Visualization of Regular Maps Scientific Visualization – Math Vis Vis – Human Computer Interaction – Visual Analytics

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Visualization Eindhoven

– information visualization – software visualization – perception – geographic visualization Visualization of Regular Maps scientific visualization – math vis vis – human computer interaction – visual analytics – parallel

Jarke J. van Wijk Eindhoven University of Technology

JCB 2016, Bordeaux Data : flow fields – trees – graphs – tables – mobile data – events – … Applications : software analysis – business cases – bioinformatics – …

Can you draw a Seifert surface?

Eindhoven 2004 Huh?

Starting MathVis Arjeh Cohen Me

Discrete geometry, Visualization algebra

Seifert surface

Eindhoven 2006

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Can you show 364 triangles in 3D? Aberdeenshire Sure! 2000 BC I’ll give you the Arjeh Cohenpattern, you can Me pick a nice shape. Discrete geometry, Visualization algebra Ok!

Athens 450 BC

Scottish Neolithic carved stone balls

8 12 20 8 12 20? 6 6 4 4

Platonic solids: perfectly symmetric Platonic solids: perfectly symmetric

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How to get more faces, How to get more faces, all perfectly symmetric? all perfectly symmetric?

Use shapes with holes Aim only at topological symmetry

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#faces?

Königsberg 1893

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Adolf Hurwitz 1859-1919

max. #faces: 3 28(genus – 1) 7

Genus Faces Genus Faces 3 56 3 56 7 168 7 168 14 364 14 364 … … … …

Can you show 364 triangles in 3D? New Orleans Sure! 2009

I’ll give you the three years later Arjeh pattern, you can Me Cohen pick a nice shape.

Ok!

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The general puzzle

Construct space models of regular maps Symmetric Tiling of Closed Surfaces: Visualization of Regular Maps • Surface topology, combinatorial group theory, graph theory, algebraic geometry, hyperbolic geometry, physics, chemistry, … Jarke van Wijk TU Eindhoven

ACM SIGGRAPH 2009, August 3-7, New Orleans

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Triangle group Regular maps

p q r T(p, q, r) = < a, b, c | a2 = b2 = c2 = ( ab ) =( bc ) = ( ca ) = I > Quotient group of triangle group T(p, q , 2): p q 2 a, b, c: reflections edges G(p, q) = < R, S | R = S = ( RS ) = … = I > ab , bc , ca : rotations vertices R: ab , rotation center polygon

R3 S : bc , rotation vertex 4 R RS : ca , rotation edge 2 2 2 M M R R S aba R−1 aba R2S b ab b ab=R I a I a bc bc=S N O N O 2 c ca c RS 2 ca= S RS

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Regular maps Symmetries of the cube

Regular map : Embedding of a graph in a closed surface, such that topologically • faces are identical • vertices are identical • edges are identical

Tiling of closed surface with maximal symmetry • 48–fold symmetry (2 pN = 2x4x6 = 48) 30 31

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Symmetric tiling sphere Genus 0: Platonic solids

Symmetric tiling : Covering of surface with q= 3 regular polygons {3, 3 } {3, 4} {3, 5} {p, q} : Schläffli symbol p= 4 At each vertex: q p -gons meet {4, 3}: cube

32 {4, 3 } {5, 3} 33

Genus 0: hosohedra Genus 1: tori

• hosohedron: faces with two edges • Tile the plane

• Define a rhombus (all sides same length)

• Project tiling {2, 4 } {2, 9} {2, 32} • Fold rhombus to torus

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Genus 1: {4 , 4} torii Genus 1: {6 , 3} torii

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Genus g ≥ 2 Tiling hyperbolic plane g shape geometry transf. tilings {3, 8} tiling 0 sphere spherical 3D {3,3}, {3,4}, {4,3}, hyperbolic plane rotation {3,5}, {5,3}, {2, n}

1 torus planar 2D {4,4}, {3,6}, {6,3} Euclidean

≥ 2 ? hyperbolic Möbius {3,7}, {4,5}, {5,4}, {4,6}, {6,4}, {5,5}, …

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Tiling hyperbolic plane Regular maps

