tiles of the tiling that is not contained in a jface for j < k. (The 0faces are the vertices and 1faces the edges ; the (n−1)faces are simply called the faces of the tiling.)
Patch (in a tiling):A set of tiles whose union is homeomorphic to an nball. See Figure 3.1.1. A spherical patch P(r, s) is the set of tiles whose intersection with the ball of radius r centered at s is nonempty, together with any additional tiles needed to complete the patch (that is, to make it homeomorphic to an nball).
Figure 3.1.1 Three patches in a tiling of the plane by squares.
:A tiling in which (i) each prototile is homeomorphic to an nball, and (ii) the prototiles are uniformly bounded (there exist r > 0 and R > 0 such that each prototile contains a ball of radius r and is contained in a ball of radius R). It is technically convenient to include a third condition: (iii) the intersection of every pair of tiles is a connected set. (A normal tiling is necessarily locally finite.)
Facetoface tiling (by polytopes):A tiling in which the faces of the tiling are also the (n−1)dimensional faces of the polytopes. (A facetoface tiling by convex polytopes is also kfacetokface for 0 ≤ k ≤ n − 1.) In dimension 2, this is an edgetoedge tiling by polygons, and in dimension 3, a facetoface tiling by polyhedra.
Dual tiling:Two tilings $\mathcal{T}\text{and}\mathcal{T}*$
are dual if there is an incidencereversing bijection between the kfaces of $\mathcal{T}$ and the (n−k)faces of $\mathcal{T}*$ (see Figure 3.1.2). Voronoi (Dirichlet) tiling:A tiling whose tiles are the Voronoi cells of a discrete set Λ of points in S. The Voronoi cell of a point p ∊ Λ is the set of all points in S that are at least as close to p as to any other point in Λ (see Chapter 23).
Delaunay (or Delone) tiling:A facetoface tiling by convex circumscribable polytopes (i.e., the vertices of each polytope lie on a sphere).
Figure 3.1.2 A Voronoi tiling (solid lines) and its Delaunay dual (dashed lines).
:A distancepreserving selfmap of S.
Symmetry group (of a tiling):The set of isometries of S that map the tiling to itself.
To say that a body tiles ${\mathbb{E}}^{n}$
usually means that there is a tiling all of whose tiles are copies of this body. The artist M.C. Escher has demonstrated how intricate such tiles can be even when n = 2. But in higher dimensions the simplest tiles—for example, cubes—can produce surprises, as the recent counterexample to Keller’s conjecture attests (see below).:A tiling with a single prototile.
rmorphic tile:A prototile that admits exactly r distinct monohedral tilings. Figure 3.2.1 shows a 5morphic tile and all its tilings, and Figure 3.2.3 shows a 1morphic tile and its tiling.
Figure 3.2.1 A pentamorphic tile.
:A body for which k copies can be assembled into a larger, similar body. (Or, equivalently, a body that can be partitioned into k congruent bodies, each similar to the original.) More formally, a krep tile is a closed set A _{1} in S with nonempty interior such that there are sets A _{2}, …, A_{k} congruent to A _{1} that satisfy
for all i ≠ j and A _{1} ∪ … ∪ A_{k} = g(A _{1}), where g is a similarity mapping. (Figure 3.2.2 shows a 3dimensional chair rep tile and the secondlevel chair. An ndimensional chair rep tile can be formed in a similar manner.)
Transitive action:A group G is said to act transitively on a set {A _{1},A _{2}, …} if the set is an orbit for G. (That is, for every pair A_{i} , A_{j} of elements of the set, there is a g_{ij} ∊ G such that g_{ij}A_{i} = A_{j} .)
Regular system of points:A discrete set of points on which an infinite group of isometries acts transitively.
Isohedral (tiling):A tiling whose symmetry group acts transitively on its tiles.
