heptagon

Regular heptagon
A regular heptagon
Edges and vertices	7
Schläfli symbol	{7}
Coxeter–Dynkin diagram
Symmetry group	Dihedral (D7)
Area (with t=edge length)
Internal angle (degrees)	128.5714286°
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, a heptagon (or septagon) is a polygon with seven sides and seven angles. In a regular heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5π/7 radians, 128.5714286 degrees. Its Schläfli symbol is {7}. The area A of a regular heptagon of side length a is given by

The heptagon is also sometimes referred to as the septagon, using "sept-" (an elision of septua-, a Latin-derived numerical prefix, rather than hepta-, a Greek-derived numerical prefix). The OED lists "septagon" as meaning "heptagonal".

Construction

A regular heptagon is not constructible with compass and straightedge but is constructible with a marked ruler and compass. This type of construction is called a Neusis construction. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge and compass construction follows from the observation that 2cos(2π/7) ≈ 1.247 is a zero of the irreducible cubic x³ + x² - 2x - 1. Consequently this polynomial is the minimal polynomial of 2cos(2π/7), whereas the degree of the minimal polynomial for a constructible number must be a power of 2.

A Neusis construction of the interior angle in a regular heptagon.

A Neusis construction of the interior angle in a regular heptagon. (method by John Horton Conway

Approximation

A decent approximation for practical use with an accuracy of is shown in the drawing. Let A lie on the circumference of the circumcircle. Draw arc BOC. Then gives an approximation for the edge of the heptagon.

Heptagrams

Two kinds of heptagrams can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor being the interval of connection.

Blue, {7/2} and green {7/3} heptagrams inside a red heptagon.

Uses

The United Kingdom currently (2009) has two heptagonal coins, the 50p and 20p pieces, and the Barbados Dollar is also heptagonal. The 20 eurocent coin has cavities placed similarly. Strictly, the shape of the coins is a curvilinear heptagon to make them curves of constant width: the sides are curved outwards so that the coin will roll smoothly in vending machines. Botswana pula coins in the denominations of 2 Pula, 1 Pula, 50 Thebe and 5 Thebe are also shaped as equilateral-curve heptagons. The Brazilian 25 cents coin has a heptagon inscribed in the coin's disk. Some old versions of Coat of arms of Georgia (country) including Soviet days had used {7/2} heptagram as an element.

Regular heptagons can tile the hyperbolic plane, as shown in this Poincaré disk model projection:

heptagonal tiling

See also

Look up Heptagon in Wiktionary, the free dictionary.

• Heptagram

External links

• Definition and properties of a heptagon With interactive animation
• Approximate construction method
• Weisstein, Eric W., "Heptagon" from MathWorld.
• Another approximate construction method
• Polygons - Heptagons
• Recently discovered and highly accurate approximation for the construction of a regular heptagon.