Fermat’s sum of squares, Pythagoras, pegboards, Fermat’s primes, regular polygons, part 1

Daydreaming again. When do polygons remind me of grids? They seem to me to show interesting holes in the way we associate shapes with numbers.

For a grid, all of the hypoteneuses of the triangles fit a pattern:

Say that hypoteneuse length is n. Then the prime factors of n^2 that are of form 4k+3 must have only even powers in the factorization.

That’s because of Fermat’s pattern for the sum of two squares and Pythagoras’ theorem for the hypoteneuse of a right triangle. In fact this same pattern is linked to the patterns of Pythagorean Triples.

For a regular polygon with n sides (pentagon=5, heptagon=7, etc), the constructible ones, with algebraic coordinates, the number of sides fit a pattern.

The prime factors of n are of form 2^{2^k}+1 and 2^{k}

There is a nice explanation of the sum of two squares pattern here:

Su, Francis E., et al. “Sums of Two Squares.” Math Fun Facts. <http://www.math.hmc.edu/funfacts>

There is a nice explanation of Gauss and Wantzel applying Fermat primes to regular polygons here:

Constructing Regular Polygons, Math Teacher Link University of Illinois at Urbana-Champaign <http://mtl.math.uiuc.edu/node/29>

But is there not some relation between these two things? In both cases we have geometric ideas of objects, but they are limited by the patterns in prime factorizations of an integer.

For the grid, there are some Quadrances (distance squared) that we cannot represent on it. For regular polygons, there are some n-gons that we cannot represent with the Descartes’ coordinates being ‘constructible numbers’, in other words, square roots combined with addition, subtraction, multiplication, and addition. Or, in other language, they are not ‘constructible’ with a straightedge and compass.

But let me ask this… how does one build an angle x if one cannot build the n-gon made of wheel-like triangle-slices out of that angle x? Does this not limit which rational angles are representable in such a system? Because every n-gon is associated with a rational angle, as the triangles fanning out from the central point, fan-like.

To my amateur’s mind, these limits are fascinating. They represent a sort of ‘unevenness’ or partiality of the systems we have constructed in our minds, such that what is seemingly a smooth even pattern, a grid, in fact, in another perspective, is full of holes and things it does not cover. The constructible n-gons, as well, display an uneven pattern where only certain rational angles are representable.

Even if one admits the Real Numbers, and posits that all things are possible, such as each point on a grid, or each angle on a circle, including irrationals and transcendentals and all of the other fascinating animals in the mathematical universe, it doesn’t wipe away the fact that these animals are not uniformly distributed. There are little oceans and islands, little clusters lying about in different places, that I never expected when I was first introduced to these ideas many years ago.

 

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