Conway symbols symbol modules O handle X cross-cap * remove a - - PowerPoint PPT Presentation

modules

symbol

handle

O

cross-cap

X

boundary

* remove a

Conway symbols

torus Klein bottle Moebius strip cylinder here are some manifolds / orbifolds:

torus Klein bottle Moebius strip cylinder

what are these manifolds?

to here

torus Klein bottle Moebius strip cylinder

which ones are nice (oriented), nasty?

torus Klein bottle Moebius strip cylinder

how many caps? xcaps? handles? boundaries?

TWO boundaries ONE boundary ONE crosscap ONE handle TWO crosscaps

* * * x

• xx

torus Klein bottle Moebius strip cylinder

what are their Conway symbols?

TWO boundaries ONE boundary ONE crosscap ONE handle TWO crosscaps

* * * x

• xx

torus Klein bottle Moebius strip cylinder

what is their geometry?

euclidean, K=0

* * * x

• xx

torus Klein bottle Moebius strip cylinder

what is their geometry?

* * * x

• xx

torus Klein bottle Moebius strip cylinder

Let’s build the universal cover of these manifolds…

The universal covers tile the euclidean plane, E^2

covers and universal cover of manifolds

“Classical Topology and Combinatorial Group Theory” John Stillwell

build the “universal cover” of o:

cover universal cover - E2 tiled by (4,4)

4 4 4 4

build the “universal cover” of oo:

universal cover - H2 tiled by (8,8)! 8

Conway symbols describe “orbifolds”

Orbifolds describe symmetric patterns

• o
• **
• xx
• *x

All these manifolds print to euclidean plane!

Zipping and unzipping manifolds is a good way to explore topology (and, symmetry…)

* * * x

• xx

torus Klein bottle Moebius strip cylinder

The torus prints a 2-periodic pattern in the euclidean plane, with symmetry p1!

Klein bottle prints a pair of glide lines (xx = pg)

boundaries/punctures (*) print mirror lines (** = pm)

Annulus prints a pair of mirrors..

generic 2D symmetries include:

• 1. reflections (e.g. p3m1)
• 2. glides (e.g. cm)

3.translations (e.g. p1)

• 4. rotations (e.g. p3)

a cone-point prints a rotation centre (e.g. 5 point)

Conway, Goodman-Strauss and Burgiel

(e.g. 3 point) a cone-point prints a rotation centre

2222

Conway, Goodman-Strauss and Burgiel

e.g. 4 order-2 cone points:

print

• rbifold

universal cover (eucl. plane)

• rbifold

universal cover (eucl. plane) glue

Conway, Goodman-Strauss and Burgiel

22*

e.g. 2 order-2 cone points + puncture:

print

• rbifold

universal cover

• rbifold

universal cover glue

3*3

Conway, Goodman-Strauss and Burgiel

intersecting mirrors = corner points

adding cone-points, corner-points to manifolds: ORBIFOLDS

generic 2D symmetries include:

• 1. reflections (e.g. p3m1)
• 2. glides (e.g. cm)

3.translations (e.g. p1)

• 4. rotations (e.g. p3)

punctures crosscaps handles ???

modules

symbol

handle

O

cross-cap

X

mirror boundary

* remove a

cone point

i

corner point

*j

Orbifold = orbit manifold (Thurston)

Orbifolds encode isometries into

• manifold singularities (cones, corners)
• topological features:

1. rotations centres induce cones 2. rotations on intersecting mirrors induce corners 3. mirror lines induce punctures 4. glide reflections (not on mirrors) induce xcaps 6. translations (not induced by other isometries) induce handles

Dissecting an Orbifold Symbol

• o.. 34.. *345 * *..xx

Punctures with corners NOTICE THAT o , *, x are TOPOLOGICAL FEATURES Cross Caps Handles Cone Points

ANY manifold= sphere + handles + xcaps + boundary! what is the equivalent index for an orbifold?

ANY orbifold is a manifold plus singular conformal structures :

sphere + handles + xcaps + boundaries + cones + corners!

what is for an orbifold?

The Orbifold Dictionary includes any combination of characters except : AB, *ab, where A≠B and a≠b

• o.. 34.. *345 * *x

• o.. 34.. *345 * *..xx

Handles Cone Points Cross Caps Mirror Strings

What is the characteristic of this orbifold?

• o 34*345 * * x x x

2-(2+2+2/3+3/4+1+2/6+3/8+4/10+1+1+1+1+1) = -1263/120

characteristic is negative, so hyperbolic…..

any allowed orbifold word with is a wallpaper group!

flat orbifolds: elliptic orbifolds:

any allowed orbifold word with is a point group!

the most interesting orbifolds are HYPERBOLIC

S.T. Hyde, S.J. Ramsden and V. Robins, “Unification and classification of 2D crystalline patterns using orbifolds”, Acta Cryst A70, 319–337 (2014)

Like the euclidean plane... the hyperbolic plane has many possible symmetry types

e.g. rotational symmetry

flat (K=0)

elliptic (K>0) hyperbolic (K<0)

Orbifold dictionary has only 3 dialects:

Elliptic Euclidean Hyperbolic

but, nearly all words are hyperbolic!

