Software Musings

Extrapolation

Posted on: 11/06/2006

David Corfield’s Towards a Philosophy of Real Mathematics spends much of its time on plausible reasoning within mathematics. He mentions a number of inductive “leaps of faith” including John MacKay’s observation about 196884 appearing both as a coefficient in the Fourier expansion of the j function and (almost) as the dimension of a representation of the monster group. That coincidence turned out to be no coincidence. He also mentions Fermat’s less successful conjecture that all Fermat numbers are prime.

One such extrapolation which I have only come across once concerns points irregularly placed around a circle. I don’t know whether this is common knowledge in mathematical circles but it certainly wasn’t common knowledge to me when my daughter, then in one of the lower forms of secondary school, brought it home. The problem was simple:

N points are placed irregularly on the circumference of a circle. Straight lines are drawn between each pair of points. Into how many pieces is the circle divided?

So, we started with N=1 and found that the circle was still in one bit. f(1) = 1.

With N=2 the circle is clearly divided into two. f(2) = 2.

It’s still pretty easy to see that f(3) = 4 and f(4) = 8 but after that the diagram gets a bit crowded. But the pattern was emerging fast. I told my daughter to check that f(5) = 16 and she did. It is.

At this point the homework exercise was effectively finished: 1, 2, 4, 8, 16 form a fairly predictable sequence and when she came back and told me that f(6) = 33 I told her to go away and draw the picture more carefully. She did and still found f(6) to be 33. I was now interested again and, after a lot of work, we found the quartic that just happens to return

1, 2, 4, 8, 16, 33

for its first few terms. Was the conjecture I made less natural than Mackay’s?

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2 Responses to "Extrapolation"

I make the sequence 1,2,4,8,16,30 or 31,57. The number of curved bits is always the same as the number of points and the sequence made up of entirely straight-sided divisions is 0,0,1,4,11,25 (or 24 for a regular hexagon),50. (2 e-mails sent 16th & 17th, no replies received) My eyes will go funny if I try the count for more than 7 points. Is the underlying sequence different if the points do not lie on a circle? I would say the naturalness of the conjecture depends entirely on how close to nature you are mathematically speaking.

I’ve just checked the sequence again (at least as far as N=6) and found that, indeed, f(6) = 31. So the sequence is

1, 2, 4, 8, 16, 31

Sorry about that.

Remember that the points have to be irregularly spaced around the circle (so that excludes at least a regular hexagon).

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The author of this blog used to be an employee of Nortel. Even when he worked for Nortel the views expressed in the blog did not represent the views of Nortel. Now that he has left, the chances are even smaller that his views match those of Nortel.
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