Can’t the nucleator work for special cases?

It’s easy to show that there are cases in which the nucleator fails to estimate perimeter. The nucleator involves measurements of radius. The mean radius is not related to perimeter. That has been known for at least 70 years.

So now I’m asked, “None of my work deals with things that look like gears. I’m not trying to estimate the perimeter of a daisy or an octopus. I’m not looking at the perimeter of a leaf. My things are all convex. They are simple blobs.  Can’t the nucleator work for those kinds of shapes?”

This is a very good question. Unfortunately, the answer is NO.

I reply, “Think about a circle. Imagine that the reference point is at the center. The measured radius is the same in all direction. That means that the mean radius is the same as any of the measured radii. The perimeter can be computed knowing the radius. Just multiply by 2 pi.”

“Think about stretching or squishing a circle and you get an ellipse. The ellipse is tricky in that the formulas for the mean length and the perimeter involve nasty math. But what you find out is that the mean radius measured from the center of the ellipse multiplied by 2 pi is the perimeter of the ellipse.”

That sure sounds encouraging doesn’t it? The only problem is that this is not true for any other point in the ellipse. It’s not true for any other point in the circle.

“So you’re saying that the perimeter can’t be figure out from the nucleator?”

“That’s the trouble with math. It tells you what works and what doesn’t work.”

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