Editing Heptagon (section)

== Regular heptagon ==
A '''[[regular polygon|regular]] heptagon''', in which all sides and all angles are equal, has [[internal angle]]s of 5π/7 [[radian]]s (128{{frac|4|7}} [[degree (angle)|degree]]s). Its [[Schläfli symbol]] is {7}.

===Area===
The area (''A'') of a regular heptagon of side length ''a'' is given by:

:<math>A = \frac{7}{4}a^2 \cot \frac{\pi}{7} \simeq 3.634 a^2.</math>

This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with [[Vertex (geometry)|vertices]] at the center and at the heptagon's vertices, and then halving each triangle using the [[apothem]] as the common side. The apothem is half the [[cotangent]] of <math>\pi/7, </math> and the area of each of the 14 small triangles is one-fourth of the apothem.

The area of a regular heptagon [[cyclic polygon|inscribed]] in a circle of [[radius]] ''R''  is <math>\tfrac{7R^2}{2}\sin\tfrac{2\pi}{7}, </math> while the area of the circle itself is <math>\pi R^2;</math> thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.

===Construction===
As 7 is a [[Pierpont prime]] but not a [[Fermat prime]], the regular heptagon is not [[Constructible polygon|constructible]] with [[compass and straightedge]] but is constructible with a marked [[ruler]] and compass. It is the smallest regular polygon with this property. 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 <math>\scriptstyle {2\cos{\tfrac{2\pi}{7}} \approx 1.247}</math> is a zero of the [[irreducible polynomial|irreducible]] [[cubic function|cubic]] {{nowrap|''x''<sup>3</sup> + ''x''<sup>2</sup> − 2''x'' − 1}}. Consequently, this polynomial is the [[minimal polynomial (field theory)|minimal polynomial]] of {{nobreak|2cos({{frac|2π|7}}),}} whereas the degree of the minimal polynomial for a [[constructible number]] must be a power of 2.

{| class=wikitable width=640
|[[File:Neusis-heptagon.png|330px]]<br>A ''neusis construction'' of the interior angle in a regular heptagon.
|[[File:01-Siebeneck-Tomahawk-Animation.gif|380px]]<br>An animation from a neusis construction with radius of circumcircle <math>\overline{OA} = 6</math>, according to [[Andrew M. Gleason]]<ref name="Gleason">{{cite journal|last=Gleason|first=Andrew Mattei|title=Angle trisection, the heptagon, and the triskaidecagon p. 186 (Fig.1) –187 |journal=The American Mathematical Monthly|date=March 1988|volume=95|issue=3 |pages=185–194|url=http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/1.pdf#3 |archiveurl=https://web.archive.org/web/20151219180208/http://apollonius.math.nthu.edu.tw/d1/ne01/jyt/linkjstor/regular/7.pdf#3 |doi= 10.2307/2323624|jstor=2323624 |archivedate=2015-12-19 |url-status=dead}}</ref> based on the [[angle trisection]] by means of the [[Tomahawk_(geometry)|tomahawk]]. This construction relies on the fact that

<math>6\cos\left(\frac{2\pi}{7}\right)=2\sqrt{7}\cos\left(\frac{1}{3}\arctan\left(3\sqrt{3}\right)\right)-1.</math>
|}

[[File:01-Siebeneck-nach Johnson.gif|thumb|left|400px|Heptagon with ''given side length'':<br />
An animation from a [[neusis construction]] with marked ruler, according to David Johnson Leisk ([[Crockett Johnson]]).]]

<br />{{clear}}

===Approximation===
An approximation for practical use with an error of about 0.2% is to use half the side of an equilateral triangle inscribed in the same circle as the length of the side of a regular heptagon. It is unknown who first found this approximation, but it was mentioned by [[Heron of Alexandria]]'s ''Metrica'' in the 1st century AD, was well known to medieval Islamic mathematicians, and can be found in the work of [[Albrecht Dürer]].<ref>{{cite journal |title=Abu'l-Jūd's Answer to a Question of al-Bīrūnī Concerning the Regular Heptagon |last=Hogendijk |first=Jan P. |year=1987 |journal=Annals of the New York Academy of Sciences |volume=500 |issue=1 |url=https://www.jphogendijk.nl/publ/Abuljud.pdf |pages=175–183 |doi=10.1111/j.1749-6632.1987.tb37202.x}}</ref><ref>G.H. Hughes, [https://arxiv.org/ftp/arxiv/papers/1205/1205.0080.pdf#12 "The Polygons of Albrecht Dürer-1525, The Regular Heptagon", Fig. 11] [https://arxiv.org/ftp/arxiv/papers/1205/1205.0080.pdf#15 the side of the Heptagon (7) Fig. 15, image on the left side], retrieved on 4 December 2015</ref> Let ''A'' lie on the circumference of the circumcircle. Draw arc ''BOC''. Then <math>\scriptstyle {BD = {1 \over 2}BC}</math> gives an approximation for the edge of the heptagon.

