Solid-coloring of objects built from 3D bricks
Joseph 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-towhole 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-explored
 3D “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 4colorable?
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
Dual graph is K4
Rhombic dodecahedron
A zonohedon: 4060 bricks
How many colors needed?
That’s It!