K12 and the Genus-6 Tiffany Lamp ISAMA 2004 ISAMA 2004, Chicago K12 and the Genus-6 Tiffany Lamp Carlo H. Séquin and Ling Xiao EECS Computer Science Division University of California, Berkeley

Graph-Embedding Problems ISAMA 2004 Graph-Embedding Problems Bob Alice Pat

On a Ringworld (Torus) this is No Problem ! ISAMA 2004 On a Ringworld (Torus) this is No Problem ! Alice Harry Bob Pat

This is Called a Bi-partite Graph ISAMA 2004 This is Called a Bi-partite Graph K3,4 Alice Bob Pat Harry “Shop”-Nodes “Person”-Nodes

A Bigger Challenge : K4,4,4 Tripartite graph ISAMA 2004 A Bigger Challenge : K4,4,4 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 “K12” ? (Unipartite) complete graph with 12 vertices. ISAMA 2004 What is “K12” ? (Unipartite) complete graph with 12 vertices. Every node connected to every other one ! In the plane: has lots of crossings…

Our Challenging Task Draw these graphs crossing-free ISAMA 2004 Our 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.

Not Just Stringing Wires in 3D … ISAMA 2004 Not Just Stringing Wires in 3D … Icosahedron has 12 vertices in a nice symmetrical arrangement; -- let’s just connect those … But we want graph embedded in a (orientable) surface !

Mapping Graph K12 onto a Surface (i.e., an orientable 2-manifold) ISAMA 2004 Mapping Graph K12 onto a Surface (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 Map ISAMA 2004 The Connectivity of Bokowski’s Map

Prof. Bokowski’s Goose-Neck Model ISAMA 2004 Prof. Bokowski’s Goose-Neck Model Can’t see the triangles – unless you have a vivid imagination.

Bokowski’s ( Partial ) Virtual Model on a Genus 6 Surface ISAMA 2004 Bokowski’s ( Partial ) Virtual Model on a Genus 6 Surface

My First Model Find highest-symmetry genus-6 surface, ISAMA 2004 My First Model Find highest-symmetry genus-6 surface, with “convenient” handles to route edges.

My Model (cont.) Find suitable locations for twelve nodes: ISAMA 2004 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) Now need to place 66 edges: Use trial and error. ISAMA 2004 My Model (3) Now need to place 66 edges: Use trial and error. Need a 3D model ! CAD model much later...

2nd Problem : K4,4,4 (Dyck’s Map) ISAMA 2004 2nd Problem : K4,4,4 (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 Graph ISAMA 2004 Another View of Dyck’s Graph Difficult to connect up matching nodes !

Folding It into a Self-intersecting Polyhedron ISAMA 2004

ISAMA 2004 Towards a 3D Model Find highest-symmetry genus-3 surface:  Klein Surface (tetrahedral frame).

Find Locations for Nodes ISAMA 2004 Find 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 the missing edges do not break the symmetry:  Inside and outside on each tetra-arm. Do not connect the nodes that lie on the same symmetry axis (same color) (or this one).

A First Physical Model Edges of graph should be nice, smooth curves. ISAMA 2004 A 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 Points ISAMA 2004 Geodesic Line Between 2 Points T S Connecting two given points with the shortest geodesic line on a high-genus surface is an NP-hard problem.

ISAMA 2004 K4,4,4 on a Genus-3 Surface LVC on subdivision surface – Graph edges enhanced

ISAMA 2004 K12 on a Genus-6 Surface

3D Color Printer (Z Corporation) ISAMA 2004 3D Color Printer (Z Corporation)

Cleaning up a 3D Color Part ISAMA 2004 Cleaning up a 3D Color Part

Finishing of 3D Color Parts ISAMA 2004 Finishing of 3D Color Parts Infiltrate Alkyl Cyanoacrylane Ester = “super-glue” to harden parts and to intensify colors.

ISAMA 2004 Genus-6 Regular Map

ISAMA 2004 Genus-6 Regular Map

ISAMA 2004 “Genus-6 Kandinsky”

Manually Over-painted Genus-6 Model ISAMA 2004 Manually Over-painted Genus-6 Model

Bokowski’s Genus-6 Surface ISAMA 2004 Bokowski’s Genus-6 Surface

Tiffany Lamps (L.C. Tiffany 1848 – 1933) ISAMA 2004 Tiffany Lamps (L.C. Tiffany 1848 – 1933)

Tiffany Lamps with Other Shapes ? ISAMA 2004 Tiffany Lamps with Other Shapes ? Globe ? -- or Torus ? Certainly nothing of higher genus !

Back to the Virtual Genus-3 Map ISAMA 2004 Back to the Virtual Genus-3 Map Define color panels to be transparent !

A Virtual Genus-3 Tiffany Lamp ISAMA 2004 A Virtual Genus-3 Tiffany Lamp

Light Cast by Genus-3 “Tiffany Lamp” ISAMA 2004 Light Cast by Genus-3 “Tiffany Lamp” Rendered with “Radiance” Ray-Tracer (12 hours)

ISAMA 2004 Virtual Genus-6 Map

Virtual Genus-6 Map (shiny metal) ISAMA 2004 Virtual Genus-6 Map (shiny metal)

Light Field of Genus-6 Tiffany Lamp ISAMA 2004 Light Field of Genus-6 Tiffany Lamp