This approximation uses <math>\scriptstyle {\sqrt{3} \over 2} \approx 0.86603 </math> for the side of the heptagon inscribed in the unit circle while the exact value is <math>\scriptstyle 2\sin{\pi \over 7} \approx 0.86777</math>.

''Example to illustrate the error:<br />
At a circumscribed circle radius ''r = 1&nbsp;m'', the absolute error of the 1st side would be ''approximately -1.7&nbsp;mm''

[[File:7-gone approx.png|240px]]

===Other approximations===

There are other approximations of a heptagon using compass and straightedge, but they are time consuming to draw.
<ref> raumannkidwai. "Heptagon." Chart. Geogebra. Accessed January 20, 2024.
     https://www.geogebra.org/classic/CvsudDWr.
</ref>
{{clear}}

=== Symmetry ===
[[File:Symmetries_of_heptagon.png|thumb|200px|Symmetries of a regular heptagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.<ref>John H. Conway, Heidi Burgiel, [[Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{ISBN|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)</ref>]]

The ''regular heptagon'' belongs to the [[dihedral symmetry|D<sub>7h</sub>]] [[point group]] ([[Schoenflies notation]]), order 28. The symmetry elements are: a 7-fold proper rotation axis C<sub>7</sub>, a 7-fold improper rotation axis, S<sub>7</sub>, 7 vertical mirror planes, σ<sub>v</sub>, 7 2-fold rotation axes, C<sub>2</sub>, in the plane of the heptagon and a horizontal mirror plane, σ<sub>h</sub>, also in the heptagon's plane.<ref>{{cite book |last1=Salthouse |first1=J.A |url=https://books.google.com/books?id=GAw4AAAAIAAJ |title=Point group character tables and related data |last2=Ware |first2=M.J. |date=1972 |publisher=Cambridge University Press |isbn=0-521-08139-4 |location=Cambridge}}</ref> 
<!-- These 4 symmetries can be seen in 4 distinct symmetries on the heptagon. [[John Horton Conway|John Conway]] labels these by a letter and group order. Full symmetry of the regular form is '''r14''' and no symmetry is labeled '''a1'''. The dihedral symmetries are divided depending on whether they pass through vertices ('''d''' for diagonal) or edges ('''p''' for perpendiculars), and '''i''' when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as '''g''' for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the '''g7''' subgroup has no degrees of freedom but can seen as [[directed edge]]s.
 -->
===Diagonals and heptagonal triangle===
{{main|Heptagonal triangle}}
[[File:Heptagrams.svg|thumb|100px|''a''=red, ''b''=blue, ''c''=green lines]]

The regular heptagon's side ''a'', shorter [[diagonal#Polygons|diagonal]] ''b'', and longer diagonal ''c'', with ''a''<''b''<''c'', satisfy<ref name=Altintas>Abdilkadir Altintas, "Some Collinearities in the Heptagonal Triangle", ''[[Forum Geometricorum]]'' 16, 2016, 249–256.http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf</ref>{{rp|Lemma 1}}

:<math>a^2=c(c-b),</math>
:<math>b^2 =a(c+a),</math>
:<math>c^2 =b(a+b),</math>
:<math>\frac{1}{a}=\frac{1}{b}+\frac{1}{c}</math>  (the [[optic equation]])

and hence

:<math> ab+ac=bc,</math>

and<ref name=Altintas/>{{rp|Coro. 2}}

:<math>b^3+2b^2c-bc^2-c^3=0, </math>
:<math>c^3-2c^2a-ca^2+a^3=0, </math>
:<math>a^3-2a^2b-ab^2+b^3=0,</math>

Thus –''b''/''c'', ''c''/''a'', and ''a''/''b'' all satisfy the [[cubic equation]] <math>t^3-2t^2-t + 1=0.</math> However, no [[algebraic expression]]s with purely real terms exist for the solutions of this equation, because it is an example of [[casus irreducibilis]].

The approximate lengths of the diagonals in terms of the side of the regular heptagon are given by

:<math>b\approx 1.80193\cdot a, \qquad c\approx 2.24698\cdot a.</math>

We also have<ref>Leon Bankoff and Jack Garfunkel, "The heptagonal triangle", ''[[Mathematics Magazine]]'' 46 (1), January 1973, 7–19.</ref>

:<math>b^2-a^2=ac,</math>

:<math>c^2-b^2=ab,</math>

:<math>a^2-c^2=-bc,</math>

and

:<math>\frac{b^2}{a^2}+\frac{c^2}{b^2}+\frac{a^2}{c^2}=5.</math>

A [[heptagonal triangle]] has [[vertex (geometry)|vertices]] coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles <math>\pi/7, 2\pi/7,</math> and <math>4\pi/7.</math> Thus its sides coincide with one side and two particular [[diagonal#Polygons|diagonals]] of the regular heptagon.<ref name=Altintas/>

===In polyhedra===

Apart from the [[heptagonal prism]] and [[heptagonal antiprism]], no convex polyhedron made entirely out of regular polygons contains a heptagon as a face.