EECS Computer Science Division University of California, Berkeley Carlo H. Séquin Art and Math Behind and Beyond the 8-fold Way

Art, Math, Magic, and the Number 8... “Eightfold Way” at MSRI by Helaman Ferguson

The Physicists’ Eightfold Way

The Noble Eightfold Path -- The way to end suffering (Siddhartha Gautama)

Siddhartha Gautama

Helaman Ferguson’s The Eightfold Way 24 (lobed) heptagons on a genus-3 surface

Visualization of Klein’s Quartic in 3D 24 heptagons on a genus-3 surface; a totally regular graph with 168 automorphisms

24 Heptagons – Forced into 3-Space u 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” ? u Petrie Polygons are “zig-zag” skew polygons that always hug a face for exactly 2 consecutive edges. u On a regular polyhedron all such Petrie paths are closed and are of the same length. u On the Klein Quartic, the length of these Petrie polygons is always eight edges.

Petrie Path on Poincaré Disk u Exactly eight zig-zag moves lead to the “same” place

My 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 Tetrus (with Pushkar Joshi) Cover of the 2007 AMS Calendar of Mathematical Imagery

24 Lizards on the Tetrus One of 12 tiles 3 different color combinations

Hyperbolic Escher Tilings All tiles are “the same”... u truly identical  from the same mold u on curved surfaces  topologically identical Tilings should be “regular”... u locally regular: all p-gons, all vertex valences v u globally regular: full flag-transitive symmetry (flag = combination: vertex-edge-face) u NOT TRUE for the Lizard Tertrus The Lizards don’t exhibit 7-fold symmetry!

Decorating the Heptagons Split into 7 equal wedges. Distort edges, while maintaining: u C 7 symmetry around the tile center, u C 2 symmetry around outer edge midpoints, u C 3 symmetry around all heptagon vertices.

Creating the Heptagonal Fish Tile Fit them together to cover the whole surface... Fundamental Domain Distorted Domain

“Infinite” Tiling on the Poincaré Disk

Genus 3 Surface with 168 fish u Every fish can map onto every other fish.

u 56 triangles u 24 vertices u genus 3 u globally regular u Petrie polygons of length 8 The Dual Surface

Why is this so special ? u A whole book has been written about it (1993). u “The most important object in mathematics...”

Maximal Amount of Symmetry u Hurwitz showed that on a surface of genus g (>1) there can be at most (g-1)*84 automorphisms. u This limit is reached for genus 3. u It cannot be reached for genus 4, 5, 6. u It can be reached again for genus 7.

Genus 3 and Genus 7 Canvas tetrahedral frame octahedral frame genus 3, 24 heptagons genus 7, 72 heptagons 168 automorphisms 504 automorphisms

Decorated Junction Elements 3-way junction 4-way junction 6 heptagons 12 heptagons

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

EIGHT 3-way Junctions u 336 Butterflies on a surface of genus 5. u Pretty, but NOT globally regular !

The 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 Models

Fundamental Domain for Genus-7 Case u A cluster of 72 heptagons gives full coverage for a surface of genus-7. u This regular hyperbolic tiling can be continued with infinitely many heptagons in the limit circle.

Genus-7 Paper Models 7-fold symmetry

The Embedding of the 1 8 -fold Way still eludes me. Perhaps at G4G1 8 in … Let’s do something pretty with the OCTA - frame: a {5,4} tiling

Genus 7 Surface with 60 Quads u Convenient to create smooth subdivision surface based on octahedral frame

{5,4} Starfish Pattern on Genus-7 u Start with 60 identical black&white quad tiles: u Color tiles consistently around joint corners u Switch to dual pattern: > 48 pentagonal starfish

Create a Smooth Subdivision Surface u Inner and outer starfish prototiles extracted, u thickened by offsetting, u sent to FDM machine...

EIGHT Tiles from the FDM Machine

White Tile Set -- 2 nd of 6 Colors

2 Outer and 2 Inner Tiles

A Whole Pile of Tiles...

The Assembly of Tiles Begins... Outer tiles Inner tiles

Assembly (cont.): 8 Inner Tiles u Forming inner part of octa-frame arm

Assembly (cont.) u 2 Hubs u + Octaframe edge 12 tiles inside view 8 tiles

About Half the Shell Assembled

The Assembled Genus-7 Object

S P A R E S

72 Lizards on a Genus-7 Surface