Tilings / Tessellations of Topological Spaces  

In General > s.a. cell complex [including simplicial]; euclidean geometry [polygon, polyhedron].
* Idea: A cell decomposition (tiling, tessellation) of a topological space M is a covering of M with a cell complex, i.e., an aggregate of cells that covers (is homeomorphic to) M without overlapping; The space is usually a manifold and often has a metric.
* History: In the XV century, 17 different types of regular tilings of the plane were used in the Alhambra; In 1891, the Russian mathematician Evgraf Fedorov proved that the number of distinct regular tilings is 17, the crystallographic groups; Between 1968 and 1984, all possible forms of tilings are classified into 19 categories; 1974, Penrose's quasiperiodic tiling; 1994, Radin and Conway's "pinwheel tiling"; 2011, John Shier's fractal tilings.
* Result: One can use the Euler formula ∑i (–1)i Ni = χ(θ) to relate the numbers of cells of different dimensionalities.
* Duality: The dual of a cell decomposition of M is also homeomorphic to M–although, since the duality Ω ↔ Ω* is an operation between abstract complexes, in general there is no natural embedding of Ω* in M.
@ References: Di Francesco et al mp/04 [determinant formulae, fully-packed loops].

Periodic or Regular Tiling / Tessellation > s.a. statistical geometry.
* Idea: A covering of the plane/space with a repeated pattern, like a mosaic, without leaving any gaps.
* Examples: The plane can be trivially tiled with squares, equilateral triangles, hexagons; Drawings by Escher of floors with lizards, butterflies, and abstract shapes; The Cairo tiling with irregular pentagons, named after the paving on several streets in Egypt's capital.
* Applications: Physics of single crystals; Getting the maximum number of parts out of a piece of sheet metal; > s.a. carbon [graphene].
@ References: Coxeter 57, Magnus 74 [non-Euclidean]; Coxeter PRS(64) [hyperbolic]; Grünbaum & Shepard 87; Adams MI(95) [knotted tiles]; Renault JCTB(08) [locally finite]; Gjerde 08 [popular level, origami tessellations].
> Online resources: see Thérèse Eveilleau page; Xavier Hubaut page.

Quasiperiodic Tiling > s.a. Quasicrystal; random walk.
* Penrose tiling: A quasiperiodic tiling of E2, with tiles of two different shapes (kites and darts); Kite angles: 3 × 72o, 144o; Dart angles: 2 × 36o, 72o, 216o; the two vertices with the large angles on darts meet with the 2 opposite 72o angles on kites.
* Penrose tiling, construction and crystals: Can be obtained from a cubic lattice in 3D, by cutting the space with a hypersurface of irrational inclination, smearing out the lattice points perpendicularly to the hypersurface and considering the induced lattice; Macroscopic crystals of this type exist (e.g., HOMgZn [@ Fisher et al PRB(99)]), but are difficult to make, because they occupy a small region of the phase diagram.
@ Penrose tiling: Penrose 74; Gardner SA(77)jan; Cotfas JPA(98), mp/04 [self-similarities]; Tasnadi mp/02 [and non-commutative algebra]; Mulvey & Resende IJTP(05) [non-commutative theory]; Battaglia & Prato CMP(10)-a0712 [Penrose kite and symplectic geometry]; Oyono-Oyono & Petite JGP(11) [C*-algebra and K-theory for Penrose hyperbolic tilings]; Boyle & Steinhardt a1608 [and Coxeter pairs].

Other Tilings and Related Topics > s.a. Delone Sets; forms; graph; Triangulation; random and voronoi tiling.
* Platonic tilings: Tilings of the plane consisting of regular periodic arrays of a single shape (such as squares, triangles, or hexagons).
* Archimedean tilings: Tilings of the plane composed of two or three different shapes, forming only one type of vertex; There are eight types.
* Aperiodic tilings: Non-periodic tilings defined by local rules.
@ With n-fold rotational symmetry: Bédaride & Fernique a1409 [and weak local rules].
@ Hierarchical tilings: Radin NAMS(95); Priebe Frank a1311-proc [general framework, fusion model for generating hierarchical tilings]
@ Aperiodic tilings: Baake & Grimm PhilMag(06)mp/05 [and invariants]; Bédaride & Fernique CMP(15)-a1309 [and surfaces in higher-dimensional spaces].
@ Other tilings: Nagel & Weiss AAP(05) [random, stable under iteration]; García & García JPA(05) [deterministic inflation rules]; Priebe Frank a0705 [substitution tilings of E2]; Dolbilin & Frettlöh EJC(10) [Böröczky tilings in hyperbolic spaces]; Lachièze-Rey AAP(11) [STIT tessellations]; Gao et al JCTA(13) [tiling of a sphere by pentagons]; Priebe Frank a1312 [with infinite local complexity]; news PhysOrg(16)dec [particles self-assemble into Archimedean tilings].
@ Combinatorial curvature: Klassert et al mp/04* [2D, and elliptic operators]; > s.a. Tetrahedron.
@ Topological invariants: Forrest et al CMP(02) [cohomology]; Gähler et al mp/05, AGT(13)-a1202 [cohomology, K-theory, and torsion]; Sadun a1406 [cohomology].
@ Counting and incidence: Aste JPA(98) [statistical properties]; Dubertret et al JPA(98) [2D, geometrical correlations]; Weiss & Cowan AAP(11) [topological relationships for tessellations of \(\mathbb R\)3 that are not facet-to-facet]; Hutchinson & Widom TCS(15)-a1306 [octagonal tilings, enumeration]; > s.a. statistical geometry.

Set T of Tilings of M and Operations on Tilings
* Structure: The set T is partially ordered by refinement, and has a σ-algebra generated by sets of the form

TK:= {θ ∈ T | edges(θ) ∩ K ≠ Ø},    for    KM compact .

* Superposition: Formed by the union of edge sets.
* Refinements: Various procedures are possible, like iterated division.
@ Space of tilings: Blackwell & Møller AAP(03) [deformed tessellations]; Sadun JMP(03)m.DS/02 [with finite local complexity, as inverse limit], m.DS/05-conf [Cech cohomology]; Bellissard et al CMP(05) [with finite pattern condition]; Priebe Frank & Sadun m.DS/07 [infinite local complexity and fault lines, as inverse limit].
@ Operations on tilings: Nagel & Weiss AAP(03) [superposition, iteration, and limits]; Maier & Schmidt AAP(03) [superposition, nesting and Bernoulli thinning].

In Physics > s.a. lattice field theory [field theories on complexes]; thermodynamics; voronoi tiling.
* Froth: A medium containing uniformly dispersed solid particles and/or gas molecules, like a soap/water mixture.
@ Froth: Aste & Rivier JPA(95) [theory, topology and curvature]; Elias et al PRE(97) [liquid magnetic froth].
@ And dynamics: Aste & Sherrington JPA(99), Davison & Sherrington JPA(00) [stochastic, glassy transition]; Holton et al CMP(05) [re tiling dynamical systems]; Kaatz et al PhyA(12) [2D, statistical mechanics]; > s.a. lattice gravity.

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