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Good reading, and a lathe problem

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  • #16
    Originally posted by Baz View Post
    Actually I don't think you can make any shape. like that. I believe the first one you can't do is quite low, like under 20 and there is a mathematical proof for which ones cannot be achieved. There was a programme on the BBC radio within the last two weeks about the mathematician who proved it.
    Everyone knows that a regular hexagon can be easily constructed in a circle using only a compass and a straightedge. With the compass set to the radius of the circle, simply walk it around the circle striking off points on the circumference. There will be exactly six points. Connect these points with straight lines and the result will be a regular hexagon.

    This raises the question of what other regular polygons can be constructed using only compass and straightedge? Obviously, three and four sides are easy and we know from above that six is possible. What about five or seven?

    Karl Friederich Gauss, the German mathematical prodigy and genius, solved the generalized problem. He proved that a regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of *distinct* Fermat primes (including none).

    A Fermat prime is a Fermat number of the form

    Fk = 2^(2^k)+1

    that is also a prime. The known Fermat primes are:

    k Fk

    0 3
    1 5
    2 17
    3 257
    4 65537

    Not all Fermat numbers are prime. The list above includes all the currently known Fermat primes. For example, for n = 5, we have the Fermat number 4,294,967,297 which has the factors 641 and 6,700,417 and so is not prime.

    Also, the restriction to *distinct* primes is important. A 9-sided polygon cannot be constructed. 9 has prime factors 3*3 and, while 3 is a Fermat prime, ALL the factors of 9 must be DISTINCT Fermat primes for it to be constructible.

    So, based on the above, a regular septagon cannot be constructed, but a pentagon can. A procedure for the latter is shown here...

    Regards, Marv

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    • #17
      Yah its a good question -- you can bisect an angle but cannot trisect it. Pentagon has been done before, but I never practiced that one. Just for the fun of it I'm thinking "What if I wanted to lay out every degree from zero to 10 degrees" I think you would have to divide the chord to do it.
      25 miles north of Buffalo NY, USA