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) caagt Toroidal azulenoids – p.1/30
Outline 1. Motivation 2. Translation to tiles 3. Tools 4. Methods 5. Results caagt Toroidal azulenoids – p.2/30
Azulenoids Azulene caagt Toroidal azulenoids – p.3/30
Azulenoids 4 n + 2 annulene with a bridging bond if a π -electron migrate towards the five membered ring then in principle two ’aromatic-sextets’ could be formed ⇒ aromatic behaviour might be expected within Hückel theory caagt Toroidal azulenoids – p.3/30
Azulenoids Consistent with this view is that it has a small dipole moment, and does indeed show some aromatic properties, under milder conditions. caagt Toroidal azulenoids – p.3/30
Question We don’t yet know whether and how the electron mobility might manifest itself among azulenes embedded within a fullerene-style network. How many variations of such networks are theoretically possible? Edward Kirby caagt Toroidal azulenoids – p.4/30
Torus caagt Toroidal azulenoids – p.5/30
Torus caagt Toroidal azulenoids – p.5/30
Torus caagt Toroidal azulenoids – p.5/30
Tiling Tiling T = set of tiles t 1 , t 2 , . . . with t i ⊂ E 2 , t i homeomorph to B (0 , 1) that satisfy the following conditions: � t = E 2 1. t ∈ T 2. ∀ t i , t j ( i � = j ) ∈ T : t ◦ i ∩ t ◦ j = ∅ ∧ t i ∩ t j ∈ {∅ , { points } , { lines }} . 3. ∀ x ∈ E 2 : x has a neighbourhood that only intersects a finite number of tiles. caagt Toroidal azulenoids – p.6/30
Tiling Tiling T = set of tiles t 1 , t 2 , . . . with t i ⊂ E 2 , t i homeomorph to B (0 , 1) that satisfy the following conditions: � t = E 2 1. t ∈ T 2. ∀ t i , t j ( i � = j ) ∈ T : t ◦ i ∩ t ◦ j = ∅ ∧ t i ∩ t j ∈ {∅ , { points } , { lines }} . 3. ∀ x ∈ E 2 : x has a neighbourhood that only intersects a finite number of tiles. Periodic tiling ⇐ ⇒ symmetry group contains two independent translations caagt Toroidal azulenoids – p.6/30
Example tiling caagt Toroidal azulenoids – p.7/30
Example tiling caagt Toroidal azulenoids – p.7/30
Barycentric subdivision For each face: one point For each edge: one point For each vertex: one point ⇒ subdivision consists of triangles caagt Toroidal azulenoids – p.8/30
Chamber system Define Σ = � σ 0 , σ 1 , σ 2 | σ 2 i = 1 � σ 0 : change the green point (vertex). σ 1 : change the red point (edge). σ 2 : change the black point (face). caagt Toroidal azulenoids – p.9/30
Chamber system Define Σ = � σ 0 , σ 1 , σ 2 | σ 2 i = 1 � σ 0 : change the green point (vertex). σ 1 : change the red point (edge). σ 2 : change the black point (face). Chamber system C of T = barycentric subdivision together with Σ caagt Toroidal azulenoids – p.9/30
Delaney/Dress graph The Delaney/Dress graph D of a periodic tiling is the set of orbits of the chambers of the chamber system of the tiling under the symmetry group. caagt Toroidal azulenoids – p.10/30
Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/30
Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/30
Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/30
Example Delaney/Dress graph caagt Toroidal azulenoids – p.12/30
Example Delaney/Dress graph caagt Toroidal azulenoids – p.12/30
Example Delaney/Dress graph ⇒ Delaney/Dress graph is not sufficient to distinguish be- tween tilings! caagt Toroidal azulenoids – p.12/30
Delaney/Dress symbol Define functions r ij : C → N ; c �→ r ij ( c ) with r ij ( c ) the smallest number for which c ( σ i σ j ) r ij ( c ) = c . caagt Toroidal azulenoids – p.13/30
Delaney/Dress symbol Define functions r ij : C → N ; c �→ r ij ( c ) with r ij ( c ) the smallest number for which c ( σ i σ j ) r ij ( c ) = c . r 02 is a constant function with value 2. r 01 ( c ) is the size of the face of T that belongs to c . r 12 ( c ) is the number of faces that meet in the vertex that belongs to c . caagt Toroidal azulenoids – p.13/30
Delaney/Dress symbol Define functions m ij : D → N ; d �→ m ij ( c ) in such a manner that the following diagram is commutative: r ij ✲ N C ✲ m ij ✲ D caagt Toroidal azulenoids – p.14/30
