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Jan 29, 2023 •278 likes •1.18k views

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)

  1. 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.1/29

  2. Outline 1. Motivation 2. Translation to tiles 3. Tools 4. Methods 5. Results caagt Toroidal azulenoids – p.2/29

  3. Azulenoids Azulene caagt Toroidal azulenoids – p.3/29

  4. Azulenoids An azulenoid is a carbon network (cubic) for which there exists a partition of the vertices into azulenes. caagt Toroidal azulenoids – p.3/29

  5. 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 caagt Toroidal azulenoids – p.4/29

  6. 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 caagt Toroidal azulenoids – p.4/29

  7. Torus caagt Toroidal azulenoids – p.5/29

  8. Torus caagt Toroidal azulenoids – p.5/29

  9. 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.6/29

  10. 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.6/29

  11. Example tiling caagt Toroidal azulenoids – p.7/29

  12. Example tiling caagt Toroidal azulenoids – p.7/29

  13. 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.8/29

  14. 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.9/29

  15. 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.9/29

  16. 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.10/29

  17. Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/29

  18. Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/29

  19. Example Delaney/Dress graph caagt Toroidal azulenoids – p.11/29

  20. Example Delaney/Dress graph caagt Toroidal azulenoids – p.12/29

  21. Example Delaney/Dress graph caagt Toroidal azulenoids – p.12/29

  22. Example Delaney/Dress graph ⇒ Delaney/Dress graph is not sufficient to distinguish be- tween tilings! caagt Toroidal azulenoids – p.12/29

  23. 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.13/29

  24. 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.13/29

  25. 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.14/29

  26. 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.14/29

  27. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.15/29

  28. Example Delaney/Dress symbol m 01 = 4 m 12 = 4 caagt Toroidal azulenoids – p.15/29

  29. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.16/29

  30. Example Delaney/Dress symbol m 01 = 6 m 12 = 3 caagt Toroidal azulenoids – p.16/29

  31. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.17/29

  32. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.17/29

  33. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.17/29

  34. Example Delaney/Dress symbol caagt Toroidal azulenoids – p.17/29

  35. Example Delaney/Dress symbol m 01 m 12 A 4 B C caagt Toroidal azulenoids – p.17/29

  36. Example Delaney/Dress symbol m 01 m 12 A 4 B C caagt Toroidal azulenoids – p.17/29

  37. Example Delaney/Dress symbol m 01 m 12 A 4 B 8 C 8 caagt Toroidal azulenoids – p.17/29

  38. Example Delaney/Dress symbol m 01 m 12 A 4 B 8 C 8 caagt Toroidal azulenoids – p.17/29

  39. Example Delaney/Dress symbol m 01 m 12 A 4 3 B 8 3 C 8 3 caagt Toroidal azulenoids – p.17/29

  40. 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.18/29

  41. 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.19/29

  42. 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.19/29

  43. 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.19/29

  44. 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.19/29

  45. 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) . . . . . . . . . . . . . . . . . . N σ 0 ( N ) σ 1 ( N ) σ 2 ( N ) m 01 ( N ) m 12 ( N ) caagt Toroidal azulenoids – p.20/29

  46. Minimal symbol caagt Toroidal azulenoids – p.21/29

  47. Minimal symbol caagt Toroidal azulenoids – p.21/29

  48. Minimal symbol add symmetry caagt Toroidal azulenoids – p.21/29

  49. Minimal symbol add symmetry ⇒ map orbits onto each other caagt Toroidal azulenoids – p.21/29

  50. Minimal symbol add symmetry ⇒ map orbits onto each other Choose two orbits c and d caagt Toroidal azulenoids – p.21/29

  51. 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.21/29

  52. 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.21/29

  53. 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.21/29

  54. Example minimal symbol caagt Toroidal azulenoids – p.22/29

  55. Example minimal symbol σ 0 σ 1 σ 2 m 01 m 12 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 caagt Toroidal azulenoids – p.22/29

  56. Example minimal symbol σ 0 σ 1 σ 2 m 01 m 12 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 caagt Toroidal azulenoids – p.22/29

  57. Example minimal symbol σ 0 σ 1 σ 2 m 01 m 12 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 caagt Toroidal azulenoids – p.22/29

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