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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
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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
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How to Make an Escher TilingHow to Make an Escher Tiling
Start from a regular tiling
Distort all equivalent edges in the same way
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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)
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““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”
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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)
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Generator for {3,7} Tilings on Genus-3Generator for {3,7} Tilings on Genus-3
Twist arms by multiples of 90 degrees ...
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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.
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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.
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Globally Regular {3,7} TilingGlobally Regular {3,7} Tiling
From genus-3 generator (use 90° twist)
Equivalent to Schulte & Wills polyhedron
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56 triangles
24 vertices
genus 3
globally regular
168 automorph.
Smoothed Smoothed Triangulated Triangulated SurfaceSurface
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Generalization of GeneratorGeneralization of Generator
Turn straight frame edges into flexible tubes
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From 3-way to 4-way JunctionsFrom 3-way to 4-way Junctions
Tetrahedral hubs
6(12)-sided arms
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6-way Junction + Three 8-sided Loops6-way Junction + Three 8-sided Loops
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Construction of Junction ElementsConstruction of Junction Elements
3-way junction
construction of
6-way junction
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Junction Elements Junction Elements Decorated with Decorated with 6, 12, 24, Heptagons6, 12, 24, Heptagons
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Assembly of Higher-Genus SurfacesAssembly of Higher-Genus Surfaces
Genus 5:8 Y-junctions
Genus 7
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Genus-5 Surface (Cube Frame)Genus-5 Surface (Cube Frame)
112 triangles, 3 butterflies each . . .
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Creating Smooth SurfacesCreating Smooth Surfaces
4-step process:
Triangle mesh
Subdivision surface
Refine until smooth
Texture-map tiling design
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Texture-Mapped Single-Color Tilings
subdivide also texture coordinates
maps pattern smoothly onto curved surface.
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What About Differently Colored Tiles ?What About Differently Colored Tiles ?
How many different tiles need to be designed ?
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24 Newts on the Tetrus (2006)24 Newts on the Tetrus (2006)
One of 12 tiles
3 different color combinations
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Use with Higher-Genus SurfacesUse with Higher-Genus Surfaces
Lack freedom to assign colors at will !
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New Escher Tile Editor
Tiles need not be just simple n-gons.
Morph edges of one boundary . . .and let all other tiles change similarly!
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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.
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Escher Tile Editor (cont.)Escher Tile Editor (cont.)
Possible to add extra decorations onto tiles
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Prototile Extraction
Flood-fill can also be used to identify all geometry that belongs to a single tile.
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Extract Prototile Geometry for RPExtract Prototile Geometry for RP
Two prototiles extracted and thickened
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Generalizing the Generator to QuadsGeneralizing the Generator to Quads
4-way junctions built around cube hubs
4-sided prismatic arms
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Genus 7 Surface with 60 QuadsGenus 7 Surface with 60 Quads
No twist
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{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
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Only Two Geometrically Different TilesOnly Two Geometrically Different Tiles
Inner and outer starfish prototiles extracted,
thickened by offsetting,
sent to FDM machine . . .
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Fresh from the FDM MachineFresh from the FDM Machine
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Red Tile Set -- 1 of 6 ColorsRed Tile Set -- 1 of 6 Colors
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2 Outer and 2 Inner Tiles2 Outer and 2 Inner Tiles
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A Whole Pile of Tiles . . .A Whole Pile of Tiles . . .
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The Assembly of Tiles Begins . . .The Assembly of Tiles Begins . . .
Outer tiles
Inner tiles
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AssemblyAssembly(cont.):(cont.):
8 Inner Tiles8 Inner Tiles
Forming inner part of octa-frame edge
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Assembly (cont.)Assembly (cont.) 2 Hubs
+ Octaframe edge
12 tiles inside view
8 tiles
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More Assembly StepsMore Assembly Steps
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More Assembly StepsMore Assembly Steps
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Assembly Gets More DifficultAssembly Gets More Difficult
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Almost Done ...Almost Done ...
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The Finished Genus-7 ObjectThe Finished Genus-7 Object
. . . I wish . . .
“work in progress . . .”
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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.
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BRIDGES, London, 2006BRIDGES, London, 2006
“Eight-fold Way” by Helaman Ferguson
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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)
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AnotherAnother View ... View ...
168 fish
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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.
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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
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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)
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PoincarPoincaréé DiscDisc
Find some numbering that repeats periodically and produces the proper Petrie length.
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Step 2: What Shape to Choose ?Step 2: What Shape to Choose ?
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Tubular Genus-7 SurfacesTubular Genus-7 Surfaces
12 x 3-way 6 x 4-way 3 x 6-way
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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
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Genus-7Genus-7Paper ModelsPaper Models
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Genus-7 Styrofoam ModelsGenus-7 Styrofoam Models
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Try Something Simpler First !Try Something Simpler First !
Banff 2007 Workshop “Teaching Math …”
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Globally Regular Tiling With 24 PentagonsGlobally Regular Tiling With 24 Pentagons
Thanks to David Richter !
Actual cardboard model
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The Dodeca-DodecahedronThe Dodeca-Dodecahedron
6 sets of 4 parallel faces:
2 large pentagons + 2 smaller pentagrams
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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
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Escher Escher TilingTiling
With texture mapping
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Another Repetitive Texture ...Another Repetitive Texture ...
3 Fish
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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.
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Other Shapes StudiedOther Shapes Studied
Lawson surface --- “Prism +4 handles”
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ExperimetsExperimets
Apply fractional Dehn twists to all these structures,
check for proper length of Petrie polygon.
No success with any of them ...
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Inspiration from Symmetry . . . Inspiration from Symmetry . . .
Look for shapes that have 3-fold and 4-fold symmetries . . .
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Truncated OctahedronTruncated Octahedron
1st try: Four hexagonal prismatic tunnels
Try different fractional Dehn twists in tunnels
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Checking Globally RegularityChecking Globally Regularity
Transfer connectivity and coloring pattern
No cigar !
These six vertices are the same as the ones on the bottom
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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 !!!
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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
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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 ?
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Thanks toThanks to
David Richter {S5 dodedadodecahedron}
John M. Sullivan {feedback on paper}
Pushkar Joshi (graduate student)
Allan Lee, Amy Wang (undergraduates)
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Questions ?Questions ?
?