Art and Math Behind and Beyond the 8-fold Way

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Art and MathBehind and Beyondthe 8-fold Way
Carlo H. Séquin
EECS Computer Science Division University of
California, Berkeley
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Art, Math, Magic, and the Number 8 ...

• Eightfold Way at MSRI by Helaman Ferguson

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The Physicists Eightfold Way
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The Noble Eightfold Path

• -- The way to end suffering (Siddhartha Gautama)

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Siddhartha Gautama
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Helaman Fergusons The Eightfold Way

• 24 (lobed) heptagons on a genus-3 surface

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Visualization of Kleins Quartic in 3D

• 24 heptagons
• on a genus-3 surface
• a totally regular graph
• with 168 automorphisms

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24 Heptagons Forced into 3-Space
Quilt by Eveline Séquin(1993), based on a
pattern obtained from Bill Thurstonturns
inside-out !

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

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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.

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Petrie Path on Poincaré Disk

• Exactly eight zig-zag moves lead to the same
  place

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My Long-standing Interest in Tilings
in the plane on the sphere
on the torus M.C. Escher
Jane Yen, 1997 Young Shon, 2002

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

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Lizard Tetrus (with Pushkar Joshi)

• Cover of the 2007 AMS Calendar of Mathematical
  Imagery

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24 Lizards on the Tetrus

• One of 12 tiles

3 different color combinations
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Hyperbolic 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
  dont exhibit 7-fold symmetry!

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Decorating 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.

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Creating the Heptagonal Fish Tile
FundamentalDomain
DistortedDomain

• Fit them together to cover the whole surface ...

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Infinite Tiling on the Poincaré Disk
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Genus 3Surface with168 fish

• Every fish can map onto every other fish.

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The Dual Surface

• 56 triangles
• 24 vertices
• genus 3
• globally regular
• Petrie polygons of length 8

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Why is this so special ?

• A whole book has been written about it(1993).
• The most important object in mathematics ...

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Maximal Amount of Symmetry

• Hurwitz showed that on a surface of genus g (gt1)
  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.

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Genus 3 and Genus 7 Canvas

• tetrahedral frame
  octahedral frame
• genus 3 , 24 heptagons genus 7,
  72 heptagons
• 168 automorphisms 504
  automorphisms

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Decorated Junction Elements

• 3-way junction 4-way
  junction
• 6 heptagons 12 heptagons

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Assembly of Genus-7 Surface

• Join zig-zag edges Genus 7
  surfaceof neighboring arms six 4-way
  junctions

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EIGHT 3-way Junctions

• 336 Butterflies on a surface of genus 5.
• Pretty, but NOTglobally regular !

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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 ?

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Genus-7 Styrofoam Models
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Fundamental 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.

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Genus-7Paper Models
7-fold symmetry
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The Embedding ofthe 18-fold Waystill eludes me.
Perhaps at G4G18 in 2028
Lets do something pretty with the OCTA -
frame a 5,4 tiling

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Genus 7 Surface with 60 Quads

• Convenient to create smooth subdivision surface
  based on octahedral frame

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5,4 Starfish Pattern on Genus-7

• Start with 60 identical blackwhite quad tiles

• Color tiles consistently around joint corners
• Switch to dual pattern
• gt 48 pentagonal starfish

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Create a Smooth Subdivision Surface

• Inner and outer starfish prototiles extracted,
• thickened by offsetting,
• sent to FDM machine . . .

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EIGHT Tiles from the FDM Machine
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White Tile Set -- 2nd of 6 Colors
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2 Outer and 2 Inner Tiles
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A Whole Pile of Tiles . . .
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The Assembly of Tiles Begins . . .

• Outer tiles

Inner tiles
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Assembly(cont.)8 Inner Tiles

• Forming inner part of octa-frame arm

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Assembly (cont.)
8 tiles

• 2 Hubs
• Octaframe edge

inside view
12 tiles
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About Half the Shell Assembled
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The Assembled Genus-7 Object
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S P A R E S
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72 Lizards on a Genus-7 Surface