Bridges 2006, London

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Bridges 2006, London

• Patterns on Kleins Quartic

Carlo H. Séquin EECS Computer Science
Division University of California, Berkeley
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What is Kleins Quartic?

• In 1878 Felix Klein discovered that the equation
  x3 y y3 z z3 x 0 (in complex projective
  coordinates)has 336-fold symmetry under the
  group of fractional linear transforms whose
  coefficients are integers and which reduce to
  the identity modulo 7.

S. Levy, in The Eightfold Way pg. ix
336 identical triangles
connect!
Hyperbolic Plane by Daina Taimina.
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Visualization of Kleins Quartic

• Project into 3D space? genus-3 surface,with 24
  heptagons.
• Divide each into 14 triangles(24 14 336).

connect!
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A Regular Graph on a Genus-3 Surface

• This symmetry group can best be visualized as a
  regular polyhedral configuration in hyperbolic
  space,consisting of 24 regular heptagons,
  joining in 56 valence-3 vertices.
• It cannot be rendered in 3-space without loosing
  most of its symmetries.
• Retain 12 symmetries in a tetrahedral shape,or
  24, if we allow self-intersections or make the
  heptagonal facets pliable (quilt!).

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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 regular polyhedron.

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Our Canvas Genus-3 Surface Tetrus

• Rendering by Hayley Iben

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Polyhedral Approximation Tetroid

• 24 non-planar heptagons 56 vertices

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Its Dual with 56 Triangles, 24 Vertices

• 12 symmetries,
• No self-intersections !

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Book (1993)

• Most important object in mathematics ...

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Eight-fold Way by Helaman Ferguson

• At MSRI on the U C Berkeley Campus (1993)

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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.

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Petrie Polygons
Zig-zag skew polygons on regular polyhedrons

• on Cube Dodecahedron Klein Quartic
• 4 (L6) 3/6 (L10)
  11/21 (L8) .

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Equatorial Weave on Dodecahedron
(Planarize the Petrie polygon, make them go up
down)

• The Icosa-Dodecahedral Dog Toy
• Gift from Marc Pelletier ? Inspired the
  following study!

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Equatorial Weave Based on 10-fold Way

• You can find this in many museum stores

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All 21 Petrie Polygons on Klein Quartic

• 6 Yellowarm loops
• 3 WhiteBorromean loops
• 12 R-G-Bshoulder loops
• all of length 8

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Equatorial Loops on Klein Quartic

• Planarize the Petrie polygons by connecting
  edge-mid-points

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My Virtual Hyperbolic Dog Toy

• 6 Yellowarm loops
• 3 WhiteBorromean loops
• 12 R-G-Bshoulder loops

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Hyperbolic Dog Toy placed on Tetrus
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Construction of Hyperbolic Dog Toy

• Partial assembly 6 Y 3 W 6 shoulder loops

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Hyperbolic Dog Toy (cont.)

• Assembly of 18 loops.

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Hyperbolic Dog Toy

• Complete assembly of all 21 loops.

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Focus of this Talk

• Since the Klein Quartic has all the properties
  of yet another Platonic solid, I can now perform
  many experiments on this special structure that I
  have previously played on the 3- and
  4-dimensional regular polytopes, and reported
  about in previous conferences.

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Topics Mentioned in Paper

• (1) Introduction of Klein Quartic
• (2) Regular Two-Manifold Meshes
• (3) Equatorial Weaves
• (4) Knot and Link Embeddings
• (5) Hamiltonian Cycles
• (6) Viae Tetrus (Ribbons)
• (7) Symmetrical Graph Embeddings
• (8) Regular Tilings and Colorings
• (9) Escher-like Tilings
• (10) Knot Tangles Dissection Puzzles

Each one of these topics turned out to be far
deeper and more difficult than first
anticipated. Each one of them could easily be
expanded into a complete 25-minute BRIDGES talk !
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Topics in Paper ? Outline of Talk

• (1) Introduction of Klein Quartic ?
• (2) Regular Two-Manifold Meshes ?
• (3) Equatorial Weaves ?
• (5) Hamiltonian Cycles
• (6) Viae Tetrus (Ribbons)
• (8) Regular Tilings and Colorings
• (9) Escher-like Tilings
• (7) Symmetrical Graph Embeddings
• (4) Knot and Link Embeddings
• (10) Yin-Yang Dissections
• (10) Borromean Rubber Bands, Knot Tangles

These use the graph structure of the Klein
symmetry group
Just use genus-3 canvas
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What else can we do with the graph structure of
this Klein group ?

