2/2/2016 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 1 2/2/2016 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 2 2/2/2016 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 64 64 #faces? Königsberg 1893 3 2/2/2016 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! 4 2/2/2016 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 26 27 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 28 29 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 5 2/2/2016 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 34 35 Genus 1: {4 , 4} torii Genus 1: {6 , 3} torii 36 37 6 2/2/2016 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}, … 38 39 Tiling hyperbolic plane Regular maps {4, 6} tiling Regular map: hyperbolic plane Cut out part of tiling hyperbolic plane For instance: 6 quads 40 41 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 42 43 7 2/2/2016 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 46 47 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 8 2/2/2016 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 …” 50 51 Tori (genus 1) ( reprise ) • Tile the plane • Take a torus • Unfold to square • Warp to a rhombus • Project tiling • Map rhombus to torus 52 53 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 9 2/2/2016 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 56 57 Basic idea Solving R2.1{3, 8}, 16 triangles tube project target shapes pattern map regular maps genus 2 warp unfold 58 59 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. 60 61 10 2/2/2016 Results Video • About 50 different space models for regular maps found automatically 62 . 63 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 66 67 11 2/2/2016 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}: 68 69 Visualization of cyclic groups Approach • Subdivide a circle into 18 intervals, • Take regular maps.