Tilings / Tessellations of Topological Spaces| 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. quasicrystals; 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];
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].
@ 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 E\(^2\)];
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 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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