Main Index Geometry Plane Geometry Geometric constructions
  Subject Index
comment on the page

Gauß-Wantzel Theorem

The Gauß-Wantzel Theorem:

A regular -gon can be constructed with compass and straightedge if and only if:

1) is a Fermat prime. ¹

2) is a power of 2.

3) is the product of a power of 2 and distinct Fermat primes.

The theorem can be rephrased in several equivalent forms:

• A regular -gon is constructible with compass and straightedge if and only if is an integer greater than 2 such that the greatest odd factor of is either 1 or a product of distinct Fermat primes.
• A regular -gon is constructible with compass and straightedge if and only if is an integer greater such that is a power of 2, where stands for the Euler totient function.
• A regular -gon is constructible with compass and straightedge if and only if the values of trigonometric function at points can be expressed in terms of finite number of addition, multiplication and square root extraction.

Carl Friedrich Gauß proved the constructibility of the regular 17-gon (called a heptadecagon) in 1796 at age 18.  He was so pleased by this  discovery that he started to write his famous ² Tagebuch [1]  and he asked to have a heptadecagon carved on his tombstone.  Under influence of this the discovery he decised for a mathematician career. Five years later, he published the whole the theory in terms of the so called Gaussian periods in his Disquisitiones Arithmeticae.

Gauß conjectured that his condition is also necessary, but he offered no proof of this fact. In 1836, Piere Laurent Wantzel [2]  proved that Gauß condition is also necessary showing that if a regular -gon is constructible for a prime , then must be a Fermat prime. Combining this result with that of Gauß and the knowledge of ancient mathematicians yield the above formulation of the theorem.

The construction for an equilateral triangle is simple and has been known since Antiquity. Constructions for the regular pentagon was known at least to ancients Greeks. It described in Euclid’s Elements (ca 300 BC), and Ptolemy’s Almagestca (AD 150). The ancient Greeks were able to construct a regular 15-gon. Inscribing these polygons in circles and repeatedly bisecting the sides they were able to construct regular polygons of , , and sides, where is the number of bisections.

Only five Fermat primes are known: , , ,   and   .

Thus, a regular 7-gon cannot be constructed using compass and straightedge, as well as a regular 9-gon 9 is the square of the Fermat prime 3. But a 15-gon is constructible because and 5 and 3 are distinct Fermat primes, etc.

The possibility to construct an -gon by compass and straightedge has deeper algebraic roots [3]  and it is connected with the fact that the extraction of roots can be done by compass and straightedge only in case of  the square roots. The most famous straightedge-and-compass problems have been proven impossible, in several cases by the results of Galois theory. For instance, Gauß in his Disquisitiones Arithmeticae, Section 345 showed that (cf. [4] , p.57)

Although Gauß proved by this formula that the regular 17-gon is constructible, he did not actually show how to do it. The first construction is due to J. Erchinger [5] .

The first explicit construction of a regular 257-gon was given by F.J. Richelot in 1832.

A construction for a regular 65537-gon was first given by J. Hermes in 1894. He spent 10 years completing the 200-page manuscript.

Notes

1	A prime is called Fermat prime if it is of the form , or equivalently of the form .

References

Cite this web-page as: