2D Hyperbolic tilings

From symmetric tilings of 2D hyperbolic space to3D euclidean crystalline patterns: EPINET

Stephen Hyde, Stuart Ramsden, Vanessa Robins

Applied Mathematics, RSPE, ANU

November 17, 2009

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Nets model structure

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Triply Periodic Minimal Surfaces.

We use TPMS as scaffolds for 3-periodic nets.

Example: the primitive cubic net is carried by

I A 3D tiling by cubes.

I The edges of an infinite polyhedron.

I A 2D tiling of Schwarz’s Primitive (P) surface.

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Hyperbolic geometry

The intrinsic geometry of a TPMS is hyperbolic.

The asymmetric unit of the P surface is a hyperbolic triangle withangles π

2 , π4 , π

6 . The primitive translational unit cell is a dodecagon.With opposite sides identified, the dodecagon glues up into agenus-3 surface.

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

... play Stu’s Escher animation

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Symmetries of the hyperbolic plane

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

The ∗246/ooo subgroup lattice

index24

index6

index12

index48

index16

index32

index8

index1

index2 index3

index4

index96

***

**xxoo **xx

**x

ooo**

o*

xxxx

xxx

xxxxxxxx

22xx

o2222

**x *2222x

o2222

22*x 22*2222

22222222

22*x

462

266

6222

2224 434 2322

*642

2*324*3*434 *2232*662 6*2*2422

2*62

2xx

22xx 22xx o22

44x

4444

4224

o22 222222

4224

222222

22*22

*22*2222xx

*2*2 *222222

22**

*22x

o22

22*22

2222x*xx22*x*xx222222

2**

o22

23x

22*3

22*2 *3232

*3x

2*3324*

*3x

3*22*4422

22*3

*22222

*6262

2*222

62x

2*222

2323

3xx

222x22222

6226

22*2**24*22 **2*2442 *2222224* 2*44

ooo

o33

222222 *3*3o3 32222*3*3

2*2222

2222*

*22*

3xx

22*22

*xx*22*22

22*22 222**4444 2*x44* o2 **22 *22x*2*244*

22**

22*2222222 22222 **22 222*2*x2**2442 222x222x 2**2xx 2*2222*222222

see http://people.physics.anu.edu.au/˜ vbr110/PDGdata/

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

131 subgroups of ∗246 that preserve the dodecagon (ooo)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Orbifold symbols

An orbifold is the quotient of a manifold by a discrete group actingon it. Orbifolds derived from 2D manifolds of constant curvature(the sphere, Euclidean plane, and hyperbolic plane) have acanonical symbol encoding their topology and orders of rotationalpoints. Conway’s notation for this is:

ooo . . . c1c2c3 . . . ∗ m1m2 . . . [∗m4m5 . . .] . . .×××

o 34 ∗ 22 ×

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Divide orbifolds into 8 classes:

*

Coxeter

simply connected

*

Hatsimply connected

- Stellate

simply connected

x Projective

simply connected

** Annulus

multiply connected

*x Möbiusmultiply connected

o Torus

multiply connected

xx Klein

multiply connected

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

MIRRORS ONLY (simple, bounded, oriented)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

MIRRORS, ROTATION CENTRES(simple, bounded, oriented)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

ROTATION CENTRES ONLY (simple, unbounded, oriented)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

SINGLE CROSS-CAP(simple, unbounded, non-oriented)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

DISJOINT MIRRORS(multiply-connected, bounded, oriented)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

MIRRORS, CROSS-CAPS(multiply-connected, bounded,

non-oriented)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

TRANSLATIONS ONLY(multiply-connected, unbounded,

oriented)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

MULTIPLE CROSS-CAPS(multiply-connected, unbounded,

non-oriented)

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Delaney-Dress tiling theoryAn algorithmic approach to encoding the symmetries and topologyof periodic tilings, using triangulations of orbifolds. Developed byDress, Huson, Delgado-Friedrichs, mid 1980’s to mid 1990’s.

See O.D.-F. (2003) Theoret. Comp. Sci. 303:431-445

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Delaney-Dress symbols

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Enumeration of D-symbols via splits and glues

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Embedding D-symbolsTo get a tiling of H2 that is compatible with the surface coveringmap we must match the combinatorics of the D-symbol to thegeometry of a specific group of isometries. There may be morethan one way to do this. e.g. Two distinct ∗25 subgroups of ∗246:

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Embedding D-symbols

Automorphisms of the D-symbol that are not ∗246 isometries. e.g.sides of different length in ∗2224:

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

EPINET — http://epinet.anu.edu.au

Results from our enumeration of tilings and nets derived fromCoxeter orbifolds are available online.

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

EPINET — http://epinet.anu.edu.au

2706 Hyperbolic tilings with 2451 net topologies6095 Surface-compatible tilings generate 14532 3-periodic nets.

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Structure relationships in Epinet

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Free tilings

Surface reticulations derived from packings of trees or lines in H2

give multi-component interwoven nets and (helical) rod packings.

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET

2D Hyperbolic tilings

Free tilings

We extend Delaney-Dress tiling theory to a quadrangulationdecomposition of the infinite-sided polygons. These newquadrangulations can be enumerated via regular D-symbols.

Stephen Hyde, Stuart Ramsden, Vanessa Robins From symmetric tilings of 2D hyperbolic space to 3D euclidean crystalline patterns: EPINET