• PART I

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CHAPTER 5 HAMILTONIAN CYCLES
(Visit all vertices exactly once)

• Such cycles exist on all Platonic solids !

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Hamiltonian Path on Tetroid Graph ?

• Can we find a Hamiltonian Cycle on the edges of
  the Klein Tetroid (using 56 out of 84 edges) ?
• What maximal symmetry can we obtain ?

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Hamiltonian Cycle on Klein Tetroid
Graph of all edges
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Hamiltonian Cycle 3D View
Vasily Volkov
On edge graph and on a smooth tetrus

• ? Let edges contract like a rubber band to form
  a smooth geodesic line.

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Is This Hamiltonian Cycle Knotted ?

• On the Platonic solids all Hamiltonian cycles
  represent the un-knot.
• They would shrink to infinitely small loops when
  realized as rubber bands.
• On higher genus surfaces the situation is more
  interesting!
• We have already seen that the geodesic line does
  not collapse.
• -- but is it knotted ?
• -- does it partition the surface ?

NO !NO !
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Surface Depicted by Hamiltonian Path

• For most Platonic solids, a tube following the
  Hamiltonian path gives a good idea of the
  overall shape (spans convex hull !).
• Is this also true for the Tetroid structure ?

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Does the Hamiltonian path fairly depict the
underlying surface ?

• Doesnt work for Tetrus Structure !
• Need a different approach ... ?

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CHAPTER 6 VIAE TETRUS(Roads on the Planet
Tetrus)

• Define a surface with a single ribbon suitably
  draped over it.
• Hamiltonian cycle on Klein Tetroid does not do a
  good job !
• Inspiration from ...

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M.C. Escher Bond of Union
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Viae Globi - Pathways on a Sphere Mathematics
and Design, Geelong, 2001.
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Virtual Design of Via Tetrus 1

• Issues
• Single ribbonYES !
• Full tetrahedral symmetry is NOT possible.
• C2 symmetry,using 2 kinds of cork screws.

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Limited Symmetry

• Multi-Eulerian path with 2 strands per arm.
• Tetrahedral symmetry is NOT possible such a
  cyclic path.
• Only C3 symmetry(around central vertex).

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Design of Via Tetrus
C2 axis
C2 axis

• Use 4 spiral tracks per arm
• the black tracks spiral thru 270,
• the red tracks spiral thru 180.

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Checking Connectivity
a 270 exchange
a 180 exchange

• Make sure there is exactly one loop !

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Design of Via Tetrus

• 3D mock-up of basic topology

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Making FDM Prototype

• Need to remove the (gray) scaffolding ...

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Via Tetrus
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CHAPTER 8 Tilings Colorings

• 24 Heptagons on a Tetrus
• At every vertex 3 heptagons join together
• Such a tiling is not possible in the plane !

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Heptagonal Tiling ...
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Poincaré Disk

• This regular hyperbolic tiling can be continued
  with infinitely many heptagons in the limit
  circle.
• A cluster of 24 heptagons gives full coverage of
  the genus-3 tetrus surface.

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Color Patterns

• How few colors can do we need to color adjacent
  heptagons differently ?
• At least 4 one for a chosen heptagon, and 3
  more to go around its perimeter.
• Number should be integer divisor of 24for nice
  symmetrical solutions.
• ? 4, 6, 8 are interesting contenders !

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8-color Tetroid

• Ea. hept. surrounded by the 7 other colors
• All 8 colors along the sides of Petrie polygon
• All possible triplets at the 56 vertices.

