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An Infinitely Colorable Set of 3D Regions

Suppose that if two regions touch, then they should have different colors. How many colors suffice? In the plane, four colors are sufficient. In 3D, the set of regions shown requires an infinite number of colors.

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"An Infinitely Colorable Set of 3D Regions" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/AnInfinitelyColorableSetOf3DRegions/
Contributed by: Jon Perry
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