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Solid-coloring of objects built from 3D bricks Joseph O’Rourke “solid-coloring” “object” “brick” … all will be explained later

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Coloring 2D Maps Famous 4-Color Theorem: Every map can be colored with at most 4 colors so that any two regions that share a positive-length boundary receive a different color: maps may be “4-colored.” A much less famous 3-Color Theorem: Every map all of whose regions are triangles may be 3-colored. A theorem of Sibley & Wagon: Every map all of whose regions are parallelograms may be 3-colored.

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=> Penrose tilings may be 3-colored

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Complex of triangles/parallelograms Best to view these “maps” as complexes constructed by gluing triangles/parallelograms whole edge-to- whole edge. In triangle complex, dual graph has maximum degree 3. [See next slide] In parallelogram complex, dual graph has maximum degree 4.

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Triangle Complex Dual graph has maximum degree 3

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Triangle Complex: 3-colorable Sketch of proof: Find a triangle with vertex v on the “boundary” of the complex. There must be at least one triangle t with an “exposed” edge e. Remove t, 3-color remainder by induction, put back. Color t with the color not used on its at most two neighbors.

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2D regions Triangle complex: 3-colorable. Parallelogram complex: 3-colorable. Convex-quadrilateral complex? 4 colors needed

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2D vs. 3D 2D coloring well-explored 3D “solid coloring”: largely unexplored

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Solid-coloring 3D “bricks” Complex built from gluing bricks of various shape types whole face-to-whole face. Color each brick so that no two that share a face have the same color. Theorems: (JOR) Objects built from tetrahedra may be 4-colored. (JOR) Objects built from d-simplices in R d may be (d+1)-colored. Suzanne Gallagher (Smith 2003): Genus-0 (no-hole) objects (i.e., balls) built from rectangular bricks may be 2-colored(!).

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Figure in proof for tetrahedra Identifying some tetrahedron with an exposed face.

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Figure in proof of 2-colorability (One “layer” of perhaps many)

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The Unknown Is every object built from rectangular bricks 3- colorable? Suzanne & JOR proved this for 1-hole objects. Is every object built from parallelepipeds 4- colorable? Is every zonohedron (which are all built from parallelepipeds) 4-colorable? How many colors are needed for objects built from convex hexahedra? Etc.

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Four parallelepiped bricks, needs 4 colors Rhombic dodecahedron Dual graph is K 4

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A zonohedon: 4060 bricks How many colors needed?

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That’s It!

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