Using a compass, draw a circle on the paper. Then, without changing the opening of the compass, draw another circle, centered somewhere in the first one. Finally, with a ruler, connect the centers of the two circles with one of the points where they intersect. The figure obtained in this way is an equilateral triangle, that is, the sides of which are all the same length.
The ancient Greeks knew how to build regular polygons of 3, 4, 5 and 15 sides using only a straightedge and compass. They also knew how to get from any regular polygon to another with double sides. Thus, they knew how to build the regular hexagon (6 sides) from the equilateral triangle. Can all regular polygons, with any number N of sides, be constructed with a straightedge and compass?
The answer is no, but this was only understood in the 18th century, when it was proved that regular 7- and 13-sided polygons cannot be constructed in this way. So what are the constructible values of N, that is, such that the regular polygon with N sides can be constructed using just ruler and compass?
The problem attracted the attention of none other than the great Carl Friedrich Gauss. In 1796 he showed how to build the regular heptadecagon (17 sides) with ruler and compass. This was a discovery that Gauss was most proud of.
In his great work “Disquisitiones Arithmeticae” he went further, concluding that for a regular polygon to be constructible it is sufficient for the number N of sides to be the product of a power of 2 by distinct Fermat primes. He also stated that this condition would be sufficient, but this was only proved by Frenchman Pierre Wantzel in 1837.
Pierre de Fermat calculated the numbers of the form 1 plus 2 to the 2n for n values from 0 to 4, found them to be prime numbers, and believed this to be true for all n values. But a few years later, Leonhard Euler pointed out that the Fermat number with n=5 is not a prime and, ironically, to date no one has found any other than the five originals he discovered.
Thus, since there are 31 distinct Fermat prime number products, the Gauss-Wantzel theorem gives 31 odd N numbers that are constructible, and this is the best result known to date.
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