Suppose you’re in a boring meeting and you start doodling. You draw a circle, and then you draw a triangle on the outside of that circle.

Next you draw a circle through the vertices of the triangle, and draw a square outside that.

Then you draw a circle through the vertices of the square, and draw a pentagon outside that.

Then you think “Will this ever stop?!”, meaning the meeting, but you could ask a similar question about your doodling: does your sequence of doodles converge to a circle of finite radius, or do they grow without bound?

An *n*-gon circumscribed on the outside of a circle of radius *r* is inscribed in a circle of radius

So if you start with a unit circle, the radius of the circle through the vertices of the *N*-gon is

and the limit as *N* → ∞ exists. The limit is known as the **polygon circumscribing constant**, and it equals 8.7000366252….

We can visualize the limit by making a plot with large *N*. The plot is less cluttered if we leave out the circles and just show the polygons. *N* = 30 in the plot below.

The reciprocal of the polygon circumscribing constant is known as the **Kepler-Bouwkamp constant**. The Kepler-Bouwkamp constant is the limiting radius if you were to reverse the process above, *inscribing* polygons at each step rather than *circumscribing* them. It would make sense to call the Kepler-Bouwkamp constant the polygon *inscribing* constant, but for historical reasons it is named after Johannes Kepler and Christoffel Bouwkamp.

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