# Heptagon

Heptagon
Rank2
TypeRegular
SpaceSpherical
Bowers style acronymHeg
Info
Coxeter diagramx7o
Schläfli symbol{7}
SymmetryI2(7), order 14
ArmyHeg
Elements
Edges7
Vertices7
Measures (edge length 1)
Circumradius${\displaystyle \frac{1}{2\sin\frac\pi7} ≈ 1.15238}$
Inradius${\displaystyle \frac{1}{2\tan\frac\pi7} ≈ 1.03826}$
Area${\displaystyle \frac{7}{4\tan\frac\pi7} ≈ 3.63391}$
Angle${\displaystyle \frac{5\pi}{7} ≈ 128.57143°}$
Central density1
Euler characteristic0
Number of pieces7
Level of complexity1
Related polytopes
DualHeptagon
ConjugatesHeptagram, great heptagram
Properties
ConvexYes
OrientableYes
NatureTame

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.[1]

Furthermore, in contrast to polygons with fewer sides, there is 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.[2] Nevertheless, regular heptagons can tile the hyperbolic plane, as in the order-3 heptagonal tiling, for example.

## Naming

The name heptagon is derived from the Ancient Greek ἑπτά (7) and γωνία (angle), referring to the number of vertices.

Other names include:

• Heg, Bowers style acronym, short for "heptagon".
• Septagon, based on Latin septum.

## 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)).

## Variations

Besides the regular heptagon, other less regular heptagons with mirror or no symmetry exist. However, onne of these polygons can tile the plain, or appear as vertex figures in higher polytopes.

## References

1. Online Encyclopedia of Integer Sequences. "A003401".
2. Semidoc (March 9, 2018). "No tiling by convex heptagons".