[PPT] - caagt Toroidal azulenoids p.1/29 Outline 1. Motivation 2. PowerPoint Presentation

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Toroidal azulenoids

Nico Van Cleemput

Nicolas.VanCleemput@UGent.be

Research group CAAGT, Department of Applied Mathematics and Computer Science, Ghent University (Joint work with Gunnar Brinkmann, Olaf Delgado-Friedrichs and Edward Kirby)

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Outline

1. Motivation
2. Translation to tiles
3. Tools
4. Methods
5. Results

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Azulenoids

Azulene

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Azulenoids

An azulenoid is a carbon network (cubic) for which there exists a partition of the vertices into azulenes.

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Question

How many variations of such networks are theoretically possible if there can only be one orbit of azulenes under the symmetry group? Edward Kirby

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Question

How many variations of such networks are theoretically possible if there can only be one orbit of azulenes under the symmetry group? Edward Kirby Toroidal, but also planar and cylindrical

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Torus

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Torus

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Tiling

Tiling T = set of tiles t1, t2, . . . with ti ⊂ E2, ti homeomorph to B(0, 1) that satisfy the following conditions: 1.

t∈T

t = E2

2. ∀ti, tj(i = j) ∈ T : t◦

i ∩ t◦ j = ∅ ∧ ti ∩ tj ∈

{∅, {points}, {lines}}.

3. ∀x ∈ E2 : x has a neighbourhood that only intersects a

finite number of tiles.

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Tiling

Tiling T = set of tiles t1, t2, . . . with ti ⊂ E2, ti homeomorph to B(0, 1) that satisfy the following conditions: 1.

t∈T

t = E2

2. ∀ti, tj(i = j) ∈ T : t◦

i ∩ t◦ j = ∅ ∧ ti ∩ tj ∈

{∅, {points}, {lines}}.

3. ∀x ∈ E2 : x has a neighbourhood that only intersects a

finite number of tiles. Periodic tiling ⇐ ⇒ symmetry group contains two independent translations

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Example tiling

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Example tiling

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Barycentric subdivision

For each face: one point For each edge: one point For each vertex: one point ⇒ subdivision consists of triangles

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Chamber system

Define Σ = σ0, σ1, σ2|σ2

i = 1

σ0 : change the green point (vertex). σ1 : change the red point (edge). σ2 : change the black point (face).

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Chamber system

Define Σ = σ0, σ1, σ2|σ2

i = 1

σ0 : change the green point (vertex). σ1 : change the red point (edge). σ2 : change the black point (face). Chamber system C of T = barycentric subdivision together with Σ

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Delaney/Dress graph

The Delaney/Dress graph D of a periodic tiling is the set of

rbits of the chambers of the chamber system of the tiling

under the symmetry group.

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Example Delaney/Dress graph

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Example Delaney/Dress graph

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Example Delaney/Dress graph

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Example Delaney/Dress graph

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Example Delaney/Dress graph

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Example Delaney/Dress graph

⇒ Delaney/Dress graph is not sufficient to distinguish be- tween tilings!

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Delaney/Dress symbol

Define functions rij : C → N; c → rij(c) with rij(c) the smallest number for which c(σiσj)rij(c) = c.

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Delaney/Dress symbol

Define functions rij : C → N; c → rij(c) with rij(c) the smallest number for which c(σiσj)rij(c) = c. r02 is a constant function with value 2. r01(c) is the size of the face of T that belongs to c. r12(c) is the number of faces that meet in the vertex that belongs to c.

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Delaney/Dress symbol

Define functions mij : D → N; d → mij(c) in such a manner that the following diagram is commutative: C rij

✲ N

D mij

✲ ✲

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Delaney/Dress symbol

Define functions mij : D → N; d → mij(c) in such a manner that the following diagram is commutative: C rij

✲ N

D mij

✲ ✲

Delaney/Dress symbol of the tiling is (D; m01, m12)

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

m01 = 4 m12 = 4

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

m01 = 6 m12 = 3

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

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Example Delaney/Dress symbol

m01 m12 A 4 B C

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Example Delaney/Dress symbol

m01 m12 A 4 B C

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Example Delaney/Dress symbol

m01 m12 A 4 B 8 C 8

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Example Delaney/Dress symbol

m01 m12 A 4 B 8 C 8

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Example Delaney/Dress symbol

m01 m12 A 4 3 B 8 3 C 8 3

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Delaney/Dress symbol

(D; m01, m12) is the Delaney/Dress symbol of a periodic tiling iff

1. D is finite
2. Σ works transitively on D
3. m01 is constant on σ0, σ1 orbits and

∀d ∈ D : d(σ0σ1)m01(d) = d

4. m12 is constant on σ1, σ2 orbits and

∀d ∈ D : d(σ1σ2)m12(d) = d

5. ∀d ∈ D : d(σ0σ2)2 = d

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Delaney/Dress symbol

K(D) =

d∈D

( 1 m01(d) + 1 m12(d) − 1 2)

1. K(D) < 0
2. K(D) = 0
3. K(D) > 0

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Delaney/Dress symbol

K(D) =

d∈D

( 1 m01(d) + 1 m12(d) − 1 2)

