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

“solid-coloring”“object”“brick”… all will be explained later

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.

=> Penrose tilings may be 3-colored

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.

Triangle Complex

Dual graph has maximum degree 3

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.

2D regions

Triangle complex: 3-colorable.Parallelogram complex: 3-colorable.Convex-quadrilateral complex?

4 colors needed

2D vs. 3D

2D coloring well-explored3D “solid coloring”: largely unexplored

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(!).

Figure in proof for tetrahedra

Identifying some tetrahedron with an exposed face.

Figure in proof of 2-colorability

(One “layer” of perhaps many)

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.

Four parallelepiped bricks,needs 4 colors

Rhombic dodecahedron

Dual graph is K4

A zonohedon: 4060 bricks

How many colors needed?

That’s It!