Delaney/Dress symbol Define functions m ij : D → N ; d �→ m ij ( c ) in such a manner that the following diagram is commutative: r ij ✲ N C ✲ m ij ✲ D Delaney/Dress symbol of the tiling is ( D ; m 01 , m 12 ) caagt Toroidal azulenoids – p.14/30
Example Delaney/Dress symbol caagt Toroidal azulenoids – p.15/30
Example Delaney/Dress symbol m 01 = 4 m 12 = 4 caagt Toroidal azulenoids – p.15/30
Example Delaney/Dress symbol caagt Toroidal azulenoids – p.16/30
Example Delaney/Dress symbol m 01 = 6 m 12 = 3 caagt Toroidal azulenoids – p.16/30
Example Delaney/Dress symbol caagt Toroidal azulenoids – p.17/30
Example Delaney/Dress symbol caagt Toroidal azulenoids – p.17/30
Example Delaney/Dress symbol caagt Toroidal azulenoids – p.17/30
Example Delaney/Dress symbol caagt Toroidal azulenoids – p.17/30
Example Delaney/Dress symbol m 01 m 12 A 4 B C caagt Toroidal azulenoids – p.17/30
Example Delaney/Dress symbol m 01 m 12 A 4 B C caagt Toroidal azulenoids – p.17/30
Example Delaney/Dress symbol m 01 m 12 A 4 B 8 C 8 caagt Toroidal azulenoids – p.17/30
Example Delaney/Dress symbol m 01 m 12 A 4 B 8 C 8 caagt Toroidal azulenoids – p.17/30
Example Delaney/Dress symbol m 01 m 12 A 4 3 B 8 3 C 8 3 caagt Toroidal azulenoids – p.17/30
Delaney/Dress symbol ( D ; m 01 , m 12 ) is the Delaney/Dress symbol of a periodic tiling iff. 1. D is finite 2. Σ works transitively on D 3. m 01 is constant on � σ 0 , σ 1 � orbits and ∀ d ∈ D : d ( σ 0 σ 1 ) m 01 ( d ) = d 4. m 12 is constant on � σ 1 , σ 2 � orbits and ∀ d ∈ D : d ( σ 1 σ 2 ) m 12 ( d ) = d 5. ∀ d ∈ D : d ( σ 0 σ 2 ) 2 = d caagt Toroidal azulenoids – p.18/30
Delaney/Dress symbol 1 m 12 ( d ) − 1 1 � K ( D ) = ( m 01 ( d ) + 2) d ∈D 1. K ( D ) < 0 2. K ( D ) = 0 3. K ( D ) > 0 caagt Toroidal azulenoids – p.19/30
Delaney/Dress symbol 1 m 12 ( d ) − 1 1 � K ( D ) = ( m 01 ( d ) + 2) d ∈D 1. K ( D ) < 0 → hyperbolic plane 2. K ( D ) = 0 3. K ( D ) > 0 caagt Toroidal azulenoids – p.19/30
Delaney/Dress symbol 1 m 12 ( d ) − 1 1 � K ( D ) = ( m 01 ( d ) + 2) d ∈D 1. K ( D ) < 0 → hyperbolic plane 2. K ( D ) = 0 → euclidean plane 3. K ( D ) > 0 caagt Toroidal azulenoids – p.19/30
Delaney/Dress symbol 1 m 12 ( d ) − 1 1 � K ( D ) = ( m 01 ( d ) + 2) d ∈D 1. K ( D ) < 0 → hyperbolic plane 2. K ( D ) = 0 → euclidean plane 4 3. K ( D ) > 0 → sphere iff. K ( D ) ∈ N caagt Toroidal azulenoids – p.19/30
Representation σ 0 σ 1 σ 2 m 01 m 12 1 σ 0 (1) σ 1 (1) σ 2 (1) m 01 (1) m 12 (1) 2 σ 0 (2) σ 1 (2) σ 2 (2) m 01 (2) m 12 (2) 3 σ 0 (3) σ 1 (3) σ 2 (3) m 01 (3) m 12 (3) . . . . . . . . . . . . . . . . . . σ 0 ( N ) σ 1 ( N ) σ 2 ( N ) m 01 ( N ) m 12 ( N ) N caagt Toroidal azulenoids – p.20/30
Canonical form caagt Toroidal azulenoids – p.21/30
Canonical form based on index priority depth-first traversal of graph caagt Toroidal azulenoids – p.21/30
Canonical form based on index priority depth-first traversal of graph canonical relabelling when m 01 (1) . . . m 12 (1) . . . σ 0 (1) . . . σ 1 (1) . . . σ 2 (1) . . . is lexicographically smallest. caagt Toroidal azulenoids – p.21/30
Minimal symbol caagt Toroidal azulenoids – p.22/30
Minimal symbol caagt Toroidal azulenoids – p.22/30
Minimal symbol add symmetry caagt Toroidal azulenoids – p.22/30
Minimal symbol add symmetry ⇒ map orbits onto each other caagt Toroidal azulenoids – p.22/30
Minimal symbol add symmetry ⇒ map orbits onto each other Choose two orbits c and d caagt Toroidal azulenoids – p.22/30
Minimal symbol add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is m ij ( c ) = m ij ( d ) ? caagt Toroidal azulenoids – p.22/30
Minimal symbol add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is m ij ( c ) = m ij ( d ) ? index priority depth-first traversal from c and d σ i ( c ) maps onto σ i ( d ) caagt Toroidal azulenoids – p.22/30
Minimal symbol add symmetry ⇒ map orbits onto each other Choose two orbits c and d Is m ij ( c ) = m ij ( d ) ? index priority depth-first traversal from c and d σ i ( c ) maps onto σ i ( d ) m ij ( · ) = m ij ( · ) ? caagt Toroidal azulenoids – p.22/30
Example minimal symbol caagt Toroidal azulenoids – p.23/30
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