Bridges 2007, San SebastianBridges 2007, San Sebastian

Symmetric Embedding of

Locally Regular Hyperbolic Tilings

Carlo H. SCarlo H. Sééquinquin

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

Goal of This StudyGoal of This Study

Make Escher-tilings on surfaces of higher genus.

in the plane on the sphere on the torus

M.C. Escher Jane Yen, 1997 Young Shon, 2002

How to Make an Escher TilingHow to Make an Escher Tiling

Start from a regular tiling

Distort all equivalent edges in the same way

Hyperbolic Escher TilingsHyperbolic Escher Tilings

All tiles are “the same” . . .

truly identical from the same mold

on curved surfaces topologically identical

Tilings should be “regular” . . .

locally regular: all p-gons, all vertex valences v

globally regular: full flag-transitive symmetry(flag = combination: vertex-edge-face)

““168 Butterflies,” D. Dunham (2002) 168 Butterflies,” D. Dunham (2002)

Locally regular {3,7} tiling on a genus-3 surfacemade from 56 isosceles triangles

“snub-tetrahedron”

E. Schulte and J. M. WillsE. Schulte and J. M. Wills

Also: 56 triangles, meeting in 24 valence-7 vertices.

But: Globally regular tiling with 168 automorphisms! (topological)

Generator for {3,7} Tilings on Genus-3Generator for {3,7} Tilings on Genus-3

Twist arms by multiples of 90 degrees ...

Dehn TwistsDehn Twists

Make a closed cut around a tunnel (hole) or around a (torroidal) arm.

Twist the two adjoining “shores” against each other by 360 degrees; and reconnect.

Network connectivity stays the same;but embedding in 3-space has changed.

Fractional Dehn TwistsFractional Dehn Twists

If the network structure around an arm or around a hole has some periodicity P,then we can apply some fractional Dehn twistsin increments of 360° / P.

This will lead to new network topologies,but may maintain local regularity.

Globally Regular {3,7} TilingGlobally Regular {3,7} Tiling

From genus-3 generator (use 90° twist)

Equivalent to Schulte & Wills polyhedron

56 triangles

24 vertices

genus 3

globally regular

168 automorph.

Smoothed Smoothed Triangulated Triangulated SurfaceSurface

Generalization of GeneratorGeneralization of Generator

Turn straight frame edges into flexible tubes

From 3-way to 4-way JunctionsFrom 3-way to 4-way Junctions

Tetrahedral hubs

6(12)-sided arms

6-way Junction + Three 8-sided Loops6-way Junction + Three 8-sided Loops

Construction of Junction ElementsConstruction of Junction Elements

3-way junction

construction of

6-way junction

Junction Elements Junction Elements Decorated with Decorated with 6, 12, 24, Heptagons6, 12, 24, Heptagons

Assembly of Higher-Genus SurfacesAssembly of Higher-Genus Surfaces

Genus 5:8 Y-junctions

Genus 7

Genus-5 Surface (Cube Frame)Genus-5 Surface (Cube Frame)

112 triangles, 3 butterflies each . . .

336

Butterflies

Creating Smooth SurfacesCreating Smooth Surfaces

4-step process:

Triangle mesh

Subdivision surface

Refine until smooth

Texture-map tiling design

Texture-Mapped Single-Color Tilings

subdivide also texture coordinates

maps pattern smoothly onto curved surface.

What About Differently Colored Tiles ?What About Differently Colored Tiles ?

How many different tiles need to be designed ?

24 Newts on the Tetrus (2006)24 Newts on the Tetrus (2006)

One of 12 tiles

3 different color combinations

Use with Higher-Genus SurfacesUse with Higher-Genus Surfaces

Lack freedom to assign colors at will !

New Escher Tile Editor

Tiles need not be just simple n-gons.

Morph edges of one boundary . . .and let all other tiles change similarly!

Escher Tile Editor (cont.)Escher Tile Editor (cont.)

Key differences:

Tiling pattern is no longer just a texture!

Tiles have a well-defined boundary,which is tracked in subdivision process.

This outline can be flood filled with color.

Escher Tile Editor (cont.)Escher Tile Editor (cont.)

Possible to add extra decorations onto tiles

Prototile Extraction

Flood-fill can also be used to identify all geometry that belongs to a single tile.

Extract Prototile Geometry for RPExtract Prototile Geometry for RP

Two prototiles extracted and thickened

Generalizing the Generator to QuadsGeneralizing the Generator to Quads

4-way junctions built around cube hubs

4-sided prismatic arms

Genus 7 Surface with 60 QuadsGenus 7 Surface with 60 Quads

No twist

{5,4} Starfish Pattern on Genus-7{5,4} Starfish Pattern on Genus-7

Polyhedral representation of an octahedral frame

108 quadrilaterals (some are half-tiles)

60 identical quad tiles:

Use dual pattern:

48 pentagonal starfish

Only Two Geometrically Different TilesOnly Two Geometrically Different Tiles

Inner and outer starfish prototiles extracted,

thickened by offsetting,

sent to FDM machine . . .

Fresh from the FDM MachineFresh from the FDM Machine

Red Tile Set -- 1 of 6 ColorsRed Tile Set -- 1 of 6 Colors

2 Outer and 2 Inner Tiles2 Outer and 2 Inner Tiles

A Whole Pile of Tiles . . .A Whole Pile of Tiles . . .

The Assembly of Tiles Begins . . .The Assembly of Tiles Begins . . .

