Heptagon

The heptagon, or heg, is a polygon with 7 sides. A regular heptagon has equal sides and equal angles.

The combining prefix is he-, as in hedip.

It has two stellations, these being the heptagram and the great heptagram.

The regular heptagon is the simplest polygon not to appear on any non-prismatic uniform polyhedron. This is partially due to its I2(7) symmetry group not being embedded in any higher fundamental Coxeter group. It's also the simplest polygon that cannot be constructed with a straightedge and a compass, as the expressions for its coordinates involve cubic roots.

Furthermore, in contrast to polygons with less sides, there's no single (convex) heptagon that can tile the plane without overlap. Intuitively, this is because the average angles around each vertex would have to be at least (15/14)×360°, a clear impossibility. This intuition may be formalized with bounds involving the Euler characteristic. Nevertheless, regular heptagons can tile the hyperbolic plane, as in the order-3 heptagonal tiling, for example.

Vertex coordinates
Coordinates for a regular heptagon of edge length 2sin(π/7), centered at the origin, are:


 * (1, 0),
 * (cos(2π/7), ±sin(2π/7)),
 * (cos(4π/7), ±sin(4π/7)),
 * (cos(6π/7), ±sin(6π/7)).

Stellations

 * 1st stellation: Heptagram
 * 2nd stellation: Great heptagram