Bridges, Pécs, 2010Bridges, Pécs, 2010

My Search for

Symmetrical Embeddings

of Regular Maps

EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley

Carlo H. Séquin

Math Math Art Art

This is a “math-first” talk !

“Art” comes into it in secondary ways:

The way I find my solutions is more an “art”than a science or a formal math procedure;

How to make the results visible is also an “art”;

Some of the resulting models can be enhancedso that they become “art-objects” on their own.

Regular Maps of Genus ZeroRegular Maps of Genus Zero

Platonic Solids Di-hedra

Hosohedra

The Symmetry of a Regular MapThe 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.

On Higher-Genus Surfaces:On Higher-Genus Surfaces:only “Topological” Symmetriesonly “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

How Many Regular Maps How Many Regular Maps on Higher-Genus Surfaces ?on Higher-Genus Surfaces ?

Two classical examples:

R2.1_{3,8} _1216 triangles

Quaternion Group [Burnside 1911]

R3.1d_{7,3} _824 heptagons

Klein’s Quartic [Klein 1888]

NomenclatureNomenclature

R3.1d_{7,3}_8

Regular mapgenus = 3# in that genus-groupthe dual configurationheptagonal facesvalence-3 verticeslength of Petrie polygon:

Schläfli symbol

“Eight-fold Way”

zig-zag path closes after 8 moves

2006: Marston Conder’s List2006: Marston Conder’s List 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”

Macbeath SurfaceMacbeath 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 ]

This is the 2nd-simplest surface for which the Hurwitz-limit of 84*(genus-1) can be achieved.

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 …

R7.1_{3,7}_18R7.1_{3,7}_18

Paper ModelsPaper Models

Styrofoam Model for R7.1_{3,7}_18Styrofoam Model for R7.1_{3,7}_18

Globally Regular Tiling of Genus 4Globally 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

A Solution for R4.2_{4,5}_6A Solution for R4.2_{4,5}_6 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

Nice Color Pattern for R4.2_{4,5}_6Nice Color Pattern for R4.2_{4,5}_6

Use 5 colors

Every color is at every vertex

Every quad is surrounded by the other 4 colors

A Graph-Embedding ProblemA Graph-Embedding Problem

Dyck’s graph = K4,4,4

Tripartite graph

Nodes of the same color are not connected.

Find surface of lowest genus in which Dyck’s graph can be drawn crossing-free

An Intuitive Approach Start with highest-symmetry genus-3 surface: “Tetrus”

Place 12 points so that the missing edges do not break symmetry:

Inside and outside on each tetra-arm.

Do not connect the nodes that lie on thesame symmetry axis(same color)(or this one).

A Tangible Physical ModelA Tangible Physical Model

3D-Print, hand-painted to enhance colors

R3.2_{3,8}_6

A Virtual Genus-3 Tiffany LampA Virtual Genus-3 Tiffany Lamp

Light Cast by Genus-3 “Tiffany Lamp”Light Cast by Genus-3 “Tiffany Lamp”

Rendered with “Radiance” Ray-Tracer (12 hours)

R2.2_{4,6}_12 R3.6_{4,8}_8

““Low-Hanging Fruit”Low-Hanging Fruit”Some early successes . . .

R4.4_{4,10}_20 and R5.7_{4,12}_12

Genus Genus 55

336

Butterflies

Only locallyregular !

Globally Regular Maps on Genus 5Globally Regular Maps on Genus 5

Emergence of a Productive ApproachEmergence 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.

{ Look how best to turn this into “Art.” }

Depiction on Poincare DiskDepiction on Poincare Disk

Use Schläfli symbol create Poincaré disk.

Relators Identify Repeated LocationsRelators 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 Complete Connectivity Connectivity InformationInformation

Triangles of the same color represent the same face.

Introduce unique labels for all edges.

Low-Genus Handle-BodiesLow-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.

Numerology, Intuition, …Numerology, Intuition, … Example: R5.10_{6,6}_4

First try:oriented cube symmetry

Second try:tetrahedral symmetry

An Valid Solution for R5.10_{6,6}_4 An Valid Solution for R5.10_{6,6}_4

Virtual model Paper model(oriented tetrahedron) (easier to trace a Petrie polygon)

2 Methods to Find Embeddings2 Methods to Find Embeddings 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!

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.

The General Text-book Method (1)The General Text-book Method (1) Convert the domain of the regular map to

a special 4g-gon with the edge sequence:( a, b, a’, b’ ) g

The General Text-book Method (2)The General Text-book Method (2)

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’.

The General Text-book Method (3)The General Text-book Method (3) Here is an explicit genus-2 net with this kind of perimeter,

and the folded-up 2-loop handle-body that results.

The General Text-book Method (4)The General Text-book Method (4)

Unfortunately, the result is not nice !

Result for R2.1_{3.8}_12

Vertices do not end upin symmetrical places.

The local edge-densityis quite non-uniform.

Jack J. van Wijk’s Method (1)Jack J. van Wijk’s Method (1)

Starts from simple regular handle-bodies, e.g. torus, “fleshed-out” hosohedron, or a 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!

Jack J. van Wijk’s Method (3)Jack J. van Wijk’s Method (3) Cool results: Derived from …

Dodecahedron 3×3 square tiles on torus

Jack J. van Wijk’s Method (2)Jack J. van Wijk’s Method (2)

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 particular map of interest.

The program has some sophisticated geometrical procedures to produce nice graphical output.

J. van Wijk’s J. van Wijk’s Method (5)Method (5)

Cool results:

Embedding of genus 29

Jack J. van Wijk’s Method (5)Jack J. van Wijk’s Method (5)

Alltogether so far, Jack has 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 . . .

Jack J. van Wijk’s Method (6)Jack J. van Wijk’s Method (6) Not so cool results: too much warping.

My solution on a Tetrus

Jack J. van Wijk’s Method (7)Jack J. van Wijk’s Method (7) Not so cool results: too much warping.

“Vertex Flower” solution

““Vertex Flowers” for Vertex Flowers” for AnyAny Genus 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

Paper Models for “Vertex Flowers”Paper Models for “Vertex Flowers”

I first found those embeddings with these paper strip models.

g = 2 g = 3

Anatomy of a Paper-Strip ModelAnatomy of a Paper-Strip Model

bendandglue

R2.5_{6,6}_2

The Regular Map R3.3_{3,12}_8The Regular Map R3.3_{3,12}_8

16 triangles,24 edges, 4 vertices (valence-12).

Deforming & Folding the Map DomainDeforming & Folding the Map Domain

Fold into a torus with two openings:connect horizontal and vertical edges of same color;

Then bring the 4 green vertices together . . .

Evolution of the Topology ModelEvolution of the Topology Model

Models for Models for R3.3_{3,12}_8R3.3_{3,12}_8

Net and paper model

Models for R3.3_{3,12}_8Models 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.

Good Solutions Can Be Re-used !Good Solutions Can Be Re-used !

The maps R3.3_{3,12}_8 and R3.5_{4,8}_8 are related.R3.3_{3,12}_8 and R3.5_{4,8}_8 are related.

Pairs of triangles turn into quadrilaterals.

Re-use of R3.3 Topology for R3.5Re-use of R3.3 Topology for R3.5

These “diagonal” edges are no longer present

Visualizing R4.2_{4,5}_6Visualizing R4.2_{4,5}_6

A transformation maintaining symmetry

R4.2_{4,5}_6 LatticeR4.2_{4,5}_6 Lattice

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

More …More …

ConclusionsConclusions “Doing math” is not just writing formulas!

It may involve paper, wires, styrofoam, glue…

Sometimes, tangible beauty may result !