Bridges 2007, San Sebastian Symmetric Embedding of Locally Regular Hyperbolic Tilings Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

Goal 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

How to Make an Escher Tiling u Start from a regular tiling u Distort all equivalent edges in the same way

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)

“168 Butterflies,” D. Dunham (2002) Locally regular {3,7} tiling on a genus-3 surface made from 56 isosceles triangles “snub-tetrahedron”

E. Schulte and J. M. Wills u Also: 56 triangles, meeting in 24 valence-7 vertices. u But: Globally regular tiling with 168 automorphisms! (topological)

Generator for {3,7} Tilings on Genus-3 u Twist arms by multiples of 90 degrees...

Dehn Twists u Make a closed cut around a tunnel (hole) or around a (torroidal) arm. u Twist the two adjoining “shores” against each other by 360 degrees; and reconnect. u Network connectivity stays the same; but embedding in 3-space has changed.

Fractional Dehn Twists u If the network structure around an arm or around a hole has some periodicity P, then we can apply some fractional Dehn twists in increments of 360° / P. u This will lead to new network topologies, but may maintain local regularity.

Globally Regular {3,7} Tiling u From genus-3 generator (use 90° twist) u Equivalent to Schulte & Wills polyhedron

u 56 triangles u 24 vertices u genus 3 u globally regular u 168 automorph. Smoothed Triangulated Surface

Generalization of Generator u Turn straight frame edges into flexible tubes

From 3-way to 4-way Junctions Tetrahedral hubs 6(12)-sided arms

6-way Junction + Three 8-sided Loops

Construction of Junction Elements 3-way junction construction of 6-way junction

Junction Elements Decorated with 6, 12, 24, Heptagons

Assembly of Higher-Genus Surfaces Genus 5: 8 Y-junctions Genus 7

Genus-5 Surface (Cube Frame) u 112 triangles, 3 butterflies each ...

336 Butterflies

Creating Smooth Surfaces 4-step process: u Triangle mesh u Subdivision surface u Refine until smooth u Texture-map tiling design

Texture-Mapped Single-Color Tilings u subdivide also texture coordinates u maps pattern smoothly onto curved surface.

What About Differently Colored Tiles ? u How many different tiles need to be designed ?

24 Newts on the Tetrus (2006) One of 12 tiles 3 different color combinations

Use with Higher-Genus Surfaces u Lack freedom to assign colors at will !

New Escher Tile Editor u Tiles need not be just simple n-gons. u Morph edges of one boundary... and let all other tiles change similarly!

Escher Tile Editor (cont.) Key differences: u Tiling pattern is no longer just a texture! u Tiles have a well-defined boundary, which is tracked in subdivision process. u This outline can be flood filled with color.

Escher Tile Editor (cont.) u Possible to add extra decorations onto tiles

Prototile Extraction u Flood-fill can also be used to identify all geometry that belongs to a single tile.

Extract Prototile Geometry for RP u Two prototiles extracted and thickened

Generalizing the Generator to Quads u 4-way junctions built around cube hubs u 4-sided prismatic arms

Genus 7 Surface with 60 Quads u No twist

{5,4} Starfish Pattern on Genus-7 u Polyhedral representation of an octahedral frame u 108 quadrilaterals (some are half-tiles) u 60 identical quad tiles: u Use dual pattern: u 48 pentagonal starfish

Only Two Geometrically Different Tiles u Inner and outer starfish prototiles extracted, u thickened by offsetting, u sent to FDM machine...

Fresh from the FDM Machine

Red Tile Set -- 1 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 edge

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

More Assembly Steps

Assembly Gets More Difficult

Almost Done...

The Finished Genus-7 Object u... I wish... u “work in progress...”

What about Globally Regular Tilings ? So far: u Method and tool set to make complex, locally regular tilings on higher-genus surfaces.

BRIDGES, London, 2006 “Eight-fold Way” by Helaman Ferguson

Visualization of Klein’s Quartic in 3D 24 heptagons on a genus-3 surface; 24x7 automorphisms 24x7 automorphisms (= maximum possible)

Another View fish

Why Is It Called: “Eight-fold Way” ? u Since it is a regular polyhedral structure, it has a set of Petrie Polygons. u These are “zig-zag” skew polygons that always hug a face for exactly 2 consecutive edges. u On a regular polyhedron you can start such a Petrie polygon from any vertex in any direction. (A good test for regularity !) u On the Klein Quartic, the length of these Petrie polygons is always eight edges.

Why Is It “Special” u The Klein quartic has the maximal number of automorphisms possible on a genus-3 surface. u A. Hurwitz showed: Upper limit is: 84(genus-1) u Can only be reached for genus 3, 7, 14,...  Temptation to try to explore the genus-7 case

My Original Plan for Bridges 2007 u Explore the genus-7 case u Make a nice sculpture model in the spirit of the “8-fold Way” u 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)

Poincaré Disc u Find some numbering that repeats periodically and produces the proper Petrie length.

Step 2: What Shape to Choose ?

Tubular Genus-7 Surfaces 12 x 3-way 6 x 4-way 3 x 6-way

Symmetrical {3,7} Maps on Genus-7 OptionJunctn valence Junctn count Junction triangles Arm prism # Arm count Arm triangles Overall triangles A –prism 3-sided B –tetra C D –cube E –octa F

Genus-7 Paper Models

Genus-7 Styrofoam Models

 Try Something Simpler First ! u Banff 2007 Workshop “Teaching Math …”

Globally Regular Tiling With 24 Pentagons u Thanks to David Richter ! Actual cardboard model

The Dodeca-Dodecahedron u 6 sets of 4 parallel faces: u 2 large pentagons + 2 smaller pentagrams

Locally Regular Maps {4,5} and {5,4} u Dual coverage of a genus 4 surface: u 30 quadrilaterals versus 24 pentagons PP > 6

Escher Tiling u With texture mapping

Another Repetitive Texture... u 3 Fish

Looking for the Globally Regular Tiling u Try to find a suitable network by applying fractional Dehn twists to the “spokes”. u Use the same amount on all arms to maintain 4-fold rotational symmetry.

Other Shapes Studied Lawson surface --- “Prism +4 handles”

Experimets u Apply fractional Dehn twists to all these structures, u check for proper length of Petrie polygon.  No success with any of them... 

Inspiration from Symmetry... u Look for shapes that have 3-fold and 4-fold symmetries...

Truncated Octahedron 1 st try: Four hexagonal prismatic tunnels Try different fractional Dehn twists in tunnels

Checking Globally Regularity u Transfer connectivity and coloring pattern  No cigar ! These six vertices are the same as the ones on the bottom

Inspiration from 8-fold Way u On Tetrus: Petrie polygons zig-zag around arms  Let Petrie polygons zig-zag around tunnel walls It works !!!

Add a Nice Coloring Pattern u Use 5 colors u Every color is at every vertex u Every quad is surrounded by the other 4 colors

Conclusions u I have not yet found my “Holy Grail” u Gained insight about locally regular tilings u Used “multi media” in my explorations Remaining question: u what are good ways to find the desired mapping to a symmetrical embedding ? u How does one search / test for global graph regularity ?

Thanks to u David Richter {S 5 dodedadodecahedron} u John M. Sullivan {feedback on paper} u Pushkar Joshi (graduate student) u Allan Lee, Amy Wang (undergraduates)

Questions ? ?