Fun With Regular Polygons

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Fun With Regular Polygons

• BY Adam De Marco

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Definition

• An n-gon (a polygon with n sides) is said to be
  regular if all n sides are congruent and all
  angles are congruent, such as in a square.

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Theorem

• For a given n gt 3, it is possible to construct a
  regular n-gon if and only if it is possible to
  construct an angle of size 360degrees/n.

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Theorem

• If n is a prime number, then a regular n-gon is
  constructible if and only if n is a Fermat Prime.

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Theorem

• A) If n is odd, then a regular n-gon is
  constructible if and only if n is a product of
  distinct Fermat Primes.
• B) In general, a regular n-gon is constructible
  if and only if n is a power of 2 times a product
  of distinct Fermat Primes.

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Homework Problem 6.5

• For which of these n is a regular n-gon
  constructible?
• A) 144
• B) 85
• C) 176
• D) 100
• E) 128