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6-color Tetroid

• Each color appears on 2 inner and 2 outer
  heptagons and once at every hub, either inside or
  outside.

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4-Color Tetroid

• Pairing of tiles always one inner and one
  outer
• ? 12 identical double tiles !

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Coloring the Dual with 56 Triangles

• Good contenders are (4?), 7, 8 colors

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7-color Dual Tetroid

• All seven colors show up around every vertex.
• One color for inner and outer hub caps.
• On the six arms one of the six other colors shows
  twice.

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8-color Dual Tetroid

• At each vertex one color is missing
• Each color shows up at one hub (in- or outside).

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Towards Archimedean Tetroids

• Blend between icosahedron dodecahedron ?
  Triacontahedron (30 rhombuses).
• Make a Tetrus with 84 quads across edges.
• 84 3x4x7 ? possible color numbers.

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3-color Archimedean Tetroid

• Vasarely look

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Archimedean Tetroid with 84 Quads
80 vertices 24 valence 3 56 valence 7

• Smoothly stretched over Tetrus surface

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CHAPTER 9 Escher Tilings

• in the plane on the sphere
  on the torus
• M.C. Escher Jane Yen, 1997
  Young Shon, 2002

Can we do this on the tetrus ?
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How To Make an Escher Tiling

• Start from a regular tiling,
• Distort all equivalent edges in the same way.

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Distorting the Fundamental Domain

• Distort the edges of a fundamental region so as
  to maintain all its symmetries
• C2 symmetry around the edge midpoints,
• C7 symmetry around the tile center, and
• C3 symmetry around the vertices.
• Distorted tiles will fit together and seamlessly
  cover the whole surface.

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Creating the Heptagonal Fish Tile
FundamentalDomain
DistortedDomain

• How do we get this onto the Tetrus shape ?

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Infinite Tiling on the Poincaré Disk

• Use texture mapping !

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Escher Fish Tiles on the Tetroid

• But how do we make this smooth ?

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Bad Subdivision Based on Heptagons

• The skewed connectivity of a Tetroid built from
  the Klein heptagons doesnt yield a smooth
  subdivision surface.

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Tetroid with 48 Quadrilaterals

• Used as control polygon for CC subdivision ...
• Yields a smooth Tetrus shape.

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48 Birds on the Tetrus
Escher tilingbased on square

• One of 12 tiles,covering 4 quads

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24 Newts on the Tetrus

• One of 12 tiles

Three different tilesfor richer colors
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Defining the Tetroid for CC-Subdivision

• Polyhedron builtwith 60 quads
• yields smooth CC subdivision surface,
• is easy to tile with minimal distortion
• but does NOT represent the desired symmetry
  group

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Tiles with the Proper Connectivity

• What a texture tile needs to look like to
  produce this pattern on the 48-quad Tetroid

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2-Step Texture Transfer

• Map 2 heptagons into square,
• 7 fish into each stretched heptagon

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Pre-distorted Fish Tile on Tetrus

• 1 tile over 4 subdiv quads

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Another View ...
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Combining Pairs of Oriented Tiles
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Another Double-Heptagon Tile

• Color marks show how edges must match up

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5 Double Tiles Assembled
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More Work To Be Done !

• Ways to design interesting Tetrus tiles
• Finding appropriate coloring schemes
• Ways to fabricate these tiles

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P A R T II Our Canvas Smooth Genus-3 Tetrus

• Rendering by Hayley Iben

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Snake-Skin Synthesis by Steven An

• Grown pixel by pixel from a small sample

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Knitted Tetrus by Chen Shen

• Fused Deposition Model

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CHAPTER 7 GRAPH EMBEDDINGS

• A genus-3 Tetrus allows nice symmetrical
  embeddings of certain non-planar graphs.

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Embedding of K4,4,4 (Dycks Map)
( ISAMA 2004 )

• 12 nodes (vertices),
• 48 links (edges),
• 32 triangular facets.
• E V F 2 2Genus
• 48 12 32 2 6 ? Genus 3

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Find Locations for Nodes

• For every node there are three that it is not
  connected to.
• Place them so that themissing edges do not
  break the symmetry
• ? Inside and outside on each tetra-arm.
• Do not connect the nodes that lie on thesame
  symmetry axis(same color)(or this one).

