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}
*N*_{i} = *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. quasicrystals; random walk.

* __Penrose tiling__: A
quasiperiodic tiling of E^{2}, 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 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];
Flicker et al PRX(20) [properties of tilings with colored edges].

**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 DCG(15)-a1409 [weak local rules];
Bédaride et al IMRN-a2012 [12-fold symmetry, cohomology].

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**Set T of Tilings of M and Operations on Tilings**

*

*T*_{K}:= {*θ* ∈
*T* | edges(*θ*) ∩ *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-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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