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Symmetry Festival, Aug. 7, 2013Symmetry Festival, Aug. 7, 2013
Symmetrical Immersions of Low-Genus Non-Orientable Regular Maps
(Inspired Guesses followed by Tangible Visualizations)(Inspired Guesses followed by Tangible Visualizations)
Carlo H. SCarlo H. Sééquinquin
EECS Computer Science DivisionEECS Computer Science DivisionUniversity of California, BerkeleyUniversity of California, Berkeley
A Very Symmetrical Object in R3A Very Symmetrical Object in R3
The Sphere
The Most Symmetrical PolyhedraThe Most Symmetrical Polyhedra
The Platonic Solids = Simplest Regular Maps
{3,4} {3,5}
{3,3}
{4,3} {5,3}
The Symmetry of a Regular MapThe Symmetry of a Regular Map
After an arbitrary edge-to-edge move, every edge can find a matching edge;the whole network coincides with itself.
All the Regular Maps of Genus ZeroAll the Regular Maps of Genus Zero
Platonic Solids Di-hedra (=dual)
Hosohedra
{3,4}
{3,5}
{3,3}
{4,3}
{5,3}
Background: Geometrical TilingBackground: Geometrical Tiling
Escher-tilings on surfaces with different genus
in the plane on the sphere on the torus
M.C. Escher Jane Yen, 1997 Young Shon, 2002
Tilings on Surfaces of Higher GenusTilings on Surfaces of Higher Genus
24 tiles on genus 3 48 tiles on genus 7
Two Types of Two Types of ““OctilesOctiles””
Six differently colored sets of tiles were used
From Regular Tilings to From Regular Tilings to Regular MapsRegular Maps
When are tiles “the same” ?
on sphere: truly identical from the same mold
on hyperbolic surfaces topologically identical(smaller on the inner side of a torus)
Tilings should be “regular” . . .
locally regular: all p-gons, all vertex valences q
globally regular: full flag-transitive symmetry(flag = combination: vertex-edge-face) Regular Map
On Higher-Genus Surfaces:On Higher-Genus Surfaces:only only ““TopologicalTopological”” Symmetries Symmetries
Regular map on torus (genus = 1)
NOT a regular map: different-length edge loops
Edges must be able to stretch and compress
90-degree rotation not possible
NOT a Regular MapNOT a Regular Map
Torus with 9 x 5 quad tiles is only locally regular.
Lack of global symmetry:Cannot turn the tile-grid by 90°.
This IS a Regular MapThis IS a Regular Map
Torus with 8 x 8 quad tiles.Same number of tiles in both directions!
On higher-genus surfaces such constraints apply to every handle and tunnel.Thus the number of regular maps is limited.
How Many Regular Maps How Many Regular Maps on Higher-Genus Surfaces ?on Higher-Genus Surfaces ?
Two classical examples:
R2.1_{3,8} _1216 triangles
Quaternion Group [Burnside 1911]
R3.1d_{7,3} _824 heptagons
Klein’s Quartic [Klein 1888]
NomenclatureNomenclature
R3.1d_{7,3}_8
Regular mapgenus = 3# in that genus-groupthe dual configurationheptagonal facesvalence-3 verticeslength of Petrie polygon:
Schläfli symbol {p,q}
“Eight-fold Way”
zig-zag path closes after 8 moves
2006: Marston Conder2006: Marston Conder’’s Lists List http://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txt
6104 Orientable regular maps of genus 2 to 101:
R2.1 : Type {3,8}_12 Order 96 mV = 2 mF = 1 Defining relations for automorphism group: [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, (R * S^-3)^2 ]
R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ]
R2.3 : Type {4,8}_8 Order 32 mV = 8 mF = 2 Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^-2 * R^2 * S^-2 ]
R2.4 : Type {5,10}_2 Order 20 mV = 10 mF = 5 Defining relations for automorphism group: [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^-5 ]
= “Relators”
R2.2_{4,6}_12 R3.6_{4,8}_8
““Low-Hanging FruitLow-Hanging Fruit””Some early successes . . .
