Compass and straightedgegeometric constructions dating back to Euclid
were capable of inscribing regular polygons of
3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, ..., sides. In 1796
(when he was 19 years old), Gauss gave a sufficient
condition for a regular -gon to be constructible, which he also conjectured (but did
not prove) to be necessary, thus showing that regular
-gons
were constructible for , 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,
48, 51, 60, 64, ... (OEIS A003401).
A complete enumeration of "constructible" polygons is given by those with central angles corresponding to so-called trigonometry
angles.
Gardner (1977) and independently Watkins (Conway and Guy 1996, Krížek et al. 2001) noticed that the number of sides for constructible polygons with
odd numbers of sides are given by the first 32 rows
of the Sierpiński sieve interpreted as binary numbers, giving 1, 3, 5, 15, 17, 51, 85, 255, ...
(OEIS A004729, Conway and Guy 1996, p. 140).
In other words, every row is a product of distinct Fermat
primes, with terms given by binary counting.
-gon to be constructible, which he also conjectured (but did
not prove) to be necessary, thus showing that regular
-gons
were constructible for 
, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,
48, 51, 60, 64, ... (OEIS A003401).
à la Gauss." Mathematica Educ. Res. 4, 31-36, 1995.Trott,
M. "
à la Gauss." §1.10.2 in