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

A Very Symmetrical Object in R3 The Sphere

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

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.

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

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

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

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

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

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

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

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

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]

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 {p,q} “Eight-fold Way” zig-zag path closes after 8 moves

2006: Marston Conder’s List u 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”

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

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

Genus 5 {3,7} 336 Butterflies Only locally regular !

Globally Regular Maps on Genus 5

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.

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

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

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

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.

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

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

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

Jack J. van Wijk’s Method (1) u Starts from simple regular handle-bodies, e.g. a torus, or a “fleshed-out”, “tube-fied” 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!

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

Jack J. van Wijk’s Method (3) 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 specific regular map of interest. u The program has some sophisticated geometrical procedures to produce nice graphical output.

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

Jack J. van Wijk’s Method (5) u Alltogether by 2010, Jack had 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...

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

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

“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

Some Models

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

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

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

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

Cross-Surface Construction

Finite Models of the Projective Plane (and their symmetries) Cross surface Steiner surface Boy surface mirror: C 2v tetrahedral cyclic: C 3

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

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

Hemi-Cube u Start with 3 perpendicular faces...

Hemi-Icosahedron u Built on Hemi-cube model

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

Embedding of Petersen Graph in Cross-Cap

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

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

Regular Maps on Non-Orientable Surfaces of Genus-2 and Genus-3 u There aren’t any !! Genus-2: Klein Bottles Genus-3: Dyck’s surface

Low-Genus Non-Orientable Regular Maps u From: Marston Conder (2012)

A Way to Make Any Surface u A sphere to start with; u A hole-punch to make punctures: Each increases Euler Characteristic by one. u We can fill these holes again with: l Disks: Decreases Euler Characteristic by one. {useless!} l Cross-Caps: Makes surface single-sided. l Boy-caps: Makes surface single-sided. l Handles (btw. 2 holes): Orientability unchanged. l Cross-Handles (btw. 2 holes): Makes surface single-sided. Euler Char. unchanged

Constructing a Surface with χ = 2 ‒ h u Punch h holes into a sphere and close them up with: handles or cross-handles cross-caps or Boy caps or Closing two holes at the same time:

Topological Diagrams for N4.1d u Diagrams from: N.S. Weed (2009, 2010); (This saves tedious work that I normally perform on the Poincaré disk.) u Other options: 4 cross-caps on a sphere... n.o.-genus = 4; Euler characteristic = ‒ 2 N4.1d: 4 hexagons, 6 val-4 vertices, 12 edges, Petrie-length=6,

Concept of a Genus-4 Surface 4 Boy caps grafted onto a sphere with tetrahedral symmetry.

Regular Maps with Tetrahedral Symmetry N4.2: 6 quads, 4 val-6 vertices. N4.2d: 4 hexagons, 6 val-4 vertices. For both: 12 edges, Petrie-length=3.

Genus-4 Surface Using 4 Boy-Caps Start with a polyhedral representation and smooth it with subdivision:

Genus-4 Surface Using 4 Boy-Caps (60°rotation between neighbors) Employ tetrahedral symmetry! ( 0°rotation between neighbors) (one more level of subdivision)

N4.1 Revisited N4.1: 6 quads, 4 val-6 vertices, 12 edges, Petrie-length=6. N4.1d: 4 hexagons, 6 val-4 vertices, 12 edges, Petrie-length=6. A cross-handle (schematic)

Regular Maps N5.1 and N5.1d N5.1: 15 quads, 12 val-5 vertices, 30 edges, Petrie-length=6. N5.1d: 12 pentagons, 15 val-4 vertices 30 edges, Petrie-length=6. Make a genus-5 surface with (oriented) tetrahedral symmetry by grafting 4 Boy caps onto the bulges of a Steiner surface.

Regular Map N5.3 (self-dual) N5.3: 6 pentagons, 6 val-5 vertices, 15 edges, Petrie-length=3. Unfolded Steiner net; a folded-up paper model; virtual Bézier model. Use again Steiner surface with 4 Boy caps added.

The Convoluted Map N5.4 (self-dual) N5.4: 3 hexagons, 3 val-6 vertices, 9 edges, mF=mV=3, Petrie-length=3. It has vertex and face multiplicities of 3! Use torus with 3 Boy caps (two views) or with 3 cross-caps.

Regular Map N6.2d u Make use of 3 cross-handle tunnels in a cube N6.2d: 6 decagons, 20 val-3 vertices, 30 edges, mF=2, Petrie-length=5. Virtual model unfolded net complete paper model

Regular Map N7.1 Match creases! N7.1: 15 quads, 10 val-6 vertices, 30 edges, Petrie-length=5. N7.1d: 10 hexagons, 15 val-4 vertices 30 edges, Petrie-length=5. A genus-7 surface. Movie !

Construction a Genus-8 Surface Concept: 8 Boy caps grafted onto sphere in octahedral positions. N8.1: 84 triangles, 36 val-6 vertices, 126 edges, Petrie-length=9.

Octa-Boy Sculpture Two half-shells made on an RP machine

Octa-Boy Sculpture The two half-shells combined

Octa-Boy Sculpture Seen from a different angle

Conclusions u Many more maps remain to be modeled. u Several puzzles among maps of genus ≤ 8. u Can this task be automated / programmed ? u Turn some interesting maps into art...

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

Orientable Regular Map of Genus-6

Light Field of Genus-6 Tiffany Lamp

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

Questions ?