Outer tiles

Inner tiles

AssemblyAssembly(cont.):(cont.):

8 Inner Tiles8 Inner Tiles

Forming inner part of octa-frame edge

Assembly (cont.)Assembly (cont.) 2 Hubs

+ Octaframe edge

12 tiles inside view

8 tiles

More Assembly StepsMore Assembly Steps

More Assembly StepsMore Assembly Steps

Assembly Gets More DifficultAssembly Gets More Difficult

Almost Done ...Almost Done ...

The Finished Genus-7 ObjectThe Finished Genus-7 Object

. . . I wish . . .

“work in progress . . .”

What about What about Globally RegularGlobally Regular Tilings ? Tilings ?

So far:

Method and tool set to make complex, locally regular tilings on higher-genus surfaces.

BRIDGES, London, 2006BRIDGES, London, 2006

“Eight-fold Way” by Helaman Ferguson

Visualization of Klein’s Quartic in 3DVisualization of Klein’s Quartic in 3D

24 heptagons 24 heptagons

on a genus-3 surface;on a genus-3 surface;

24x7 automorphisms24x7 automorphisms

(= maximum possible)(= maximum possible)

AnotherAnother View ... View ...

168 fish

Why Is It Called: “Eight-fold Way” ?Why Is It Called: “Eight-fold Way” ?

Since it is a regular polyhedral structure, it has a set of Petrie Polygons.

These are “zig-zag” skew polygons that always hug a face for exactly 2 consecutive edges.

On a regular polyhedron you can start such a Petrie polygon from any vertex in any direction.(A good test for regularity !)

On the Klein Quartic, the length of these Petrie polygons is always eight edges.

Why Is It “Special”Why Is It “Special”

The Klein quartic has the maximal number of automorphisms possible on a genus-3 surface.

A. Hurwitz showed: Upper limit is: 84(genus-1)

Can only be reached for genus 3, 7, 14, ...

Temptation to try to explore the genus-7 case

My Original Plan for Bridges 2007My Original Plan for Bridges 2007

Explore the genus-7 case

Make a nice sculpture modelin the spirit of the “8-fold Way”

This requires 2 steps:

A) figure out the complete connectivity (map mesh on the Poincaré disc)

B) embed it on a genus-7 surface(while maximizing 3D symmetry)

PoincarPoincaréé DiscDisc

Find some numbering that repeats periodically and produces the proper Petrie length.

Step 2: What Shape to Choose ?Step 2: What Shape to Choose ?

Tubular Genus-7 SurfacesTubular Genus-7 Surfaces

12 x 3-way 6 x 4-way 3 x 6-way

Symmetrical {3,7} Maps on Genus-7Symmetrical {3,7} Maps on Genus-7

Option Junctnvalence

Junctn count

Junction triangles

Arm prism #

Arm count

Arm triangles

A –prism 3-sided

3 12 24 4 18 144

B –tetra 4 6 24 6 12 144

C 5 4 28 7 10 140

D –cube 6 3 24 8 9 144

E –octa 8 2 24 9 8 144

F 14 1 0 12 7 168

Genus-7Genus-7Paper ModelsPaper Models

Genus-7 Styrofoam ModelsGenus-7 Styrofoam Models

Try Something Simpler First !Try Something Simpler First !

Banff 2007 Workshop “Teaching Math …”

Globally Regular Tiling With 24 PentagonsGlobally Regular Tiling With 24 Pentagons

Thanks to David Richter !

Actual cardboard model

The Dodeca-DodecahedronThe Dodeca-Dodecahedron

6 sets of 4 parallel faces:

2 large pentagons + 2 smaller pentagrams

Locally Regular Maps {4,5} and {5,4}Locally Regular Maps {4,5} and {5,4}

Dual coverage of a genus 4 surface:

30 quadrilaterals versus 24 pentagons

PP > 6

Escher Escher TilingTiling

With texture mapping

Another Repetitive Texture ...Another Repetitive Texture ...

3 Fish

Looking for the Globally Regular TilingLooking for the Globally Regular Tiling

Try to find a suitable network by applying fractional Dehn twists to the “spokes”.

Use the same amount on all arms to maintain 4-fold rotational symmetry.

Other Shapes StudiedOther Shapes Studied

Lawson surface --- “Prism +4 handles”

ExperimetsExperimets

Apply fractional Dehn twists to all these structures,

check for proper length of Petrie polygon.

No success with any of them ...

Inspiration from Symmetry . . . Inspiration from Symmetry . . .

Look for shapes that have 3-fold and 4-fold symmetries . . .

Truncated OctahedronTruncated Octahedron

1st try: Four hexagonal prismatic tunnels

Try different fractional Dehn twists in tunnels

Checking Globally RegularityChecking Globally Regularity

Transfer connectivity and coloring pattern

No cigar !

These six vertices are the same as the ones on the bottom

Inspiration from Inspiration from 8-fold Way8-fold Way On Tetrus: Petrie polygons zig-zag around arms

Let Petrie polygons zig-zag around tunnel walls

It works !!!

Add a Nice Coloring PatternAdd a Nice Coloring Pattern

Use 5 colors

Every color is at every vertex

Every quad is surrounded by the other 4 colors

ConclusionsConclusions

I have not yet found my “Holy Grail”

Gained insight about locally regular tilings

Used “multi media” in my explorations

Remaining question:

what are good ways to find the desired mapping to a symmetrical embedding ?

How does one search / test for global graph regularity ?

Thanks toThanks to

David Richter {S5 dodedadodecahedron}

John M. Sullivan {feedback on paper}

Pushkar Joshi (graduate student)

Allan Lee, Amy Wang (undergraduates)

Questions ?Questions ?

?