R4.4_{4,10}_20 and R5.7_{4,12}_12
A Tangible Physical ModelA Tangible Physical Model
3D-Print, hand-painted to enhance colors
R3.2_{3,8}_6
Genus 5Genus 5
{3,7}
336
Butterflies
Only locallyregular !
Globally Regular Maps on Genus 5Globally Regular Maps on Genus 5
Emergence of a Productive ApproachEmergence of a Productive Approach
Depict map domain on the Poincaré disk; establish complete, explicit connectivity graph.
Look for likely symmetries and pick a compatible handle-body.
Place vertex “stars” in symmetrical locations.
Try to complete all edge-interconnections without intersections, creating genus-0 faces.
Clean-up and beautify the model.
Depiction on Poincare DiskDepiction on Poincare Disk
Use Schläfli symbol create Poincaré disk.
{5,4}
Relators Identify Repeated LocationsRelators Identify Repeated Locations
Operations:R = 1-”click” ccw-rotation around face center; r = cw-rotation.S = 1-”click” ccw-rotation around a vertex; s = cw-rotation.
R3.4_{4,6}_6
Relator:
R s s R s s
Complete Complete Connectivity Connectivity InformationInformation
Triangles of the same color represent the same face.
Introduce unique labels for all edges.
Low-Genus Handle-BodiesLow-Genus Handle-Bodies
There is no shortage of nice symmetrical handle-bodies of low genus.
This is a collage I did many years ago for an art exhibit.
Numerology, Intuition, …Numerology, Intuition, … Example: R5.10_{6,6}_4
First try:oriented cube symmetry
Second try:tetrahedral symmetry
A Valid Solution for R5.10_{6,6}_4 A Valid Solution for R5.10_{6,6}_4
Virtual model Paper model(oriented tetrahedron) (easier to trace a Petrie polygon)
OUTLINEOUTLINE
Just an intro so far; by now you should understand what regular maps are.
Next, I will show some nice results.
Then go to non-orientable surfaces,which have self-intersections,and are much harder to visualize!
Jack J. van WijkJack J. van Wijk’’s Method (1)s Method (1) Starts from simple regular handle-bodies, e.g.
a torus, or a “fleshed-out”, “tube-fied” Platonic solid.
Put regular edge-pattern on each connector arm:
Determine the resulting edge connectivity,and check whether this appears in Conder’s list.If it does, mark it as a success!
Jack J. van Wijk’s Method (2)Jack J. van Wijk’s Method (2) Cool results: Derived from …
a dodecahedron 3×3 square tiles on torus
Jack J. van WijkJack J. van Wijk’’s Method (3)s Method (3)
For any such regular edge-configuration found, a wire-frame can be fleshed out, and the resulting handle-body can be subjected to the same treatment.
It is a recursive approach that may yield an unlimited number of results; but you cannot predict which ones you will find and which will be missing.
You cannot (currently) direct that system to give you a solution for a specific regular map of interest.
The program has some sophisticated geometrical procedures to produce nice graphical output.
J. van WijkJ. van Wijk’’s s Method (4)Method (4)
Cool results:
Embedding of genus 29
Jack J. van WijkJack J. van Wijk’’s Method (5)s Method (5)
Alltogether by 2010, Jack had found more than 50 symmetrical embeddings.
But some simple maps have eluded this program, e.g. R2.4, R3.3, and: the Macbeath surface R7.1 !
Also, in some cases, the results don’t look as good as they could . . .
Jack J. van WijkJack J. van Wijk’’s Method (6)s Method (6)
Not so cool result for R3.8: too much warping: My solution on a Tetrus:
Jack J. van WijkJack J. van Wijk’’s Method (7)s Method (7) Not so cool results:
too much warping:“Vertex Flower” solution
““Vertex FlowersVertex Flowers”” for for AnyAny Genus Genus
This classical pattern is appropriate for the 2nd-last entry in every genus group.