Anisohedral tile:A prototile that admits monohedral tilings but no isohedral tilings. In Figure 3.2.3, the prototile admits a unique nonisohedral tiling; the shaded tiles are each surrounded differently, from which it follows that no isometry can map one to the other (and the tiling to itself). This tiling is periodic, however (see Section 3.3).
Figure 3.2.2 A 3dimensional chair rep tile and a secondlevel chair in which seven copies surround the first.
Figure 3.2.3 An anisohedral tile (due to R. Penrose) and its unique tiling in which tiles are surrounded in two different ways.
:Define C ^{0}(P) = P. Then C^{k} (P), the kth corona of P, is the set of all tiles Q ∊ T for which there exists a path of tiles P = P _{0}, P _{1}, …, P_{m} = Q with m ≤ k in which ${P}_{i}\cap {P}_{i+1}\ne \overline{)0}$
, i = 0, 1, …, m − 1. Lattice:The group of integral linear combinations of n linearly independent vectors in S. A point orbit of a lattice, often called a point lattice , is a particular case of a regular system of points.
Translation tiling:A monohedral tiling of S in which every tile is a translate of a fixed prototile. See Figure 3.2.4.
Lattice tiling:A monohedral tiling on whose tiles a lattice of translation vectors acts transitively. Figure 3.2.4 is not a lattice tiling since it is invariant by multiples of just one vector.
nparallelotope:A convex npolytope that tiles ${\mathbb{E}}^{n}$
by translation.
Figure 3.2.4 A translation nonlattice tiling.
:A maximal subset of parallel (n−2)faces of a parallelotope in ${\mathbb{E}}^{n}$
. The number of (n−2)faces in a belt is its length. Center of symmetry (for a set A in ${\mathbb{E}}^{n}$ ):A point a ∊ A such that A is invariant under the mapping x → 2a−x; the mapping is called central inversion and an object that has a center of symmetry is said to be centrosymmetric .
Stereohedron:A convex polytope that is the prototile of an isohedral tiling. A Voronoi cell of a regular system of points is a stereohedron.
Linear expansive map:A linear transformation all of whose eigenvalues have modulus greater than one.
Figure 3.2.5 Ammann’s 3corona tile cannot be surrounded by a fourth corona. 4corona and 5corona tiles also exist.
Periodic tilings have been studied intensely, in part because their applications range from ornamental design to crystallography, and in part because many techniques (algebraic, geometric, and combinatorial) are available for studying them.
:A tiling, not necessarily monohedral, whose symmetry group contains an ndimensional lattice. This definition can be adapted to include “subperiodic” tilings (those whose symmetry groups contain 1 ≤ k < n linearly independent vectors) and tilings of other spaces (for example, cylinders). Tilings in Figures 3.2.1, 3.2.3, 3.3.1, and 3.3.3 are periodic.
Fundamental domain (generating region) for a periodic tiling:A minimal subset of S whose orbit under the symmetry group of the tiling is the whole tiling. A fundamental domain may be a tile (Figure 3.2.1), a subset of a single tile (Figure 3.3.1), or a subset of tiles (two shaded tiles in Figure 3.2.3).
Orbifold (of a tiling of S):The manifold obtained by identifying points of S that are in the same orbit under the action of the symmetry group of the tiling.
Free tiling:A tiling whose symmetry group acts freely and transitively on the tiles.
kisohedral (tiling):A tiling whose tiles belong to k transitivity classes under the action of its symmetry group. Isohedral means 1isohedral (Figures 3.2.1, 3.3.1, and 3.3.3). The tiling in Figure 3.2.3 is 2isohedral.
Equitransitive (tiling by polytopes):A tiling in which each combinatorial class of tiles forms a single transitivity class under the action of the symmetry group of the tiling.
kisogonal (tiling):A tiling whose vertices belong to k transitivity classes under the action of its symmetry group. Isogonal means 1isogonal.
kuniform (tiling of a 2dimensional surface):A kisogonal tiling by regular polygons.
Uniform (tiling for n > 2):An isogonal tiling with congruent edges and uniform faces.
