1. 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

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

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

4. 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

5. 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.

6. 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…

7. 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.

8. 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 !

9. 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 – =  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]

10. The Connectivity of Bokowski’s Map
ISAMA 2004
The Connectivity of Bokowski’s Map

11. 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.

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

13. 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.

14. 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.

15. 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...

16. 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 – =  Genus = 3

17. Another View of Dyck’s Graph
ISAMA 2004
Another View of Dyck’s Graph
Difficult to connect up matching nodes !

18. Folding It into a Self-intersecting Polyhedron
ISAMA 2004

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

20. 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).

21. 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.

22. 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.

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

24. ISAMA 2004
K12 on a Genus-6 Surface

25. 3D Color Printer (Z Corporation)
ISAMA 2004
3D Color Printer (Z Corporation)

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

27. 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.

28. ISAMA 2004
Genus-6 Regular Map

29. ISAMA 2004
Genus-6 Regular Map

30. ISAMA 2004
“Genus-6 Kandinsky”

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

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

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

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

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

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

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

38. ISAMA 2004
Virtual Genus-6 Map

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

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