Estimating perimeter with the nucleator

Gear in metal lid along with notes on a napkin

Gear inside a metal lid

A while back I was asked why the nucleator does not estimate perimeter. My answer at the time was off the cuff – the math doesn’t work out. I knew that this was correct, but had not really looked into the matter.

The person asked me again if it were possible to use the nucleator to estimate perimeter. I needed a quick demonstration that it was simply not possible to use the nucleator probe for that purpose.

I drew a circle on a piece of paper.

“See this circle”, I said. “The circle is interesting because it has the least perimeter for the area that is enclosed. A circle is easy to draw, because all you have to do is use a compass or string to keep the radius fixed and swing around a point to draw the circle.”

I saw a glimmer of understanding, a glimmer of what I was about to do.

“Think of a gear,” I began. “Let me draw a gear inside of the circle. It is pretty clear that the radius of the gear is always smaller than the radius of the circle. Despite it being smaller it should be clear that I can make the perimeter of the gear as long as I want by getting as jaggy as I want.”

“That’s really obvious”, was the response.

In a few strokes of a pen I was able to demonstrate that the radius of the object was not related to the perimeter.

“OK?”, was the next thing I heard.

Sure it was true that the radius did not seem to be related to the perimeter if the measurements were taken from the center of the figure, but I could see what was coming next.

“You measured the radius from the center, didn’t you?”

“That’s right”, I said.

“But isn’t the nucleator unbiased on average?”

Here was the problem: why is it claimed that the nucleator is unbiased?

“There are two types of sterological probes. Most people are familiar with the probes like the counting frame or points. The other type of probe is part of local stereology. Local does not mean for small things. It means that the probe is based on the use of reference points. In the case of the nucleator the reference point is the nucleolus. The nucleator must be unbiased for the estimates made at every possible point, not the estimates made across all points.”

“To be unbiased the nucleator has to be unbiased at the center, and at every other point that is chosen. So if the nucleator is not unbiased at the center, then it is not unbiased.”

I almost missed that important qualifier.

“I meant to say unbiased in estimating perimeter.”

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