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Symmetrical Immersions: Tangible Visualizations of Regular Maps

Explore the world of symmetrical immersions through tangible visualizations of low-genus non-orientable regular maps, inspired by intuitive guesses. Delve into the symmetrical properties of objects like spheres and platonic solids in R3, examining the intricate symmetry of regular maps on different surfaces such as the torus. Discover the connection between regular tilings and maps, and unravel the mysteries of orientable and non-orientable surfaces through beautiful depictions on the Poincaré disk. Join this journey to unlock the secrets of geometrical symmetries and regular map configurations.

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Symmetrical Immersions: Tangible Visualizations of Regular Maps

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  1. Symmetry Festival, Aug. 7, 2013 Carlo H. Séquin EECS Computer Science Division University of California, Berkeley Symmetrical Immersions of Low-Genus Non-Orientable Regular Maps (Inspired Guesses followed by Tangible Visualizations)

  2. A Very Symmetrical Object in R3 The Sphere

  3. The Most Symmetrical Polyhedra {5,3} {4,3} {3,3} The Platonic Solids = Simplest Regular Maps {3,4} {3,5}

  4. The Symmetry of a Regular Map • After an arbitrary edge-to-edge move, every edge can find a matching edge;the whole network coincides with itself.

  5. All the Regular Maps of Genus Zero {3,3} Hosohedra {4,3} {3,4} Platonic Solids {5,3} {3,5} Di-hedra (=dual)

  6. Background: Geometrical Tiling Escher-tilings on surfaces with different genus in the plane on the sphere on the torus M.C. Escher Jane Yen, 1997 Young Shon, 2002

  7. Tilings on Surfaces of Higher Genus 24 tiles on genus 3 48 tiles on genus 7

  8. Six differently colored sets of tiles were used Two Types of “Octiles”

  9. From Regular Tilings to Regular Maps When are tiles “the same” ? • on sphere: truly identical  from the same mold • on hyperbolic surfaces  topologically identical(smaller on the inner side of a torus) Tilings should be “regular” . . . • locally regular: all p-gons, all vertex valences q • globally regular: full flag-transitive symmetry(flag = combination: vertex-edge-face) Regular Map

  10. On Higher-Genus Surfaces:only “Topological” Symmetries Edges must be able to stretch and compress Regular map on torus (genus = 1) NOT a regular map: different-length edge loops 90-degree rotation not possible

  11. NOT a Regular Map • Torus with 9 x 5 quad tiles is only locally regular. • Lack of global symmetry:Cannot turn the tile-grid by 90°.

  12. This IS a Regular Map • Torus with 8 x 8 quad tiles.Same number of tiles in both directions! • On higher-genus surfaces such constraints apply to every handle and tunnel.Thus the number of regular maps is limited.

  13. How Many Regular Maps on Higher-Genus Surfaces ? Two classical examples: R2.1_{3,8} _12 16 trianglesQuaternion Group [Burnside 1911] R3.1d_{7,3} _824 heptagonsKlein’s Quartic [Klein 1888]

  14. Nomenclature Regular map genus = 3 # in that genus-group the dual configuration heptagonal faces valence-3 vertices length of Petrie polygon:  R3.1d_{7,3}_8 “Eight-fold Way” zig-zag path closes after 8 moves Schläfli symbol {p,q}

  15. 2006: Marston Conder’s List 6104 Orientable regular maps of genus 2 to 101: R2.1 : Type {3,8}_12 Order 96 mV = 2 mF = 1 Defining relations for automorphism group: [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, (R * S^-3)^2 ] R2.2 : Type {4,6}_12 Order 48 mV = 3 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^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: [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^-5 ] = “Relators” • http://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txt

  16. “Low-Hanging Fruit” Some early successes . . . R2.2_{4,6}_12 R3.6_{4,8}_8 R4.4_{4,10}_20 and R5.7_{4,12}_12

  17. A Tangible Physical Model • 3D-Print, hand-painted to enhance colors R3.2_{3,8}_6

  18. Genus 5 {3,7} 336 Butterflies Only locallyregular !

  19. Globally Regular Maps on Genus 5

  20. Emergence of a Productive Approach • Depict map domain on the Poincaré disk; establish complete, explicit connectivity graph. • Look for likely symmetries and pick a compatible handle-body. • Place vertex “stars” in symmetrical locations. • Try to complete all edge-interconnections without intersections, creating genus-0 faces. • Clean-up and beautify the model.

  21. Depiction on Poincare Disk {5,4} • Use Schläfli symbol  create Poincaré disk.

  22. R3.4_{4,6}_6 Relator:R s s R s s Relators Identify Repeated Locations Operations: R = 1-”click” ccw-rotation around face center; r = cw-rotation. S = 1-”click” ccw-rotation around a vertex; s = cw-rotation.