1. K(D) < 0 → hyperbolic plane
2. K(D) = 0
3. K(D) > 0

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Delaney/Dress symbol

K(D) =

d∈D

( 1 m01(d) + 1 m12(d) − 1 2)

1. K(D) < 0 → hyperbolic plane
2. K(D) = 0 → euclidean plane
3. K(D) > 0

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Delaney/Dress symbol

K(D) =

d∈D

( 1 m01(d) + 1 m12(d) − 1 2)

1. K(D) < 0 → hyperbolic plane
2. K(D) = 0 → euclidean plane
3. K(D) > 0 → sphere iff

4 K(D) ∈ N

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Representation

σ0 σ1 σ2 m01 m12

1 σ0(1) σ1(1) σ2(1) m01(1) m12(1)

2 σ0(2) σ1(2) σ2(2) m01(2) m12(2)

3 σ0(3) σ1(3) σ2(3) m01(3) m12(3)

. . . . . . . . . . . . . . . . . .

N σ0(N) σ1(N) σ2(N) m01(N) m12(N)

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Minimal symbol

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Minimal symbol

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Minimal symbol

add symmetry

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Minimal symbol

add symmetry ⇒ map orbits onto each other

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Minimal symbol

add symmetry ⇒ map orbits onto each other Choose two orbits c and d

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Minimal symbol

add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is mij(c) = mij(d)?

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Minimal symbol

add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is mij(c) = mij(d)? index priority depth-first traversal from c and d

σi(c) maps onto σi(d)

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Minimal symbol

add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is mij(c) = mij(d)? index priority depth-first traversal from c and d

σi(c) maps onto σi(d) mij(·) = mij(·)?

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Example minimal symbol

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 B C C D B 4 3 C D D C E 4 3 D E E F D 8 3 E F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 B C C D B 4 3 a D D C E 4 3 a E E F D 8 3 E F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 B C C D B 4 3 a D D C E 4 3 a E E F D 8 3 E F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 A B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 F

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 c B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 c

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 c B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 c

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Example minimal symbol

σ0 σ1 σ2 m01 m12 A A B F 8 3 c B B A C 8 3 b C C D B 4 3 a D D C E 4 3 a E E F D 8 3 b F F E A 8 3 c

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Example minimal symbol

σ0 σ1 σ2 m01 m12 a a a b 4 3 b b c a 8 3 c c a b 8 3

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Translation

Restrictions azulenoid: 1 orbit of azulenes every atom part of exactly one azulene in that orbit

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Translation

Restrictions azulenoid: 1 orbit of azulenes every atom part of exactly one azulene in that orbit Restrictions Delaney/Dress symbol: ∃σ0σ1 orbit O :        r01(O) = 8 ∀σ1σ2 orbit V : O ∩ V = ∅ ∀d ∈ O : σ2(d) / ∈ O

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Translation

Restrictions azulenoid: 1 orbit of azulenes every atom part of exactly one azulene in that orbit Restrictions Delaney/Dress symbol: ∃σ0σ1 orbit O :        r01(O) = 8 ∀σ1σ2 orbit V : O ∩ V = ∅ ∀d ∈ O : σ2(d) / ∈ O ∀σ1σ2 orbit V : r12(V ) = 3

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Translation

Restrictions azulenoid: 1 orbit of azulenes every atom part of exactly one azulene in that orbit Restrictions Delaney/Dress symbol: ∃σ0σ1 orbit O :        r01(O) = 8 ∀σ1σ2 orbit V : O ∩ V = ∅ ∀d ∈ O : σ2(d) / ∈ O ∀σ1σ2 orbit V : r12(V ) = 3

d∈D

( 1 m01(d) + 1 m12(d) − 1 2) = 0

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Method

Octagon and the different vertex orbits

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Method

Octagon and the different vertex orbits

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s Replace octagon with azulene

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s Replace octagon with azulene

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s Replace octagon with azulene

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Method

Octagon and the different vertex orbits Calculate and assign remaining m01 values Assign remaining σ0’s Replace octagon with azulene

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Visualisation

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Visualisation

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Visualisation

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Visualisation

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Results

m01 values # strings # symbols 1 4 4 4 4 4 6 24 24 21 6 2 4 4 4 4 4 8 12 24 42 42 3 4 4 4 4 4 8 16 16 21 48 4 4 4 4 4 4 10 10 20 21 5 4 4 4 4 4 12 12 12 7 44 6 4 4 4 4 6 6 8 24 105 7 4 4 4 4 6 6 12 12 54 2 8 4 4 4 4 6 8 8 12 105 12 9 4 4 4 4 8 8 8 8 10 160 10 4 4 4 6 6 6 6 12 35 6 11 4 4 4 6 6 6 8 8 70 38 12 4 4 6 6 6 6 6 6 4 25

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Results

383 symbols of tilings containing octagons

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Results

383 symbols of tilings containing octagons ⇓ 1274 minimal unmarked azulenoids 1324 minimal marked azulenoids

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Marked vs. Unmarked

structures that have an azulenoid set vs. structures with their azulenoid set

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Marked vs. Unmarked

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Marked vs. Unmarked

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Translation only

ne orbit of azulenes under the subgroup of translations
r

all the azulenes have the same orientation

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Translation only

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