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].

**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].

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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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