Editing Constructible polygon (section)

== Compass and straightedge constructions ==

[[Compass and straightedge construction]]s are known for all known constructible polygons. If ''n''&nbsp;=&nbsp;''pq'' with ''p''&nbsp;=&nbsp;2 or ''p'' and ''q'' [[coprime]], an ''n''-gon can be constructed from a ''p''-gon and a ''q''-gon.
*If ''p''&nbsp;=&nbsp;2, draw a ''q''-gon and [[bisection|bisect]] one of its central angles. From this, a 2''q''-gon can be constructed.
*If ''p''&nbsp;>&nbsp;2, inscribe a ''p''-gon and a ''q''-gon in the same circle in such a way that they share a vertex. Because ''p'' and ''q'' are coprime, there exists integers ''a'' and ''b'' such that ''ap'' + ''bq'' = 1. Then 2''a''π/''q'' + 2''b''π/''p'' = 2π/''pq''. From this, a ''pq''-gon can be constructed.
Thus one only has to find a compass and straightedge construction for ''n''-gons where ''n'' is a Fermat prime.
*The construction for an equilateral [[triangle]] is simple and has been known since [[Ancient history|antiquity]]; see [[Equilateral triangle]].
*Constructions for the regular [[pentagon]] were described both by [[Euclid]] (''[[Euclid's Elements|Elements]]'', ca. 300&nbsp;BC), and by [[Ptolemy]] (''[[Almagest]]'', ca. 150&nbsp;AD).
*Although Gauss ''proved'' that the regular [[heptadecagon|17-gon]] is constructible, he did not actually ''show'' how to do it. The first construction is due to Erchinger, a few years after Gauss's work.
*The first explicit constructions of a regular [[257-gon]] were given by [[Magnus Georg Paucker]] (1822)<ref>{{cite journal |author=Magnus Georg Paucker |title=Geometrische Verzeichnung des regelmäßigen Siebzehn-Ecks und Zweyhundersiebenundfünfzig-Ecks in den Kreis |language=German |journal=Jahresverhandlungen der Kurländischen Gesellschaft für Literatur und Kunst |volume=2 |year=1822 | pages=160–219|url=https://books.google.com/books?id=aUJRAAAAcAAJ}}</ref> and [[Friedrich Julius Richelot]] (1832).<ref>{{cite journal |author=Friedrich Julius Richelot |title=De resolutione algebraica aequationis x<sup>257</sup> = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata |language=Latin |journal=Journal für die reine und angewandte Mathematik |volume=9 |year=1832 | pages=1–26, 146–161, 209–230, 337–358 |url=http://www.digizeitschriften.de/resolveppn/PPN243919689_0009 |doi=10.1515/crll.1832.9.337| s2cid=199545940 }}</ref>
*A construction for a regular [[65537-gon]] was first given by [[Johann Gustav Hermes]] (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.<ref>{{cite journal | author=Johann Gustav Hermes |title=Über die Teilung des Kreises in 65537 gleiche Teile |language=German |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse | location=Göttingen | year=1894 |volume=3 |pages=170–186 |url=http://www.digizeitschriften.de/resolveppn/GDZPPN002496585}}</ref>

===Gallery===
[[File:Regular_Pentadecagon_Inscribed_in_a_Circle.gif]]
[[File:Regular Heptadecagon Using Carlyle Circle.gif|257px]]
[[File:Regular 257-gon Using Carlyle Circle.gif]]
[[File:Regular 65537-gon First Carlyle Circle.gif|257px]]<BR>
From left to right, constructions of a [[pentadecagon|15-gon]], [[heptadecagon|17-gon]], [[257-gon]] and [[65537-gon]]. Only the first stage of the 65537-gon construction is shown; the constructions of the 15-gon, 17-gon, and 257-gon are given completely.