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Regular heptagon

A regular heptagon
Edge and vertices 7
Schläfli symbol {7}
Coxeter–Dynkin diagram
Symmetry group Dihedral (D7)
Area
(with t=edge length)
A=\frac{7}{4}t^2 \cot \frac{\pi}{7}
\simeq 3.63391 t^2.
Internal angle
(degree)
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 degree. Its Schläfli symbol is {7}. The area A of a regular heptagon of side length a is given by
A = \frac{7}{4}a^2 \cot \frac{\pi}{7} \simeq 3.63391 a^2.


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 angletrisector. 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 \approx 0.2% is shown in the drawing. Let A lie on the circumference of the circumcircle. Draw arc BOC. Then BD = {1 \over 2}BC 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 Kingdommarker currently (2009) has two heptagonal coins, the 50p and 20p pieces, and the Barbadosmarker 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 Brazilianmarker 25 cents coin has a heptagon inscribed in the coin's disk. Some old versions of Coat of arms of Georgia 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



External links




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