Solid-coloring of objects built from 3D bricks Joseph O’Rourke
description
Transcript of Solid-coloring of objects built from 3D bricks Joseph O’Rourke
![Page 1: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/1.jpg)
Solid-coloring of objects built from 3D bricksJoseph O’Rourke
“solid-coloring”“object”“brick”… all will be explained later
![Page 2: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/2.jpg)
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.
![Page 3: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/3.jpg)
=> Penrose tilings may be 3-colored
![Page 4: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/4.jpg)
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.
![Page 5: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/5.jpg)
Triangle Complex
Dual graph has maximum degree 3
![Page 6: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/6.jpg)
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.
![Page 7: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/7.jpg)
2D regions
Triangle complex: 3-colorable.Parallelogram complex: 3-colorable.Convex-quadrilateral complex?
4 colors needed
![Page 8: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/8.jpg)
2D vs. 3D
2D coloring well-explored3D “solid coloring”: largely unexplored
![Page 9: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/9.jpg)
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(!).
![Page 10: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/10.jpg)
Figure in proof for tetrahedra
Identifying some tetrahedron with an exposed face.
![Page 11: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/11.jpg)
Figure in proof of 2-colorability
(One “layer” of perhaps many)
![Page 12: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/12.jpg)
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.
![Page 13: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/13.jpg)
Four parallelepiped bricks,needs 4 colors
Rhombic dodecahedron
Dual graph is K4
![Page 14: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/14.jpg)
A zonohedon: 4060 bricks
How many colors needed?
![Page 15: Solid-coloring of objects built from 3D bricks Joseph O’Rourke](https://reader036.fdocuments.us/reader036/viewer/2022062812/568163a1550346895dd49fd5/html5/thumbnails/15.jpg)
That’s It!