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K4,4,4 on a Genus-3 Surface

• Graph edges plus colored facets

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A Physical Model
3D-Print model enamel paint
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A Virtual Genus-3 Tiffany Lamp
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Light Cast by Genus-3 Tiffany Lamp

• Rendered by Ling Xiao with Radiance Ray-Tracer
  (12 hours)

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CHAPTER 4 KNOT EMBEDDINGS

• What (interesting) knots can be embedded in a
  genus-3 surface ?
• Which is the simplest knot that needs at least a
  genus-3 surface ?

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A Primer on Knot Theory

• NOT a knot Un-knot Trefoil
  knot

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Table of Simplest Knots

• Colin Adams The Knot Book
• A Knot Zoo http//www.pims.math.ca/knotplot/zoo/z
  oo1.html

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Analysis of Knot 63

• What genus is needed to embed this knot ?

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Knot 63

• I thought this needed a genus-3 surface ...

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My Physical Playground

• work with string, sticky tape ...

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Understanding Knot 63

• Later found out
• This is a rational knot
• All rational knots have Bridge Number 2
• Embedding genus is always ? Bridge Number
• ? thus 63 is embeddable in a genus-2 surface
  !BUT HOW ??

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Fitting Knot 63 onto 2-Handle Shape

• Unwind the brown loops

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Partitioning the Surface

• Try to split the surface into two parts with a
  link/knot that cannot be embedded in a surface
  of lower genus.
• The domain must have multiple boundaries.
• This cannot be a single knot.

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Table of Simplest Links

• Colin Adams The Knot Book
• A Knot Zoo http//www.pims.math.ca/knotplot/zoo/z
  oo1.html

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Link 622

• Partitions the genus-3 surface into 2 domains

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Link 622 Redrawn with 7 Crossings

• Can also be fit on a genus-2 surface !!
• ? More on this at a future BRIDGES conference.

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CHAPTER 10 DECOMPOSITIONS

• Two Platonics solids split into congruent parts.
• How can the tetrus / tetroid be partitioned ?

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Puzzles Based on Heptagon Tiling

• 12 inner and 12 outer heptagon tiles

• 12 identical double heptagon tiles

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CS 285 Design Problem 3D Yin-Yang
Do this in 3D !

• What this might mean ...
• Subdivide a sphere into two halves.

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3D Yin-Yang (Sphere Dissection) BRIDGES,
Winfield KS, 1999.
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Decomposition Into More Than 2 Parts
In Korea, the 3-part taeguk symbolizes heaven,
earth, and humanity.
Found on Internet
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Yin-Yang Tetrus Decomposition ?

• Split object into four wavy parts, inspired by
  2D Yin-Yang figure!
• Each part is a hub with 3 tentacles reaching out
  along each of the 3 attached tetrahedral edges.

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Tetrus Dissection into 4 Parts

• CS 285 Project of Catherine Bendebury
  (2006)(third try)

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4-Way Tetrus Yin-Yang Assembled
NEEDS MORE WORK !

• Catherine Bendebury (2006)

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CHAPTER X BORROMEAN BANDS

• Borromean Rings
• No two loops interlink !

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Linkage of the Borromean Rubberbands

• 6 rubberbands always 3 join in a vertex

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Borromean Rubberbands on Tetrus

• 6 edges, 4 vertices, symmetrically placed on
  tetrus surface
• How can this be done ?

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CHAPTER 10 KNOT TANGLES

• 12 Pentafoils describing a Sphere(ArtMathX,
  Boulder, CO, 2005)

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Same Principle Applied to Klein Quartic

• Distortions on tetrus distroy that beauty !

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A Special Knot to Fit the Heptagons

• Distortions still cause implementation
  difficulties ...

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Poincaré Double-Lace
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Final Remarks

• Many loose ends !
• More on knot embeddings, tangles, dissections,
  ...at future Bridges conferences ...