# Heptagon

Heptagon | |
---|---|

Rank | 2 |

Type | Regular |

Space | Spherical |

Bowers style acronym | Heg |

Info | |

Coxeter diagram | x7o |

Schläfli symbol | {7} |

Symmetry | I2(7), order 14 |

Army | Heg |

Elements | |

Vertex figure | Dyad, length 2cos(π/7) |

Edges | 7 |

Vertices | 7 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Area | |

Angle | |

Central density | 1 |

Euler characteristic | 0 |

Number of pieces | 7 |

Level of complexity | 1 |

Related polytopes | |

Dual | Heptagon |

Conjugates | Heptagram, great heptagram |

Properties | |

Convex | Yes |

Orientable | Yes |

Nature | Tame |

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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.

## Stellations[edit | edit source]

- 1st stellation: Heptagram.
- 2nd stellation: Great heptagram.

## References[edit | edit source]

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

## External links[edit | edit source]

- Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

- Wikipedia Contributors. "Heptagon".