Spherical Tilings by Congruent Quadrangles Yohji Akama 1 Nico Van - PowerPoint PPT Presentation

Spherical Tilings by Congruent Quadrangles Yohji Akama 1 Nico Van Cleemput 2 1 Mathematical Institute Graduate School of Science Tohoku University 2 Combinatorial Algorithms and Algorithmic Graph Theory Department of Applied Mathematics, Computer Science and Statistics Ghent University Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Spherical tilings edges are parts of great circles edge-to-edge tiling vertex degree ≥ 3 Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Spherical tilings by congruent polygons all faces the same size ⇓ only triangles, quadrangles or pentagons Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Chemical applications 1 1 Leonard R. MacGillivray, “Design Rules: A Net and Archimedean Polyhedra Score Big for Self-Assembly”, in: Angewandte Chemie International Edition 51.5 (2012), pp. 1110–1112, ISSN : 1521-3773, DOI : 10.1002/anie.201107282 , URL : http://dx.doi.org/10.1002/anie.201107282 . Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Spherical tilings by congruent triangles Classification of spherical tilings by congruent triangles completed by Davies 2 and Ueno-Agaoka 3 . 2 H.L. Davies, “Packings of spherical triangles and tetrahedra”, in: Proc. Colloquium on Convexity (Copenhagen, 1965) , Kobenhavns Univ. Mat. Inst., 1967, pp. 42–51. 3 Y. Ueno and Y. Agaoka, “Classification of tilings of the 2-dimensional sphere by congruent triangles”, in: Hiroshima Math. J. 32.3 (2002), pp. 463–540. Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

The next step: classification of spherical tilings by congruent quadrangles Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles aaaa abab aabb aaab aabc abac abcd Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles a aaaa a a abab a aabb aaab aabc abac abcd Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles b aaaa a a abab aabb b aaab aabc abac abcd Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles aaaa abab a a aabb aaab b b aabc abac abcd Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles aaaa abab aabb b a a aaab aabc a abac abcd Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles aaaa abab aabb b aaab c a aabc a abac abcd Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles aaaa abab aabb aaab aabc b a a abac abcd c Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles aaaa abab aabb aaab aabc b abac c abcd a d Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

In every quadrangulation of the sphere, there exists a vertex of degree 3. Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

abab , abac b a a Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

abab , abac a a b a a Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

abab , abac a a ✌ b a a Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

abcd b a c d Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

abcd a or c b a c d Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

abcd a or c ✌ b a c d Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles aaaa abab aaab aabb aabc abac abcd Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

aaaa aabb a a a a a a a b b a b b Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Classification of spherical tilings by congruent rhombi, kites and daggers completed by Akama-Sakano 4 . 4 Y. Akama and Y. Sakano, “Classification of spherical tilings by congruent rhombi (kites, darts)”, In preparation. Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Types of quadrangles aaaa abab aaab aabb aabc abac abcd Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Type 2 quadrangles b α δ a a γ β a Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Even number of tiles Assignment of side lengths corresponds to perfect matching in dual a a a b a a a Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Concave type 2 quadrangles Ambiguity of inner angles 5 Edge which is not a geodesic 5 5 Yohji Akama and K. Nakamura, “Spherical tilings by congruent quadrangles over pseudo-double wheels ( II ) the ambiguity of the inner angles”, Preprint, 2012. Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Convex type 2 quadrangles 0 < α , β , γ , δ < π Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Some restrictions on the angles α + δ < π + β α + δ < π + γ α = δ ⇔ β = γ ( 1 − cos β ) cos 2 α − ( 1 − cos β )( 1 − cos γ ) cos α cos δ +( 1 − cos γ ) cos 2 δ + cos β cos γ + sin α sin β sin γ sin δ = 1 Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Area of a tile S = α + β + γ + δ − 2 π S = 4 π F Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Area of a tile S = α + β + γ + δ − 2 π S = 4 π F α + β + γ + δ − 2 π = 4 π F Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Generation of spherical tilings by congruent convex quadrangles of type 2 Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Generate quadrangulations of the sphere Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Generate perfect matchings for the dual of the quadrangulation Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Generate perfect matchings for the dual of the quadrangulation Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Generate perfect matchings for the dual of the quadrangulation Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Generate perfect matchings for the dual of the quadrangulation Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Filter out quadrangulations for which the dual has no perfect matching? Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Every edge in the dual of a quadrangulation belongs to a perfect matching of the dual. 6 6 C. D. Carbonera and Jason F. Shepherd, On the existence of a perfect matching for 4-regular graphs derived from quadrilateral meshes. Tech. rep., UUSCI-2006-021, SCI Institute Technical Report, 2006. Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Number of perfect matchings in the dual of quadrangulations 600 12 vertices 14 vertices 16 vertices 18 vertices 500 20 vertices 400 Number of graphs 300 200 100 0 0 50 100 150 200 250 Number of perfect matchings Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

Generate angle assignments: 2 F − 1 possibilities α β γ δ Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

α α γ β α δ α γ δ β γ β β γ γ δ α β δ γ β β δ δ γ δ α γ α α δ β β β γ α α δ δ γ Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

α α γ β α δ α γ δ β γ β β γ γ δ α β δ γ β β δ δ γ δ α γ α α δ β β β γ α α δ δ γ Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

α + β + δ = 2 α α γ β α δ α γ δ β γ β β γ γ δ α β δ γ β β δ δ γ δ α γ α α δ β β β γ α α δ δ γ Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

α + β + δ = 2 α α + β + δ = 2 α γ β α δ α γ δ β γ β β γ γ δ α β δ γ β β δ δ γ δ α γ α α δ β β β γ α α δ δ γ Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

α + β + δ = 2 α α + β + δ = 2 α γ α + γ + δ = 2 β α δ α γ δ β γ β β γ γ δ α β δ γ β β δ δ γ δ α γ α α δ β β β γ α α δ δ γ Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

α + β + δ = 2 α α + β + δ = 2 α γ α + γ + δ = 2 α + 2 β + δ = 2 β α δ α γ δ β γ β β γ γ δ α β δ γ β β δ δ γ δ α γ α α δ β β β γ α α δ δ γ Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles

α + β + δ = 2 α α + β + δ = 2 α γ α + γ + δ = 2 α + 2 β + δ = 2 β α δ . . . α γ δ β γ γ + 2 δ = 2 β β γ γ δ α β δ γ β β δ δ γ δ α γ α α δ β β β γ α α δ δ γ Akama, Van Cleemput Spherical Tilings by Congruent Quadrangles