Flag of a tiling (of S):An ordered (n+1)tuple (X _{0}, X _{1}, …, X_{n} ), with X_{n} a tile and X_{k} a kface for 0 ≤ k ≤ n − 1, in which X _{ i−1} ⊂ X_{i} for i = 1, …, n.
Regular tiling (of S):A tiling $\mathcal{T}$
whose symmetry group is transitive on the flags of $\mathcal{T}$ . (For n > 2, these are also called regular honeycombs.) See Figure 3.3.3. kcolored tiling:A tiling in which each tile has a single color, and k different colors are used. Unlike the case of map colorings, in a colored tiling adjacent tiles may have the same color.
Perfectly kcolored tiling:A kcolored tiling for which each element of the symmetry group G of the uncolored tiling effects a permutation of the colors. The ordered pair (G, Π), where Π is the corresponding permutation group, is called a kcolor symmetry group.
The mathematical study of tilings (like most mathematical investigations) has been accompanied by the development and use of a variety of notations for classification of different “types” of tilings and tiles. Far from being merely names by which to distinguish types, these notations tell us the investigators’ point of view and the questions they ask. Notation may tell us the global symmetries of the tiling, or how each tile is surrounded, or the topology of its orbifold. Notation makes possible the computer implementation of investigations of combinatorial questions about tilings.
Periodic tilings are classified by symmetry groups and, sometimes, by their skeletons (of vertices, edges, …, (n−1)faces). The groups are known as crystallographic groups; up to isomorphism, there are 17 in ${\mathbb{E}}^{2}$
and 219 in ${\mathbb{E}}^{3}$ . For ${\mathbb{E}}^{2}$ and ${\mathbb{E}}^{3}$ , the most common notation for the groups has been that of the International Union of Crystallography (IUCr) [Hah83]. This is crossreferenced to earlier notations in [Sch78]. Recently developed notations include DelaneyDress symbols [Dre87] and orbifold notation for n = 2 [Con92, CH02] and for n = 3 [CDHT01].:Encodes lattice type and particular symmetries of the tiling. In Figure 3.3.1, the lattice unit diagram at the right encodes the symmetries of the tiling and the IUCr symbol p31m indicates that the highestorder rotation symmetry in the tiling is 3fold, that there is no mirror normal to the edge of the lattice unit, and that there is a mirror at 60° to the edge of the lattice unit. These symbols are augmented to denote symmetry groups of perfectly 2colored tilings.
DelaneyDress symbol (for tilings of Euclidean, hyperbolic, or spherical space of any dimension):Associates an edgecolored and vertexlabeled graph derived from a chamber system (a formal barycentric subdivision) of the tiling. In Figure 3.3.2, the nodes of the graph represent distinct triangles A, B, C, D in the chamber system, and colored edges (dashed, thick, or thin) indicate their adjacency relations. Numbers on the nodes of the graph show the degree of the tile that contains that triangle and the degree of the vertex of the tiling that is also a vertex of that triangle.
Figure 3.3.1 An isohedral tiling with standard IUCr lattice unit shaded; a halfleaf is a fundamental domain. The classification symbols are for the symmetry group of the tiling.
Figure 3.3.2 A chamber system of the tiling in Figure 3.3.1 determines the graph that is its DelaneyDress symbol.
:Encodes properties of the orbifold induced by the symmetry group of a periodic tiling of the Euclidean plane or hyperbolic plane, or a finite tiling of the surface of a sphere; introduced by Conway. In Figure 3.3.1, the first 3 in the orbifold symbol 3*3 for the symmetry group of the tiling indicates there is a 3fold rotation center (gyration point) that becomes a cone point in the orbifold, while *3 indicates that the boundary of the orbifold is a mirror with a corner where three mirrors intersect.
See Table 3.3.1 for the IUCr and orbifold notations for ${\mathbb{E}}^{2}$
.
