By George Hart for the Museum of Mathematics

A complete graph is what mathematicians call a collection of items in which every pair is connected. If the items are spaced evenly around a circle and the connections are shown as straight lines, the lines form an attractive pattern of concentric circles.

This is a complete graph with eleven vertices. We made it at Mathcamp 2010 using plastic surveyor’s tape. There is a simple algorithm for constructing it in which people stand in a circle, and pass the roll from one to the next while counting aloud, wrapping it around their left wrist at the proper intervals.

A similar algorithm, executed by people standing in two straight lines, gives this large parabola. The construction steps for both of these figures can be seen in the additional photos here.

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