ISAMA 2004, ChicagoISAMA 2004, Chicago

K12 and the Genus-6 Tiffany Lamp

Carlo H. Séquin and Ling Xiao

EECS Computer Science DivisionUniversity of California, Berkeley

Graph-Embedding ProblemsGraph-Embedding Problems

Bob Alice Pat

On a Ringworld (Torus) this is No Problem !On a Ringworld (Torus) this is No Problem !

Bob

Alice

Pat

Harry

This is Called a Bi-partite GraphThis is Called a Bi-partite Graph

Bob

Alice

Pat

Harry

“Person”-Nodes “Shop”-Nodes

K3,4

A Bigger Challenge : KA Bigger Challenge : K44,,44,4,4

Tripartite graph

A third set of nodes: E.g., access to airport, heliport, ship port, railroad station. Everybody needs access to those…

Symbolic view:= Dyck’s graph

Nodes of the same color are not connected.

What is “What is “KK1212” ?” ?

(Unipartite) complete graph with 12 vertices.

Every node connected to every other one !

In the plane:has lots of crossings…

Our Challenging TaskOur Challenging Task

Draw these graphs crossing-free

onto a surface with lowest possible genus,e.g., a disk with the fewest number of holes;

so that an orientable closed 2-manifold results;

maintaining as much symmetry as possible.

Icosahedron has 12 vertices in a nice symmetrical arrangement; -- let’s just connect those …

But we want graph embedded in a (orientable) surface !

Not Just Stringing Wires in 3D …Not Just Stringing Wires in 3D …

Mapping Graph KMapping Graph K1212 onto a Surface onto a Surface

(i.e., an orientable 2-manifold)(i.e., an orientable 2-manifold) Draw complete graph with 12 nodes (vertices)

Graph has 66 edges (=border between 2 facets)

Orientable 2-manifold has 44 triangular facets

# Edges – # Vertices – # Faces + 2 = 2*Genus

66 – 12 – 44 + 2 = 12 Genus = 6

Now make a (nice) model of that !

There are 59 topologically different ways in which this can be done ! [Altshuler et al. 96]

The Connectivity of Bokowski’s MapThe Connectivity of Bokowski’s Map

Prof. Bokowski’s Goose-Neck ModelProf. Bokowski’s Goose-Neck Model

Bokowski’s Bokowski’s ( Partial ) ( Partial )

Virtual Model Virtual Model on a on a

Genus 6 Genus 6 SurfaceSurface

My First ModelMy First Model

Find highest-symmetry genus-6 surface,

with “convenient” handles to route edges.

My Model (cont.)My Model (cont.)

Find suitable locations for twelve nodes:

Maintain symmetry!

Put nodes at saddle points,

because of 11 outgoing edges, and 11 triangles between them.

My Model (3)My Model (3)

Now need to place 66 edges:

Use trial and error.

Need a 3D model !

CAD model much later...

22ndnd Problem : K Problem : K4,4,44,4,4 (Dyck’s Map) (Dyck’s Map)

12 nodes (vertices),

but only 48 edges.

E – V – F + 2 = 2*Genus

48 – 12 – 32 + 2 = 6 Genus = 3

Another View of Dyck’s GraphAnother View of Dyck’s Graph

Difficult to connect up matching nodes !

Folding It into a Self-intersecting PolyhedronFolding It into a Self-intersecting Polyhedron

Towards a 3D ModelTowards a 3D Model Find highest-symmetry genus-3 surface:

Klein Surface (tetrahedral frame).

Find Locations for NodesFind Locations for Nodes Actually harder than in previous example,

not all nodes connected to one another. (Every node has 3 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).

A First Physical ModelA First Physical Model

Edges of graph should be nice, smooth curves.

Quickest way to get a model: Painting a physical object.

Geodesic Line Between 2 PointsGeodesic Line Between 2 Points

Connecting two given points with the shortest geodesic line on a high-genus surface is an NP-hard problem.

T

S

““Pseudo Geodesics”Pseudo Geodesics”

Need more control than geodesics can offer.

Want to space the departing curves from a vertex more evenly, avoid very acute angles.

Need control over starting and ending tangent directions (like Hermite spline).

LVC Curves (instead of MVC)LVC Curves (instead of MVC)

Curves with linearly varying curvaturehave two degrees of freedom: kA kB,

Allows to set two additional parameters,i.e., the start / ending tangent directions.

A

B

CURVATURE

kA

kB

ARC-LENGTH

Path-Optimization Towards LVCPath-Optimization Towards LVC Start with an approximate path from S to T.

Locally move edge crossing points ( C ) so as to even out variation of curvature:

T

CS

CV

For subdivision surfaces: refine surface and LVC path jointly !

KK4,4,44,4,4 on a Genus-3 Surface on a Genus-3 Surface

LVC on subdivision surface – Graph edges enhanced

KK1212 on a Genus-6 Surface on a Genus-6 Surface

3D Color Printer3D Color Printer (Z Corporation)(Z Corporation)

Cleaning up a 3D Color PartCleaning up a 3D Color Part

Finishing of 3D Color PartsFinishing of 3D Color Parts

Infiltrate Alkyl Cyanoacrylane Ester = “super-glue” to harden parts and to intensify colors.

Genus-6 Regular MapGenus-6 Regular Map

Genus-6 Regular MapGenus-6 Regular Map

““Genus-6 Kandinsky”Genus-6 Kandinsky”

Manually Over-painted Genus-6 ModelManually Over-painted Genus-6 Model

Bokowski’s Genus-6 SurfaceBokowski’s Genus-6 Surface

Tiffany Lamps Tiffany Lamps (L.C. Tiffany 1848 – 1933)(L.C. Tiffany 1848 – 1933)

Tiffany Lamps with Other Shapes ?Tiffany Lamps with Other Shapes ?

Globe ? -- or Torus ?

Certainly nothing of higher genus !

Back to the Virtual Genus-3 MapBack to the Virtual Genus-3 Map

Define color panels to be transparent !

A Virtual Genus-3 Tiffany LampA Virtual Genus-3 Tiffany Lamp

Light Cast by Genus-3 “Tiffany Lamp”Light Cast by Genus-3 “Tiffany Lamp”

Rendered with “Radiance” Ray-Tracer (12 hours)

Virtual Genus-6 Map Virtual Genus-6 Map

Virtual Genus-6 Map (shiny metal)Virtual Genus-6 Map (shiny metal)

Light Field of Genus-6 Tiffany LampLight Field of Genus-6 Tiffany Lamp