Solid-coloring of objects built from 3D bricks Joseph O’Rourke

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

2. 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.

3. => Penrose tilings may be 3-colored

4. 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.

5. Triangle Complex Dual graph has maximum degree 3

6. 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.

7. 2D regions Triangle complex: 3-colorable. Parallelogram complex: 3-colorable. Convex-quadrilateral complex? 4 colors needed

8. 2D vs. 3D 2D coloring well-explored 3D “solid coloring”: largely unexplored

9. 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 Rd 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(!).

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

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

12. 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.

13. Four parallelepiped bricks,needs 4 colors Dual graph is K4 Rhombic dodecahedron

14. A zonohedon: 4060 bricks How many colors needed?

15. That’s It!