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

Carlo H. Séquin

Art and MathArt and MathBehind and BeyondBehind and Beyond

the 8-fold Waythe 8-fold Way

Art, Math, Magic, and the Number 8 ...Art, Math, Magic, and the Number 8 ...

“Eightfold Way” at MSRI by Helaman Ferguson

The Physicists’ The Physicists’ Eightfold WayEightfold Way

The Noble Eightfold PathThe Noble Eightfold Path

-- The way to end suffering (Siddhartha Gautama)

Siddhartha GautamaSiddhartha Gautama

Helaman Ferguson’s Helaman Ferguson’s The Eightfold WayThe Eightfold Way

24 (lobed) heptagons on a genus-3 surface

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;

a totally regular graph a totally regular graph

with 168 automorphismswith 168 automorphisms

24 Heptagons – Forced into 3-Space24 Heptagons – Forced into 3-Space

Retains 12 (24) symmetries of the original 168 automorphisms of the Klein polyhedron.

Quilt by:Eveline Séquin(1993),

based on a pattern obtained from Bill Thurston;

turns inside-out !

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

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

On a regular polyhedron all such Petrie paths are closed and are of the same length.

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

Petrie Path on Poincaré DiskPetrie Path on Poincaré Disk

Exactly eight zig-zag moves lead to the “same” place

My Long-standing Interest in TilingsMy Long-standing Interest in Tilings

Can we do Escher-tilings on higher-genus surfaces?

in the plane on the sphere on the torus

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

Lizard TetrusLizard Tetrus (with Pushkar Joshi)(with Pushkar Joshi)

Cover of the 2007 AMS Calendar of Mathematical Imagery

24 Lizards on the Tetrus24 Lizards on the Tetrus

One of 12 tiles

3 different color combinations

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)

NOT TRUE for the Lizard TertrusThe Lizards don’t exhibit 7-fold symmetry!

Decorating the HeptagonsDecorating the Heptagons

Split into 7 equal wedges.

Distort edges,while maintaining:

C7 symmetry around the tile center,

C2 symmetry around outer edge midpoints,

C3 symmetry around all heptagon vertices.

Creating the Heptagonal Fish TileCreating the Heptagonal Fish Tile

Fit them together to cover the whole surface ...

FundamentalDomain

DistortedDomain

““Infinite” Tiling Infinite” Tiling on the on the PoincarPoincaréé DiskDisk

Genus 3Genus 3Surface Surface withwith168 fish168 fish

Every fish can map onto every other fish.

56 triangles

24 vertices

genus 3

globally regular

Petrie polygons of length 8

The Dual The Dual SurfaceSurface

Why is this Why is this so special ?so special ?

A whole book has been written about it(1993).

“The most important object in mathematics ...”

Maximal Amount of SymmetryMaximal Amount of Symmetry

Hurwitz showed that on a surface of genus g (>1)

there can be at most (g-1)*84 automorphisms.

This limit is reached for genus 3.

It cannot be reached for genus 4, 5, 6.

It can be reached again for genus 7.

Genus 3 and Genus 7 CanvasGenus 3 and Genus 7 Canvas

tetrahedral frame octahedral frame

genus 3 , 24 heptagons genus 7, 72 heptagons

168 automorphisms 504 automorphisms

Decorated Junction ElementsDecorated Junction Elements

3-way junction 4-way junction

6 heptagons 12 heptagons

Assembly of Genus-7 SurfaceAssembly of Genus-7 Surface

Join zig-zag edges Genus 7 surface:of neighboring arms six 4-way junctions

EIGHT 3-way Junctions

336 Butterflies on a surface of genus 5.

Pretty, but NOTglobally regular !

The Genus-7 CaseThe Genus-7 Case

Can do similar decorations

-- but NOT globally regular!

Perhaps the Octahedral frame

does NOT have the best symmetry.

Try to use surface with 7-fold symmetry ?

Genus-7 Styrofoam ModelsGenus-7 Styrofoam Models

Fundamental Domain for Genus-7 CaseFundamental Domain for Genus-7 Case

A cluster of 72 heptagons gives full coveragefor a surface of genus-7.

This regular hyperbolic tiling can be continued with infinitely many heptagons in the limit circle.

Genus-7Genus-7Paper ModelsPaper Models

7-fold symmetry

The Embedding ofThe Embedding ofthe the 1188-fold Way-fold Waystill eludes me.still eludes me.

Perhaps at G4G18 in 2028 …

Let’s do something pretty

with the OCTA - frame:

a {5,4} tiling

Genus 7 Surface with 60 QuadsGenus 7 Surface with 60 Quads

Convenient to create smooth subdivision surface based on octahedral frame

{5,4} {5,4} StarfishStarfish Pattern on Genus-7 Pattern on Genus-7 Start with 60 identical

black&white quad tiles:

Color tiles consistently around joint corners

Switch to dual pattern:

> 48 pentagonal starfish

Create a Smooth Subdivision SurfaceCreate a Smooth Subdivision Surface

Inner and outer starfish prototiles extracted,

thickened by offsetting,

sent to FDM machine . . .

EIGHT Tiles from the FDM MachineEIGHT Tiles from the FDM Machine

White Tile Set -- 2White Tile Set -- 2ndnd of 6 Colors 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 arm

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

+ Octaframe edge

12 tiles inside view

8 tiles

About Half the Shell AssembledAbout Half the Shell Assembled

The Assembled Genus-7 ObjectThe Assembled Genus-7 Object

S P A R E SS P A R E S

72 Lizards on a Genus-7 Surface72 Lizards on a Genus-7 Surface