{4, 6} tiling Regular map: hyperbolic plane Cut out part of tiling hyperbolic plane

For instance: 6 quads

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Regular maps Regular maps

Regular map: Regular map: Cut out part of tiling Cut out part of hyperbolic plane tiling hyperbolic plane For instance: 6 quads, and match edges M. Conder (2006): enumerated all regular maps for g ≤ 101

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Conder’s list Conder’s list

R2.1 : Type {3,8}_12 Order 96 mV = 2 mF = 1 Defining relations for automorphism group: R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, (R * S^-3)^2 ] Defining relations for automorphism group:

R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ] Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ]

R2.3 : Type {4,8}_8 Order 32 mV = 8 mF = 2 Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^-2 * R^2 * S^-2 ]

R2.4 : Type {5,10}_2 Order 20 mV = 10 mF = 5 Defining relations for automorphism group: • Rg.i: genus g, member i [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^-5 ]

…… • complete definition topology

R101.55 : Type {204,204}_2 Order 816 mV = 204 mF = 204 Self-dual Defining relations for automorphism group: (connectivity) [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^92 * S^-1 * R^3 * T * S^2 * T * R^16 * S^-89 * R ]

R101.56 : Type {404,404}_2 Order 808 mV = 404 mF = 404 Self-dual Defining relations for automorphism group: [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^98 * T * S^2 * T * R^10 * T * R^-3 * T * R^4 * S^-85 ] • No cue on possible 3D geometry

Total number of maps in list above: 3378 44 45

Conder’s list A perfect puzzle

R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 Defining relations for automorphism group: Find a space model of a regular map: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ] • Easy to understand The challenge: • Scalable Given the complete topology, • Many approaches possible find a space model : an • Fascinating embedding of faces, edges and • Many puzzles to be cracked vertices in 3D space • Few known solutions

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Genus 3 Genus 3

• Helaman Ferguson, 1993 • Carlo Séquin, 2006 • The Eightfold Way • Klein’s surface • Hurwitz genus 3 • 24 heptagons • Klein’s surface • tetrahedral frame • 24 heptagons

Carlo Séquin, 2009: many more, found manually by exploiting symmetries 48 49

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Work of Carlo Séquin An addictive puzzle

“During the last few nights I woke up at 3am with some ideas, and sometimes they worked, and sometimes they evaporated in daylight!”

“… , that is why I had to physically Carlo Séquin remove all signs of these puzzles Berkeley from my desk and ’lock them up in a vault’, so that I would not be constantly distracted from the duties that I HAVE to fulfill …”

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Tori (genus 1) ( reprise )

• Tile the plane • Take a torus • Unfold to square • Warp to a rhombus • Project tiling • Map rhombus to torus

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Approach for g ≥ 2 Nice genus g shape?

• Tile the hyperbolic plane • Take a nice genus g shape …. …. • Unfold to cut out • Warp to match shape Solid shape with holes? • Project tiling Where to place holes or • Map cut out to nice shape handles to get maximal symmetry? … For g = 6, 13, 17, …? Sphere with handles? 54 55

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Tubified regular maps Tubified regular maps

• Take a regular map • Take a regular map • Turn edges into tubes • Turn edges into tubes • Remove faces • Remove faces

• edges → tubes • edges → tubes • vertices → junctions • vertices → junctions • faces → holes • faces → holes • triangles → ¼ tubes • triangles → ¼ tubes

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Basic idea Solving R2.1{3, 8}, 16 triangles

tube project

target shapes pattern map regular maps genus 2 warp

unfold

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Solving R5.1{3, 8}, 24 octagons Tubified regular maps, reprise

tube 1. Take a regular map project 2. Turn edges into tubes 3. Remove faces 4. Map a regular map map genus 5 5. Goto step 2 again unfold and warp 1.2.3.4.

