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Circle Puzzle

Evelyn Sander
geometry.college, geometry.puzzles, Mon, 28 Nov 1994.

Can you find a way to cover all of R^3 with disjoint geometric circles? In other words, can you place disjoint geometric circles in R^3 so that every point in R^3 is on one and only one of the geometric circles?

A geometric circle is the set of points in a fixed plane that lie a fixed positive distance from a center point that lies in the same plane. (Notice that circle refers to the curve, and not to the disk it bounds.)

To give the idea, here is a way to fill all of R^3 except the vertical axis: label the vertical axis the z-axis. At each fixed height (z = constant), put a circle of every positive radius centered at the z-axis. Since circles of radius of zero are not allowed, this set of circles does not cover the z-axis.

In this example, nearby points in R^3 (off the z-axis) are contained in circles with nearby centers and nearby radii -- in other words this example shows a continuous method of filling R^3 minus the z-axis with disjoint geometric circles. However, for this puzzle, it is not necessary to fill R^3 in a continuous manner. In fact, the solution that I give will have some discontinuities.

Two Solutions to the Circle Puzzle

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Created: May 19 1995 --- Last modified: Jun 18 1996