Hi all,
From before,
According to Wikipedia (constructible polygon article), there are infinitely many constructible polygons, but only 31 with an odd number of sides are known.
5 Fermat primes are known.
I worked out why 31 different regular polygons with an odd number of sides are constructible.
We have nCk, read n choose k, defined as
nCk = n!/(k!*(nk)!)
So we want combinations of 5 things taken 1,2,3,4, and 5 at a time without repetition. Hence
5C1 = 5
5C2 = 10
5C3 = 10
5C4 = 5 and
5C5 = 1
So 5+10+10+5+1 = 31.
So we see that there are 31 ways of, among 5 things, taking 1,2,3,4 or all 5 of them without repetition.
And all is right with the world.
Regards,
Matt
