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Me!
Well, the constructions for the pentagon are not difficult to explain, but not so immediate to prove. I was used to prove one of them in a third year class of university course in mathematics, and it required to master complex numbers ;indeed, the key fact here is that the vertices of the regular polygon with n edges can be seen as the n-roots of 1, i.e.they correspond to complex numbers z= (a + b i) such that z times z times z ... n times gives 1. Here you should know the i times i gives -1.
The important fact here is to undestand that we are talking about "precise" constructions made just with ruler and compass.
There is a famous theorem of Gauss stating that just few polygons are constructible that way - I remember 3,5, 17,255... there is a rule...
the other construction you can find are approximate, in the sense that the error is so small that they are good for applications (for example, building gears)
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What about a hexagon? That one's trivial.
TEd
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yes, indeed.
I was speaking just about the possible values of n which are prime numbers.
The real statement is that the "constructible" polygons have n = 2 times 2...times 2 (any number of times) times again p1 times p2...times pw, where p1,p2,...,pw are distinct primes of the kind I was saying before. I could write the rule, if I had a mathematical editor.
So, 6 = 2 times 3 is ok.
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The only known Primes of this form are 3, 5, 17, 257, and 65537.
A clear - I suppose- statement is well written in
http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/MathWorld/math/math/c/c615.htm
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" A Portia come to judgement!"
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The only known Primes of this form are 3, 5, 17, 257, and 65537.
I note that these are all primes of the form 2^n+1, where n is a power of two.
OK I looked at the web page and I see that my little insight was explicitly stated.
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I still remember that e to the pi i = minus one. But only that.
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The only known Primes of this form are 3, 5, 17, 257, and 65537.
I note that these are all primes of the form 2^n+1, where n is a power of two.
Are these the Mersenne Primes? Something like (2^(2^n))+1? I think I remember something about the series failing at n>3, with the counterexample for n=4 being publically and painstakingly rendered on the blackboard in front of the assembled multitudes, one of the factors being 671...Does that sound familiar to anyone? But I don't recall ever coming across the numbers in the context of constructing regular polygons.
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