Conditions For Constructibility
Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not?
Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:
- 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.
Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as the Gauss–Wantzel theorem.
Read more about this topic: Constructible Polygon
Famous quotes containing the word conditions:
“To get it right, be born with luck or else make it. Never give up. Get the knack of getting people to help you and also pitch in yourself. A little money helps, but what really gets it right is to never—I repeat—never under any conditions face the facts.”
—Ruth Gordon (1896–1985)