Orbifold dictionary has only 3 dialects:

There are many many 2D orbifolds

(far more than wallpaper or point groups!)

Need a classification

Conway Symbols Have Pretty Much All Topologies

Sort via

• simply vs. multiply connected
• nice vs. nasty
• bounded vs. unbounded
• added corners / cones

C H S P T M A

• simply connected
• multiply connected

coxeter

KALEIDOTILE

stellate

hat

Any orbifold whose Euler characteristic is a euclidean isometry

• rbifold feature

Euler characteristic sphere (Identity) (1) 2 handle (translation)

• 2

puncture (mirror) *

• 1

cross-cap (glide) x

• 1

cone (rotation) A corner (mirror + rot’n) *k

euclidean orbifolds = wallpaper groups

• rbifold feature

Euler characteristic sphere (Identity) (1) 2 handle (translation)

• 2

puncture (mirror) *

• 1

cross-cap (glide) x

• 1

cone (rotation) A corner (mirror + rot’n) *k

Traditional description

*235 235 233 3*2 *234 *233 234

Orbifold description

• rbifold gives simple group-subgp information:

*235 235 233 3*2 *234 *233 234

cost 1/60 1/30 1/24 1/12 1/6

group order = 2/ index subgroup (H) in group (G) = (H)/ (G)

Any orbifold whose Euler characteristic is negative will print a pattern in the hyperbolic plane

• rbifold feature

Euler characteristic sphere (Identity) (1) 2 handle (translation)

• 2

puncture (mirror) *

• 1

cross-cap (glide) x

• 1

cone (rotation) A corner (mirror + rot’n) *k

since any combination of these characters is an allowed orbifold (except *ik and AB where i≠j and A≠B) nearly all 2D symmetries are hyperbolic!

61

mirrors *237, *245, *334, *2223 mirrors and rotations 3*4, 4*3, 32*, 3*22 rotations (cone points) 239, 24(12), 33(35), 2229 glides (cross-caps) (+ corners) 79x non-intersecting mirrors *2* cross-caps and mirrors 6*89x translations

• multiple cross-caps

xxxxx

possible “orbifolds” on the hyperbolic plane

Rank orbifolds by area of orbifold - from most to least symmetric

Generic hyperbolic orbifolds do not map into 3D euclidean space to form discrete symmetry groups….

…. but many do! AND the most symmetric examples build 2- and 3-periodic crystalline hyperbolic surfaces!

Only some hyperbolic orbifolds map to 3D flat space...

The smallest orbifold that “fits” 3D euclidean space (?) The smallest hyperbolic orbifold

*246 symmetry

So this is the smallest homogeneous hyperbolic grain size allowed in 3d space!

P surface is a 2d tiling

• f 246 triangles in H^2

P-Surface: *246 orbifold

coxeter orbifold (cf. *333, *235)

Hyperbolic

plane

D P Gyroid

H^2 and P, D, Gyroid all share *246 2d hyperbolic symmetry

These Triply Periodic Minimal Surfaces are likely the “least frustrated” embeddings of 2d hyperbolic space in 3d euclidean space!

Known to be the most symmetric three-periodic surfaces:

Gyroid is everywhere in materials science and biology….

• M. Evans
• see Angew. Chemie Int. Ed., 47, 7996-8000 (2008)

Nature Materials, November 2008, “A twisted tale”

These hyperbolic orbifolds map to P , D, G surfaces in euclidean 3-space to give 3D space groups:

3-periodic “crystallographic hyperbolic orbifolds” via genus-3

Why bother to probe 3d euclidean space via 2d hyperbolic space? Resulting crystallographic structures can be very complex in 3d space, but emerge from very simple 2d hyperbolic patterns

“Penny packing” (3,6) Hyperbolic penny packings (3,7)

The smallest (sub)orbifolds of *237 that “fit” 3D euclidean space

Ideal “unfrustrated” structure in H2 “Frustrated” structure in E3 regular (3,7) irregular (3,7)

2 2’ 2” 2” 2’’’ 2’

Graphene is a euclidean 2d material

can draw as universal cover of the 2222 euclidean orbifold

2 2’ 2” 2” 2’’’ 2’

HYPERBOLIC Graphene

2222 euclidean orbifold

3 2’ 2” 2” 2’’’

2223 HYPERBOLIC orbifold OR 23x HYPERBOLIC orbifold

• M. Cramer Pedersen

REMARK: Each loop is independent, so 3(g-1) loop-lengths

add Dehn twist angle (2πN): ....3(g-1) angles

Uniform curvature manifold defined by 6(g-1) parameters

(Fenchel-Nielsen coordinates)

Many topologically different hyperbolic (7,3) graphenes have been constructed theoretically.

• Structural Chemistry, 28 (1) 113—122
• Stephen Hyde and Martin Cramer Pedersen,

in prep.

Flatten with intercalated graphite regions: SCHWARZITES

• “Unconventional magnetism in all-

carbon nanofoam”, Phys Rev B, 70, 054407 (2004)

S Ramsden