17 Nov 2009 ... 2D Hyperbolic tilings. Triply Periodic Minimal Surfaces. We use TPMS as
scaffolds for 3-periodic nets. Example: the primitive cubic net is ...
2D Hyperbolic tilings
From symmetric tilings of 2D hyperbolic space to 3D 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 c
2D Hyperbolic tilings
Nets model structure
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
Triply Periodic Minimal Surfaces. We use TPMS as scaffolds for 3-periodic nets. Example: the primitive cubic net is carried by
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A 3D tiling by cubes.
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The edges of an infinite polyhedron.
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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 c
2D Hyperbolic tilings
Hyperbolic geometry The intrinsic geometry of a TPMS is hyperbolic.
The asymmetric unit of the P surface is a hyperbolic triangle with angles π2 , π4 , π6 . The primitive translational unit cell is a dodecagon. With opposite sides identified, the dodecagon glues up into a genus-3 surface.
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
... play Stu’s Escher animation
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
Symmetries of the hyperbolic plane
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings index1 *642
The ∗246/ooo subgroup lattice index3
index2
*2422
462
*662
*434
6*2
4*3
2*32
*2232
266
434
*3232
3*22
23x
2322
2*33
index6
index4
24*
4224
*4444
44x
222x
2442
44*
44*
2**
*22x
*2*2
2xx
2xx
22*22
222x
*2442
2*44
4*22
**2
22*2
**2
o2
**22
2*x
*2*2
2*x
22*22
2224
2*222
*4422
24*
22*2
*22222
*22x
22222
22222
4224
22*22
22222
2*222
*22222
*222222
222*
**22
*222222
22*22
2*2222
2**
222x
222*
2**
2*2222
*22*
22*22
2323
6222
2*62
*3x
*6262
62x
22*3
*3x
index12
22*3 index8
4444
22xx
o22
22xx
22xx
o22
*22*22
*xx
222222
*22*22
222222
o22
22**
22**
22xx
222222
xxx
2222*
22*x
**x
o*
o22
**x
***
*2222x
222222
*xx
22*2222
22*x
22*x
*xx
2222x
6226
o3
*3*3
3xx
index24
32222
*3*3 index16
o2222
xxxx
oo
o**
oo
**xx
o2222
index48
see http://people.physics.anu.edu.au/˜ vbr110/PDGdata/ Stephen Hyde, Stuart Ramsden, Vanessa Robins
3xx
22222222
**xx
xxxx
xxxx
o33 index32
ooo index96
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
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 c
2D Hyperbolic tilings
Orbifold symbols An orbifold is the quotient of a manifold by a discrete group acting on it. Orbifolds derived from 2D manifolds of constant curvature (the sphere, Euclidean plane, and hyperbolic plane) have a canonical symbol encoding their topology and orders of rotational points. Conway’s notation for this is: ooo . . . c1 c2 c3 . . . ∗ m1 m2 . . . [∗m4 m5 . . .] . . . × ××
o
34 ∗ 22
Stephen Hyde, Stuart Ramsden, Vanessa Robins
×
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
Divide orbifolds into 8 classes:
Coxeter
* Hat simply connected -
Stellate
x
Projective
**
Annulus
*x
Möbius multiply connected
o
Torus
xx
Klein
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
MIRRORS ONLY (simple, bounded, oriented)
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
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 c
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 c
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 c
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 c
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 c
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 c
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 c
2D Hyperbolic tilings
Delaney-Dress tiling theory An algorithmic approach to encoding the symmetries and topology of periodic tilings, using triangulations of orbifolds. Developed by Dress, 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 c
2D Hyperbolic tilings
Delaney-Dress symbols
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
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 c
2D Hyperbolic tilings
Embedding D-symbols To get a tiling of H2 that is compatible with the surface covering map we must match the combinatorics of the D-symbol to the geometry of a specific group of isometries. There may be more than 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 c
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 c
2D Hyperbolic tilings
EPINET — http://epinet.anu.edu.au Results from our enumeration of tilings and nets derived from Coxeter orbifolds are available online.
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
EPINET — http://epinet.anu.edu.au 2706 Hyperbolic tilings with 2451 net topologies 6095 Surface-compatible tilings generate 14532 3-periodic nets.
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
2D Hyperbolic tilings
Structure relationships in Epinet
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
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 c
2D Hyperbolic tilings
Free tilings We extend Delaney-Dress tiling theory to a quadrangulation decomposition of the infinite-sided polygons. These new quadrangulations can be enumerated via regular D-symbols.
Stephen Hyde, Stuart Ramsden, Vanessa Robins
From symmetric tilings of 2D hyperbolic space to 3D euclidean c