All of these maps have exactly two vertices and two faces bounded by 2(g+1) edges.
g = 1 g = 2 g = 3 g = 4 g = 5
Some ModelsSome Models
New FocusNew Focus
Now we want to construct such models for non-orientable surfaces, like Klein bottles.
Unfortunately, there exist no regular maps on the Klein bottle !
But there are several regular maps on the simplest non-orientable surface: the Projective Plane.
The Projective PlaneThe Projective Plane
-- Equator projects to infinity.-- Walk off to infinity -- and beyond … come back from opposite direction: mirrored, upside-down !
The Projective Plane is a Cool Thing!The Projective Plane is a Cool Thing!
It is single-sided:Flood-fill paint flows to both faces of the plane.
It is non-orientable:Shapes passing through infinity get mirrored.
A straight line does not cut it apart!One can always get to the other side of that line by traveling through infinity.
It is infinitely large! (somewhat impractical)It would be nice to have a finite model with the same topological properties . . .
Trying to Make a Finite ModelTrying to Make a Finite Model
Let’s represent the infinite plane with a very large square.
Points at infinity in opposite directions are the same and should be merged.
Thus we must glue both opposing edge pairs with a 180º twist.
Can we physically achieve this in 3D ?
Cross-Surface ConstructionCross-Surface Construction
Finite Models of the Projective PlaneFinite Models of the Projective Plane
(and their symmetries)
Cross surface Steiner surface Boy surface
mirror: C2v tetrahedral cyclic: C3
The Hemi-Platonic PolyhedraThe Hemi-Platonic Polyhedra
Cube Octahedron Dodecahedron Icosahedron
Hemi-Cube Hemi-Octa-h. Hemi-Dodeca-h. Hemi-Icosa-h.
Q
Hemi-OctahedronHemi-Octahedron
Make a polyhedral model of Steiner’s surface.
Need 4 copies of this!
Hemi-CubeHemi-Cube
Start with 3 perpendicular faces . . .
Hemi-IcosahedronHemi-Icosahedron
Built on Hemi-cube model
Hemi-DodecahedronHemi-Dodecahedron
Built on Hemi-cube model with suitable face partitioning.
Movie_HemiDodeca.mp4
Hemi-Hosohedra & Hemi-DihedraHemi-Hosohedra & Hemi-Dihedra
All wedge slices pass through intersection line.
N = 12 N = 2 : self-dual
Hemi-Hosohedra with Higher SymmetryHemi-Hosohedra with Higher Symmetry
Get more symmetry by using a cross-surface with a higher-order self-intersection line.
N = 12 N = 60
Regular Maps on Non-Orientable Regular Maps on Non-Orientable Surfaces of Genus-2 and Genus-3Surfaces of Genus-2 and Genus-3
There aren’t any !!
Genus-2: Klein Bottles Genus-3: Dyck’s surface
Low-Genus Non-Orientable Regular MapsLow-Genus Non-Orientable Regular Maps
From: Marston Conder (2012)
A Way to Make A Way to Make AnyAny Surface Surface
A sphere to start with;
A hole-punch to make punctures:Each increases Euler Characteristic by one.
We can fill these holes again with: Disks: Decreases Euler Characteristic by one. {useless!}
Cross-Caps: Makes surface single-sided.
Boy-caps: Makes surface single-sided.
Handles (btw. 2 holes): Orientability unchanged.
Cross-Handles (btw. 2 holes): Makes surface single-sided.
Euler Char. unchanged
Constructing a Surface with Constructing a Surface with χχ = 2 ‒ = 2 ‒ hh Punch h holes into a sphere and close them up with:
handles or cross-handles
cross-caps or Boy caps or
Closing two holes at the same time:
Topological Diagrams for N4.1dTopological Diagrams for N4.1d
Diagrams from: N.S. Weed (2009, 2010);(This saves tedious work that I normally perform on the Poincaré disk.)