IUCr 
ORBIFOLD 
IUCr 
ORBIFOLD 

p1 
o or o1 
p3 
333 
pg 
×× or 1×× 
p31m 
3*3 
cm 
*× or 1*× 
p3m1 
*333 
pm 
** or 1** 
p4 
442 
p2 
2222 
p4g 
4*2 
pgg 
22× 
p4m 
*442 
pmg 
22* 
p6 
632 
cmm 
2*22 
p6m 
*632 
pmm 
*2222 


Isohedral tilings of ${\mathbb{E}}^{2}$
fall into 11 combinatorial classes, typified by the Laves nets (Figure 3.3.3). The Laves net for the tiling in Figure 3.3.1 is [3.6.3.6]; this gives the vertex degree sequence for each tile. In an isohedral tiling, every tile is surrounded in the same way. Grünbaum and Shephard provide an incidence symbol for each isohedral type by labeling and orienting the edges of each tile [GS79]. Figure 3.3.4 gives the incidence symbol for the tiling in Figure 3.3.1. The tile symbol a ^{+} a ^{−} b ^{+} b ^{−} records the cycle of edges of a tile and their orientations with respect to the (arrowed) first edge (+ indicates the same, − indicates opposite orientation). The adjacency symbol b ^{−} a ^{−} records for each different letter edge of a single tile, beginning with the first, the edge it abuts in the adjacent tile and their relative orientations (now − indicates same, + opposite). These symbols can be augmented to adjacency symbols to denote kcolor symmetry groups. Earlier, Heesch devised signatures for the 28 types of tiles that could be fundamental domains of isohedral tilings without reflection symmetry [HK63]; this signature system was extended in [BW94].Figure 3.3.3 The 11 Laves nets. The three regular tilings of E 2 are at the top of the illustration.
Figure 3.3.4 Labeling and orienting the edges of the isohedral tiling in Figure 3.3.1 determines its GrünbaumShephard incidence symbol.
Nonperiodic tilings are found everywhere in nature, from cracked glazes to biological tissues to real crystals. In a remarkable number of cases, such tilings exhibit strong regularities. For example, many such tilings have simplicial duals. Others repeat on increasingly larger scales. An even larger class of tilings are those now called repetitive, in which every bounded configuration appearing anywhere in the tiling is repeated infinitely many times throughout it (see below). Aperiodic tilings—those whose prototile sets admit only nonperiodic tilings—are particularly interesting. They were first introduced to prove the Undecidability Theorem (Section 3.1). Later, after Penrose found pairs of aperiodic prototiles (see Figure 3.4.1), they became popular in recreational mathematical circles. Their deep mathematical properties were first studied by Penrose, Conway, de Bruijn, and others. After the discovery of “quasicrystals” in 1984, aperiodic tilings became the focus of intense research. The basic ideas of this rapidly developing subject are only introduced here; they are discussed in more detail in Chapter 62.
Figure 3.4.1 Portions of Penrose tilings of the plane (a) by rhombs; (b) by kites and darts. The matching rules that force nonperiodicity are not shown (see Chapter 62).
:A tiling with no translation symmetry.
Hierarchical tiling:A tiling whose tiles can be composed into larger tiles, called levelone tiles, whose levelone tiles can be composed into leveltwo tiles, and so on ad infinitum. In some cases it is necessary to partition the original tiles before composition.
Selfsimilar tiling:A hierarchical tiling for which the larger tiles are copies of the prototiles (all enlarged by a constant expansion factor λ). krep tiles are the special case when there is just one prototile (Figure 3.2.2).
Uniquely hierarchical tiling:A tiling whose jlevel tiles can be composed into (j+1)level tiles in only one way (j = 0, 1, …).
Composition rule (for a hierarchical tiling):The equations T′_{i} = m _{ i1} T _{1} ∪ … ∪ m_{ik}T_{k} , i = 1, …, k, that describe the numbers m_{ij} of each prototile T_{j} in the next higher level prototile T′_{i} . These equations define a linear map whose matrix has i, j entry m_{ij} .
