1. Bridges, Pécs, 2010 My Search for Symmetrical Embeddings of Regular Maps EECS Computer Science Division University of California, Berkeley Carlo H. Séquin

2. Math  Art This is a “math-first” talk ! “Art” comes into it in secondary ways: u The way I find my solutions is more an “art” than a science or a formal math procedure; u How to make the results visible is also an “art”; u Some of the resulting models can be enhanced so that they become “art-objects” on their own.

3. Regular Maps of Genus Zero Platonic SolidsDi-hedra Hosohedra

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

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

6. How Many Regular Maps on Higher-Genus Surfaces ? Two classical examples: R2.1_{3,8} _12 16 triangles Quaternion Group [Burnside 1911] R3.1d_{7,3} _8 24 heptagons Klein’s Quartic [Klein 1888]

7. Nomenclature R3.1d_{7,3}_8R3.1d_{7,3}_8 Regular map genus = 3 # in that genus-group the dual configuration heptagonal faces valence-3 vertices length of Petrie polygon:  Schläfli symbol “Eight-fold Way” zig-zag path closes after 8 moves

8. 2006: Marston Conder’s List u http://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txt 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”

9. Macbeath Surface Entry for the Macbeath surface of genus 7: R7.1 : Type {3,7}_18 Order 1008 mV = 1 mF = 1 Defining relations for automorphism group: [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, S^-7, S^-2 * R * S^-3 * R * S^-2 * R^-1 * S^2 * R^-1 * S^2 * R^-1 * S^-2 * R * S^-1 ] u This is the 2 nd -simplest surface for which the Hurwitz-limit of 84*(genus-1) can be achieved. u For my 2006 Bridges talk I wanted to make a nice sculptural model of this surface …  I simply could not find a solution ! Even though I tried really hard …

10. R7.1_{3,7}_18 Paper Models

11. Styrofoam Model for R7.1_{3,7}_18

12. Globally Regular Tiling of Genus 4 But this is not an embedding! Faces intersect heavily! Actual cardboard model (Thanks to David Richter) Conder: R4.2d_{5,4}_6

13. A Solution for R4.2_{4,5}_6 u Inspiration R3.1: Petrie polygons zig-zag around arms.  R4.2: Let Petrie polygons zig-zag around tunnel walls. It works !!! A look into a tunnel

14. Nice Color Pattern for R4.2_{4,5}_6 u Use 5 colors u Every color is at every vertex u Every quad is surrounded by the other 4 colors

15. A Graph-Embedding Problem u Dyck’s graph = K 4,4,4 u Tripartite graph u Nodes of the same color are not connected. Find surface of lowest genus in which Dyck’s graph can be drawn crossing-free

16. An Intuitive Approach u Start with highest-symmetry genus-3 surface: “Tetrus” u Place 12 points so that the missing edges do not break symmetry: u  Inside and outside on each tetra-arm. u Do not connect the nodes that lie on the same symmetry axis (same color) (or this one).

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

18. A Virtual Genus-3 Tiffany Lamp

19. Light Cast by Genus-3 “Tiffany Lamp” Rendered with “Radiance” Ray-Tracer (12 hours)

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

21. Genus 5 336 Butterflies Only locally regular !

22. Globally Regular Maps on Genus 5

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

24. Depiction on Poincare Disk u Use Schläfli symbol  create Poincaré disk.

25. 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. R3.4_{4,6}_6 Relator: R s s R s s

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

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

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

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

30. 2 Methods to Find Embeddings u A general “text-book” method for embedding a network in a handle-body of appropriate genus. But this will not yield any nice regular solutions! u The computer-search by Jack J. van Wijk, which found more than 50 good embeddings. But not clear which solutions will emerge; some simple cases could not be found! Some solutions are more twisted than they need be.

31. The General Text-book Method (1) u Convert the domain of the regular map to a special 4g-gon with the edge sequence: ( a, b, a’, b’ ) g

32. The General Text-book Method (2) u Now each sequence of 4 edges: ( a, b, a’, b’ ) is first closed into a tube by joining a to a’, then into a closed handle by joining b to b’.

33. The General Text-book Method (3) u Here is an explicit genus-2 net with this kind of perimeter, and the folded-up 2-loop handle-body that results.

34. The General Text-book Method (4) u Unfortunately, the result is not nice ! u Result for R2.1_{3.8}_12 u Vertices do not end up in symmetrical places. u The local edge-density is quite non-uniform.

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

36. Jack J. van Wijk’s Method (3) u Cool results: Derived from … Dodecahedron 3×3 square tiles on torus

37. Jack J. van Wijk’s Method (2) u 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. u 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. u You cannot (currently) direct that system to give you a solution for a particular map of interest. u The program has some sophisticated geometrical procedures to produce nice graphical output.

38. J. van Wijk’s Method (5) u Cool results: Embedding of genus 29

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

40. Jack J. van Wijk’s Method (6) u Not so cool results: too much warping. My solution on a Tetrus

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

42. “Vertex Flowers” for Any Genus u This classical pattern is appropriate for the 2 nd -last entry in every genus group. u 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

43. Paper Models for “Vertex Flowers” I first found those embeddings with these paper strip models. g = 2 g = 3

44. Anatomy of a Paper-Strip Model bend and glue R2.5_{6,6}_2

45. The Regular Map R3.3_{3,12}_8 u 16 triangles, 24 edges, 4 vertices (valence-12).

46. Deforming & Folding the Map Domain u Fold into a torus with two openings: connect horizontal and vertical edges of same color; u Then bring the 4 green vertices together...

47. Evolution of the Topology Model

48. Models for R3.3_{3,12}_8 Net and paper model

49. Models for R3.3_{3,12}_8 Original clean paper model Alternative model in which the four vertices have been moved to the middle of the handles.

50. Good Solutions Can Be Re-used ! R3.3_{3,12}_8 and R3.5_{4,8}_8 are related. u The maps R3.3_{3,12}_8 and R3.5_{4,8}_8 are related. Pairs of triangles turn into quadrilaterals.

51. Re-use of R3.3 Topology for R3.5 These “diagonal” edges are no longer present

52. Visualizing R4.2_{4,5}_6 u A transformation maintaining symmetry

53. R4.2_{4,5}_6 Lattice

54. R4.2_{4,5}_6 Lattice (wide angle)

55. More …

56. Conclusions u “Doing math” is not just writing formulas! u It may involve paper, wires, styrofoam, glue… u Sometimes, tangible beauty may result !