Other options: 4 cross-caps on a sphere . . .
n.o.-genus = 4; Euler characteristic = ‒2
N4.1d: 4 hexagons, 6 val-4 vertices, 12 edges, Petrie-length=6,
Concept of a Genus-4 SurfaceConcept of a Genus-4 Surface
4 Boy caps grafted onto a sphere with tetrahedral symmetry.
Regular Maps with Tetrahedral SymmetryRegular Maps with Tetrahedral Symmetry
N4.2: 6 quads,4 val-6 vertices.
N4.2d: 4 hexagons,6 val-4 vertices.
For both: 12 edges, Petrie-length=3.
Genus-4 Surface Using 4 Boy-CapsGenus-4 Surface Using 4 Boy-Caps
Start with a polyhedral representationand smooth it with subdivision:
Genus-4 Surface Using 4 Boy-CapsGenus-4 Surface Using 4 Boy-Caps
(60°rotation between neighbors)
Employ tetrahedral symmetry!
( 0°rotation between neighbors)(one more level of subdivision)
N4.1 RevisitedN4.1 Revisited
N4.1: 6 quads, 4 val-6 vertices , 12 edges, Petrie-length=6.
N4.1d: 4 hexagons, 6 val-4 vertices, 12 edges, Petrie-length=6.
A cross-handle (schematic)
Regular Maps N5.1 and N5.1dRegular Maps N5.1 and N5.1d
N5.1: 15 quads, 12 val-5 vertices,30 edges, Petrie-length=6.
N5.1d: 12 pentagons, 15 val-4 vertices30 edges, Petrie-length=6.
Make a genus-5 surface with (oriented) tetrahedral symmetryby grafting 4 Boy caps onto the bulges of a Steiner surface.
Regular Map N5.3 (self-dual)Regular Map N5.3 (self-dual)
N5.3: 6 pentagons, 6 val-5 vertices, 15 edges, Petrie-length=3.
Unfolded Steiner net; a folded-up paper model; virtual Bézier model.
Use again Steiner surface with 4 Boy caps added.
The Convoluted Map N5.4 (self-dual)The Convoluted Map N5.4 (self-dual)
N5.4: 3 hexagons, 3 val-6 vertices, 9 edges, mF=mV=3, Petrie-length=3.
It has vertex and face multiplicities of 3!
Use torus with 3 Boy caps (two views) or with 3 cross-caps.
Regular Map N6.2dRegular Map N6.2d
Make use of 3 cross-handle tunnels in a cube
N6.2d: 6 decagons, 20 val-3 vertices, 30 edges, mF=2, Petrie-length=5.
Virtual model unfolded net complete paper model
Regular Map N7.1Regular Map N7.1
Match creases!
N7.1: 15 quads, 10 val-6 vertices,30 edges, Petrie-length=5.
N7.1d: 10 hexagons, 15 val-4 vertices30 edges, Petrie-length=5.
A genus-7 surface.
Movie !
Construction a Genus-8 SurfaceConstruction a Genus-8 Surface
Concept:
8 Boy capsgrafted onto spherein octahedral positions.
N8.1: 84 triangles, 36 val-6 vertices,126 edges, Petrie-length=9.
Octa-Boy SculptureOcta-Boy Sculpture
Two half-shells made on an RP machine
Octa-Boy SculptureOcta-Boy Sculpture
The two half-shells combined
Octa-Boy SculptureOcta-Boy Sculpture
Seen from a different angle
ConclusionsConclusions
Many more maps remain to be modeled.
Several puzzles among maps of genus ≤ 8.
Can this task be automated / programmed ?
Turn some interesting maps into art . . .
Light Cast by Genus-3 Light Cast by Genus-3 ““Tiffany LampTiffany Lamp””
Rendered with “Radiance” Ray-Tracer (12 hours)
Orientable Regular Map of Genus-6Orientable Regular Map of Genus-6
Light Field of Genus-6 Tiffany LampLight Field of Genus-6 Tiffany Lamp
EpilogEpilog
“Doing math” is not just writing formulas!
It may involve paper, wires, styrofoam, glue…
Sometimes, tangible beauty may result !
Questions ?