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Results Video

• About 50 different space models for regular maps found automatically

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Visualization of Regular Maps: The Chase Continues Paris 2014 Jarke J. van Wijk Five years later Eindhoven University of Technology

IEEE SciVis, 2014, Paris

Results 2014 Visualization of cyclic groups

• More generic approach for regular shapes • Subdivide a circle into 18 intervals, – 45 new space models for regular maps found given a circle, subdivided in 12 intervals

• New smoothing approach • Source: C 18 – Better quality of models • Target: C 12

(Lots of) details: see paper

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Visualization of cyclic groups Visualization of cyclic groups

Source C 18 : Target: C 12 : Source C 18 : Target: C 12 :

• C9 × {1, 2} • C6 × {1, 2} • C9 × {1, 2} • C6 × {1, 2}

• C6 × {1, 2, 3} • C4 × {1, 2, 3} • C6 × {1, 2, 3} • C4 × {1, 2, 3}

• C3 × {1, 2, 3, 4, 5, 6} • C3 × {1, 2, 3, 4} • C3 × {1, 2, 3, 4, 5, 6} • C3 × {1, 2, 3, 4}

• C2 × {1, 2, 3, 4, 5, 6, 7, 8, 9} • C2 × {1, 2, 3, 4, 5, 6} • C2 × {1, 2, 3, 4, 5, 6, 7, 8, 9} • C2 × {1, 2, 3, 4, 5, 6}

•C6: greatest common subgroup • Match sets: – Source {1, 2, 3} – Target {1, 2}:

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Visualization of cyclic groups Approach

• Subdivide a circle into 18 intervals, • Take regular maps. Produce face-transitive given a circle, subdivided in 12 intervals variations by enumerating subgroups; • Make target shapes. Produce face-transitive • Source: C maps by duplicating regular maps, 18 enumerating subgroups, punching, gluing • Target: C 12 • Match variations regular map and target.

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Face-transitive variations Face-transitive variations

GS: given source group 2 Factorization:

4 2 GS = H SAS HS: subgroup GS

AS: subset G S, tile of Fuchsian map here (0; 4, 2, 2)

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Face-transitive variations Fuchsian groups:

2 Hyperbolic plane can be tesselated with face- 4 2 transitive polygons, characterized by genus and rotational symmetry boundary points (Poincaré, 1882)

Approach

R2 • Take regular maps. Produce face-transitive 2 8 variations by enumerating subgroups; e2 e R e e M 1 1 8 M 7 7 1 2 • Make target shapes. Produce face-transitive R3 3 M M 7 8 1 maps by duplicating regular maps,

e3 M6 M3 e6 enumerating subgroups, punching, gluing; • Match variations regular map and target. M5 M4 4 5 6 R4 e4 e5 R5

Producing target surface Producing target surfaces

3 GT: group target surface 2 2 Factorization: 2 2 GT = H TAT

HT: subgroup GT

AT: subset G T, tile of Fuchsian map here (0; 3, 2, 2, 2, 2)

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Approach Approach

• Take regular maps. Produce face-transitive • Match variations regular map and target: variations by enumerating subgroups; • Make target shapes. Produce face-transitive Given lists of factorizations of regular maps maps by duplicating regular maps, and alternative target surfaces: enumerating subgroups, punching, gluing • Find matches of G S = HSAS and GT = H TAT, • Match variations regular map and target: such that HS = HT and AS = AT – same type of face, same group • Find corresponding polygons in the – geometric match in hyperbolic plane hyperbolic plane

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R9.16 {5, 6} on tetrahedron R9.6 {4, 8} on torus 2,2

R4.2 {4, 5} on hosohedron-3 R9.16 {5, 6} on tetrahedron

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R17.20 {6, 6} on torus 2,0 R9.6 {4, 8} on torus 2,2

R7.1 {3, 7} on 7-hosohedron

The genus 7 Hurwitz/MacBeath surface…

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{3,7} {7,3}

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Video

Holes Faces Holes Faces 3 56 done 3 56 done 7 168 done 7 168 done 14 364 … 14 364 todo … … … … … …

Some place Thank you! 20XX

The chase ain’t over yet