Heptagon

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

The regular heptagon is the simplest polygon not to appear on any non-prismatic uniform polyhedron. This is partially due to their I2(7) symmetry groups not being embedded in any higher fundamental Coxeter group. It's also the simplest polygons that cannot be constructed with a straightedge and a compass. .

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)).