heptagram

A heptagram is a seven-pointed star polygon. In geometry it denotes a star polygon with seven vertices obtained by connecting every k-th vertex of a regular heptagon, where k is relatively prime to 7. The two regular heptagrams are denoted by the Schläfli symbols {7/2} and {7/3}. The {7/2} star connects each vertex to the vertex two steps away, while the {7/3} connects every third vertex. These two figures share the same seven-point set but produce different crossing patterns. Both have seven points and dihedral symmetry of order 14 (D7).

Construction and properties: Place seven equally spaced points on a circle. For {7/2}, join i to i+2 (mod 7); for {7/3}, join i to i+3. The resulting lines intersect to form a self-intersecting polygon with a seven-point outline. The central angular spacing between consecutive vertices is 360/7 degrees, and the figures exhibit rotational symmetry by 360/7 degrees and reflectional symmetry through seven axes.

Variants and usage: In addition to the regular forms, seven-pointed stars appear in decorative arts, heraldry, and various cultural traditions. In esoteric and occult contexts, the term heptagram is used for seven-pointed symbolic designs, including unicursal forms intended to be drawn in one continuous stroke. In mathematics, heptagrams relate to the broader study of star polygons {n/k}, where gcd(n,k)=1 and n=7 in these cases.

See also: Star polygon; Seven-pointed star.