We know that regular polygons can be constructed when they have 3, 4, 5, 6, 8, 10, 12, 15, 16, or 20 sides. We also know that if they have 7, 9, 11, 13, 14, 18, or 19 sides, they cannot be constructed. By examining these numbers, we can see that there is one missing which is the number 17. Until Carl Friedrich Gauss, the construction of a 17-gon (short form) was unknown. However, on March 30th, Guass found a Euclidean construction for a regular polygon with 17 sides. Guass examined two simple properties of the number 17. First, its only divisors are itself anf one; second, it is one greater than a power of two: 17=2^4+1
"If you were a genius like Guass, you could see why those two unassuming
statements implied the existence of a construction, using a straightedge and
compass, of the regular seventeen-sided polygon. If you were any of the other
great mathematicians who had lived between 500 BCE and 1796, you would not even have got a sniff of any connections. We know this because they didn't."
Gauss thought his discovery of the 17-gon was so significant that he asked to have one carved on his tomb stone.
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