Constructible regular polygons (Gauss–Wantzel)

Solved, 1837GeometryAlgebra

Statement

Determine for which integers $n \ge 3$ a regular $n$ -gon can be constructed in finitely many steps using only an idealized compass and an unmarked straightedge.

Carl Friedrich Gauss discovered the constructibility of the regular $17$ -gon on March 30, 1796, and in Section VII of his Disquisitiones Arithmeticae (1801) proved that a regular $n$ -gon is constructible whenever $\varphi(n)$ is a power of $2$ — equivalently, when $n = 2^k p_1 \cdots p_m$ for distinct Fermat primes $p_i = 2^{2^{r_i}} + 1$ . Gauss explicitly stated that this condition is also necessary, and Pierre Laurent Wantzel published the complete proof of necessity in 1837 by showing that constructibility requires the cyclotomic field $\mathbb{Q}(\zeta_n)$ to have degree $\varphi(n) = 2^s$ .

Posed by, 300 BCEEuclid of Alexandria Solved by, 1837Carl Friedrich Gauss

Although the algebraic criterion $n = 2^k p_1 \cdots p_m$ completely reduces constructibility to Fermat primes, it remains a famous open problem in number theory whether there exist any Fermat primes beyond the five known values $F_0 = 3$ , $F_1 = 5$ , $F_2 = 17$ , $F_3 = 257$ , and $F_4 = 65{,}537$ ; consequently, only $2^5 - 1 = 31$ odd constructible regular polygons are currently known. If one enlarges the toolkit to include angle trisection (via neusis with a marked ruler, conic intersections, or paper folding), Andrew M. Gleason (1988) proved that a regular $n$ -gon is constructible if and only if $n = 2^k 3^l p_1 \cdots p_m$ where each $p_i$ is a distinct Pierpont prime of the form $2^u 3^v + 1$ , which makes the regular $7$ -gon, $9$ -gon, and $13$ -gon constructible.

References

Carl Friedrich Gauss (1801). Disquisitiones Arithmeticae
Pierre Laurent Wantzel (1837). Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas
Andrew M. Gleason (1988). Angle trisection, the heptagon, and the triskaidecagon · DOI:10.1080/00029890.1988.11971989

History of the problem

300 BCEEuclid's Elements (Book IV) presents compass-and-straightedge constructions of regular polygons with $3$ , $4$ , $5$ , and $15$ sides, together with angle bisection to double the number of sides of any constructible polygon.Euclid of Alexandria
1796Nineteen-year-old Carl Friedrich Gauss proves on March 30 that the regular $17$ -gon (heptadecagon) is constructible by expressing $\cos(2\pi/17)$ in terms of nested square roots using Gaussian periods, the first advance on the problem in over two millennia.Carl Friedrich Gauss
1801Gauss publishes Disquisitiones Arithmeticae, proving that the cyclotomic equation $x^n - 1 = 0$ is solvable by quadratic radicals whenever $n = 2^k p_1 \cdots p_m$ for distinct Fermat primes $p_i = 2^{2^{r_i}} + 1$ , and asserting that no other regular $n$ -gons are constructible.Carl Friedrich Gauss
1837Pierre Laurent Wantzel proves the necessity of Gauss's criterion, establishing that a regular $n$ -gon with $n$ divisible by an odd prime square $p^2$ or a non-Fermat odd prime $p$ cannot be constructed with compass and straightedge.
1894Johann Gustav Hermes completes a decade-long explicit calculation for the regular $65{,}537$ -gon ( $F_4 = 2^{2^4} + 1$ ), following Friedrich Julius Richelot's 1832 construction of the regular $257$ -gon ( $F_3 = 2^{2^3} + 1$ ).