Relatively dense configuration:A configuration C of tiles in a tiling for which there exists a radius r_{C} such that every ball of radius r_{C} in the tiling contains a copy of C.
Repetitive:A tiling in which every bounded configuration of tiles is relatively dense in the tiling.
Local isomorphism class:A family of tilings such that every bounded configuration of tiles that appears in any of them appears in all of the others. (For example, the uncountably many Penrose tilings with the same prototile set form a single local isomorphism class.)
Projected tiling:A tiling obtained by the canonical projection method (see Chapter 62).
Aperiodic prototile set:A prototile set that admits only nonperiodic tilings; see Figure 3.4.1.
Aperiodic tiling:A tiling with an aperiodic prototile set.
Matching rules:A list of rules for fitting together the prototiles of a given prototile set.
Mutually locally derivable tilings:Two tilings are mutually locally derivable if the tiles in either tiling can, through a process of decomposition into smaller tiles, or regrouping with adjacent tiles, or a combination of both processes, form the tiles of the other (see Figure 3.4.2).
Complex Perron number:An algebraic integer that is strictly larger in modulus than its Galois conjugates (except for its complex conjugate).
Figure 3.4.2 The Penrose tilings by kites and darts and by rhombs are mutually locally derivable.
Figure 3.4.3 Conway’s biprism consists of two prisms fused at a common rhombus face. Small angle of rhombus is acos(3/4) ≈ 41.4°; diagonal of prism ≈ 2.87. When assembled, the vertices of the rhombus that is a common face of the two prisms are the poles of two 2fold rotation axes.
Does there exist a prototile in ${\mathbb{E}}^{2}$
that is aperiodic? Does there exist a convex prototile for ${\mathbb{E}}^{3}$ that is aperiodic without restriction?There is a vast literature on tilings (or dissections) of bounded regions (such as rectangles and boxes, polygons, and polytopes) by tiles to satisfy particular conditions. This and much of the recreational literature focuses on tilings by tiles of a particular type, such as tilings by rectangles, tilings by clusters of ncubes (polyominoes—see Chapter 15–and polycubes) or nsimplices (polyiamonds in ${\mathbb{E}}^{2}$
), or tilings by recognizable animate figures. In the search for new ways to produce tiles and tilings, both mathematicians (such as P.A. MacMahon [Mac21]) and amateurs (such as M.C. Escher [Sch90]) have contributed to the subject. Recently the search for new shapes that tile a given bounded region S has produced knotted tiles, toroidal tiles, and twisted tiles. Kuperberg and Adams have shown that for any given knot K, there is a monohedral tiling of ${\mathbb{E}}^{3}$ (or of hyperbolic 3space, or of spherical 3space) whose prototile is a solid torus that is knotted as K. Also, Adams has shown that, given any polyhedral submanifold M with one boundary component in ${\mathbb{E}}^{n}$ , a monohedral tiling of ${\mathbb{E}}^{n}$ can be constructed whose prototile has the same topological type as M [Ada95].Other directions of research seek to broaden the definition of prototile set: in new contexts, the tiles in a tiling may be homothetic (rather than congruent) images of tiles in a prototile set, or be topological images of tiles in a prototile set. For example, a tiling of ${\mathbb{E}}^{n}$
by polytopes in which every tile is combinatorially isomorphic to a fixed convex npolytope (the combinatorial prototile) is said to be monotypic . It has been shown that in ${\mathbb{E}}^{2}$ , there exist monotypic facetoface tilings by convex ngons for all n ≥ 3; in ${\mathbb{E}}^{3}$ , every convex 3polytope is the combinatorial prototile of a monotypic tiling [Sch84a]. Many (but not all) classes of convex 3polytopes admit monotypic facetoface tilings [DGS83, Sch84b].The following surveys are useful, in addition to the references below.