  23. Triangles of the same color represent the same face. Introduce unique labels for all edges. Complete Connectivity Information

  24. Low-Genus Handle-Bodies • There is no shortage of nice symmetrical handle-bodies of low genus. • This is a collage I did many years ago for an art exhibit.

  25. Numerology, Intuition, … • Example: R5.10_{6,6}_4 First try:oriented cube symmetry Second try:tetrahedral symmetry

  26. Virtual model Paper model A Valid Solution for R5.10_{6,6}_4 (oriented tetrahedron) (easier to trace a Petrie polygon)

  27. OUTLINE • Just an intro so far; by now you should understand what regular maps are. • Next, I will show some nice results. • Then go to non-orientablesurfaces,which have self-intersections,and are much harder to visualize!

  28. Jack J. van Wijk’s Method (1) • Starts from simple regular handle-bodies, e.g. a torus, or a “fleshed-out”, “tube-fied” Platonic solid. • Put regular edge-pattern on each connector arm: • Determine the resulting edge connectivity,and check whether this appears in Conder’s list.If it does, mark it as a success!

  29. Jack J. van Wijk’s Method (2) a dodecahedron 3×3 square tiles on torus • Cool results: Derived from …

  30. Jack J. van Wijk’s Method (3) • For any such regular edge-configuration found, a wire-frame can be fleshed out, and the resulting handle-body can be subjected to the same treatment. • It is a recursive approach that may yield an unlimited number of results; but you cannot predict which ones you will find and which will be missing. • You cannot (currently) direct that system to give you a solution for a specific regular map of interest. • The program has some sophisticated geometrical procedures to produce nice graphical output.

  31. J. van Wijk’s Method (4) • Cool results: Embedding of genus 29

  32. Jack J. van Wijk’s Method (5) • Alltogether by 2010, Jack had found more than 50 symmetrical embeddings. • But some simple maps have eluded this program, e.g. R2.4, R3.3, and: the Macbeath surface R7.1 ! • Also, in some cases, the results don’t look as good as they could . . .

  33. Jack J. van Wijk’s Method (6) My solution on a Tetrus: • Not so cool result for R3.8: too much warping:

  34. Jack J. van Wijk’s Method (7) “Vertex Flower” solution • Not so cool results: too much warping:

  35. “Vertex Flowers” for Any Genus • This classical pattern is appropriate for the 2nd-last entry in every genus group. • All of these maps have exactly two vertices and two faces bounded by 2(g+1) edges. g = 1 g = 2 g = 3 g = 4 g = 5

  36. Some Models

  37. New Focus • Now we want to construct such models for non-orientable surfaces, like Klein bottles. • Unfortunately, there exist no regular maps on the Klein bottle ! • But there are several regular maps on the simplest non-orientable surface: the Projective Plane.

  38. The Projective Plane -- Equator projects to infinity. -- Walk off to infinity -- and beyond …come back from opposite direction: mirrored, upside-down !

  39. The Projective Plane is a Cool Thing! • It is single-sided:Flood-fill paint flows to both faces of the plane. • It is non-orientable:Shapes passing through infinity get mirrored. • A straight line does not cut it apart!One can always get to the other side of that line by traveling through infinity. • It is infinitely large! (somewhat impractical)It would be nice to have a finite model with the same topological properties . . .

  40. Trying to Make a Finite Model • Let’s represent the infinite plane with a very large square. • Points at infinity in opposite directions are the same and should be merged. • Thus we must glue both opposing edge pairs with a 180º twist. Can we physically achieve this in 3D ?

  41. Cross-Surface Construction

  42. Finite Models of the Projective Plane (and their symmetries) Cross surface Steiner surface Boy surface mirror: C2v tetrahedral cyclic: C3

  43. Q The Hemi-Platonic Polyhedra Cube Octahedron Dodecahedron Icosahedron Hemi-Cube Hemi-Octa-h. Hemi-Dodeca-h. Hemi-Icosa-h.

  44. Hemi-Octahedron • Make a polyhedral model of Steiner’s surface. Need 4 copies of this!

  45. Hemi-Cube • Start with 3 perpendicular faces . . .

  46. Hemi-Icosahedron • Built on Hemi-cube model

  47. Hemi-Dodecahedron • Built on Hemi-cube model with suitable face partitioning. • Movie_HemiDodeca.mp4

  48. Embedding of Petersen Graph in Cross-Cap Konrad.Polthier@fu-berlin.de

  49. Hemi-Hosohedra & Hemi-Dihedra • All wedge slices pass through intersection line. N = 2 : self-dual N = 12

  50. Hemi-Hosohedra with Higher Symmetry • Get more symmetry by using a cross-surface with a higher-order self-intersection line. N = 12 N = 60

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