Rings of Callicrates

Last Updated: September 12, 2002

Clifford Pickover described a variety of magic circles, including one he called Rings of Callicrates (Figure 4.24 in Pickover's book). The original ring (created by Michael Keith) has the digits of pi on the outer node points:

This type of ring structure falls into the type I call simple, and other orderings of the numbers are possible:

The above two sets of circles have two rings of 10 circles each. There is nothing special about having 10 circles; rings with more or less circles are easily constructed. Here are examples with 12 and 15 circles:

Further, we can add additional layers of rings, although we must be careful with the connectivity to ensure that each ring has the same number of intersections: