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ISAMA 2004, Chicago K 12 and the Genus-6 Tiffany Lamp Carlo H. Séquin and Ling Xiao EECS Computer Science Division University of California, Berkeley

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Graph-Embedding Problems Bob Alice Pat

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

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This is Called a Bi-partite Graph Bob Alice Pat Harry Person-Nodes Shop-Nodes K 3,4

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A Bigger Challenge : K 4,4,4 u Tripartite graph u A third set of nodes: E.g., access to airport, heliport, ship port, railroad station. Everybody needs access to those… u Symbolic view: = Dycks graph u Nodes of the same color are not connected.

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What is K 12 ? u (Unipartite) complete graph with 12 vertices. u Every node connected to every other one ! u In the plane: has lots of crossings…

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Our Challenging Task Draw these graphs crossing-free l onto a surface with lowest possible genus, e.g., a disk with the fewest number of holes; l so that an orientable closed 2-manifold results; l maintaining as much symmetry as possible.

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

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Mapping Graph K 12 onto a Surface (i.e., an orientable 2-manifold) u Draw complete graph with 12 nodes (vertices) u Graph has 66 edges (=border between 2 facets) u Orientable 2-manifold has 44 triangular facets u # Edges – # Vertices – # Faces + 2 = 2*Genus u 66 – 12 – = 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]

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The Connectivity of Bokowskis Map

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Prof. Bokowskis Goose-Neck Model

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Bokowskis ( Partial ) Virtual Model on a Genus 6 Surface

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My First Model u Find highest-symmetry genus-6 surface, u with convenient handles to route edges.

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My Model (cont.) u Find suitable locations for twelve nodes: u Maintain symmetry! u Put nodes at saddle points, because of 11 outgoing edges, and 11 triangles between them.

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My Model (3) u Now need to place 66 edges: u Use trial and error. u Need a 3D model ! u CAD model much later...

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2 nd Problem : K 4,4,4 (Dycks Map) u 12 nodes (vertices), u but only 48 edges. u E – V – F + 2 = 2*Genus u 48 – 12 – = 6 Genus = 3

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Another View of Dycks Graph u Difficult to connect up matching nodes !

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Folding It into a Self-intersecting Polyhedron

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

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Find Locations for Nodes u Actually harder than in previous example, not all nodes connected to one another. (Every node has 3 that it is not connected to.) u Place them so that the missing edges do not break the symmetry: u Inside and outside on each tetra-arm. u Do not connect the nodes that lie on the same symmetry axis (same color) (or this one).

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A First Physical Model u Edges of graph should be nice, smooth curves. Quickest way to get a model: Painting a physical object.

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Geodesic Line Between 2 Points u Connecting two given points with the shortest geodesic line on a high-genus surface is an NP-hard problem. T S

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Pseudo Geodesics u Need more control than geodesics can offer. u Want to space the departing curves from a vertex more evenly, avoid very acute angles. u Need control over starting and ending tangent directions (like Hermite spline).

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LVC Curves (instead of MVC) u Curves with linearly varying curvature have two degrees of freedom: k A k B, u Allows to set two additional parameters, i.e., the start / ending tangent directions. A B CURVATURE kAkA kBkB ARC-LENGTH

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Path-Optimization Towards LVC u Start with an approximate path from S to T. u Locally move edge crossing points ( C ) so as to even out variation of curvature: T C S C V u For subdivision surfaces: refine surface and LVC path jointly !

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K 4,4,4 on a Genus-3 Surface LVC on subdivision surface – Graph edges enhanced

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K 12 on a Genus-6 Surface

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3D Color Printer (Z Corporation)

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Cleaning up a 3D Color Part

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

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Genus-6 Regular Map

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Genus-6 Kandinsky

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Manually Over-painted Genus-6 Model

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Bokowskis Genus-6 Surface

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

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

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

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A Virtual Genus-3 Tiffany Lamp

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

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Virtual Genus-6 Map

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Virtual Genus-6 Map (shiny metal)

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Light Field of Genus-6 Tiffany Lamp

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