The 4-way nucleator, when clicked inside of a convex profile creates 4 lengths – one for each ray of the nucleator. In this case we examine the circle, the simplest of shapes.

A circle is simple, because it is isotropic. Rotate the circle and nothing changes. The circle looks the same. That makes many questions about the circle easy to answer.

Here an arbitrary point is selected inside of the circle. Four rays start at the arbitrary point and intercept the circle. Because the circle is isotropic it is possible to simplify the math and use rays parallel to the coordinate axes. Also, without loss of generality let’s select the arbitrary point from the first quadrat.

These two simplifications mean that the lengths show in the drawings, a, b, c, and d, can be used to identify the 4 intercepts as (0, a), (b, 0), (0, -c), and (-d, 0). The axes cross at the arbitrarily chosen point. That point has coordinates (0,0). The intercepts on the y-axis are at a and -c. The x-axis intercepts are at -d and b.

So the question is whether or not the information given here is capable of estimating the circumference of the circle? Yes. Let’s see how that is done.

<insert math>

In general, the nucleator cannot be used to estimate perimeter. There is at least 1 case in which the nucleator does work. Are there other cases? The answer to that is unfortunately no. No shape other than a circle has a perimeter that can be estimated by the nucleator without bias.