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 (including none). (A Fermat prime is a prime number of the form ) 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. Equivalently, a regular n-gon is constructible if and only if the cosine of its common angle is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. Regular skew polygons A regular skew polygon in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniform antiprism. All edges and internal angles are equal. More generally regular skew polygons can be defined in n-space. Examples include the Petrie polygons, polygonal paths of edges that divide a regular polytope into two halves, and seen as a regular polygon in orthogonal projection. In the infinite limit regular skew polygons become skew apeirogons. Regular star polygons A non-convex regular polygon is a regular star polygon. The most common example is the pentagram, which has the same vertices as a pentagon, but connects alternating vertices. For an n-sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as {n/m}. If m is 2, for example, then every second point is joined. If m is 3, then every third point is joined. The boundary of the polygon winds around the center m times. The (non-degenerate) regular stars of up to 12 sides are: Pentagram – {5/2} Heptagram – {7/2} and {7/3} Octagram – {8/3} Enneagram – {9/2} and {9/4} Decagram – {10/3} Hendecagram – {11/2}, {11/3}, {11/4} and {11/5} Dodecagram – {12/5} m and n must be coprime, or the figure will degenerate. The degenerate regular stars of up to 12 sides are: Tetragon – {4/2} Hexagons – {6/2}, {6/3} Octagons – {8/2}, {8/4} Enneagon – {9/3} Decagons – {10/2}, {10/4}, and {10/5} Dodecagons – {12/2}, {12/3}, {12/4}, and {12/6} Depending on the precise derivation of the