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.
* Result: Can use the Euler formula _i (–1)ⁱ N_i = () 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 * is an operation between abstract complexes, in general there is no natural embedding of * 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.
* Applications: Physics of single crystals; getting the maximum number of parts out of a piece of sheet metal.
@ References: Coxeter 57, Magnus 74 [non-Euclidean]; Coxeter PRS(64) [hyperbolic]; Grünbaum & Shepard 87; Adams MI(95) [knotted tiles]; Renault JCTB(08) [locally finite].

Quasiperiodic Tiling > s.a. Quasicrystal.
* Penrose tiling: A quasiperiodic tiling of E², with tiles of two different shapes (kites and darts); Kite angles: 3 72^o, 144^o; Dart angles: 2 36^o, 72^o, 216^o; the two vertices with the large angles on darts meet with the 2 opposite 72^o 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 a0712 [Penrose kite and symplectic geometry].

Other Tilings and Related Topics > s.a. forms; graph; Triangulation.
@ Other tilings: Radin NAMS(95) [hierarchical]; Nagel & Weiss AAP(05) [random, stable under iteration]; García & García JPA(05) [deterministic inflation rules]; Priebe Frank a0705 [substitution tilings of E²]; > s.a. random and voronoi tiling.
@ 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 [cohomology, K-theory, and torsion].
@ Counting and incidence: Aste JPA(98) [statistical properties]; Dubertret et al JPA(98) [2D, geometrical correlations]; Baake & Grimm PhilMag(06)mp/05 [aperiodic tilings, and invariants]; > 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 -algebra generated by sets of the form

T_K:= {  T | edges() K Ø},    for    K M 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-in [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]; > s.a. lattice gravity.

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