Editing Constructible polygon

{{short description|Regular polygon that can be constructed with compass and straightedge}}
[[Image:Pentagon construct.gif|thumb|Construction of a regular pentagon]]
In [[mathematics]], a '''constructible polygon''' is a [[regular polygon]] that can be [[Compass and straightedge constructions|constructed with compass and straightedge]]. For example, a regular [[pentagon]] is constructible with compass and straightedge while a regular [[heptagon]] is not. There are infinitely many constructible polygons, but only 31 with an [[odd number]] of sides are known.

==Conditions for constructibility==
[[File:Constructible polygon set.svg|thumb|300px|Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)]]
[[File:HeptadecagonConstructionAni.gif|thumb|Construction of the regular {{nowrap|17-gon}}]]
Some regular polygons are easy to construct with compass and straightedge; others are not. The [[Greek mathematics|ancient Greek mathematicians]] knew how to construct a regular polygon with 3, 4, or 5 sides,<ref name=Bold/>{{rp|p. xi}} and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.<ref name=Bold>Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969).</ref>{{rp|pp. 49–50}} This led to the question being posed: is it possible to construct ''all'' regular polygons with compass and straightedge? If not, which ''n''-gons (that is, [[polygon]]s with ''n'' edges) are constructible and which are not?

{{anchor|1=Gauss–Wantzel theorem}}[[Carl Friedrich Gauss]] proved the constructibility of the regular [[heptadecagon|17-gon]] in 1796. Five years later, he developed the theory of [[Gaussian period]]s in his ''[[Disquisitiones Arithmeticae]]''. This theory allowed him to formulate a [[sufficient condition]] for the constructibility of regular polygons. Gauss stated without proof that this condition was also [[necessary condition|necessary]],<ref>{{cite book |last1=Gauss |first1=Carl Friedrich |title=Disquisitiones arithmeticae |date=1966 |publisher=Yale University Press |location=New Haven and London |pages=458–460 |url=https://archive.org/details/disquisitionesar0000carl/ |access-date=25 January 2023}}</ref> but never published his proof.

A full proof of necessity was given by [[Pierre Wantzel]] in 1837. The result is known as the '''Gauss–Wantzel theorem''': A regular ''n''-gon can be constructed with compass and straightedge [[if and only if]] ''n'' is the product of a [[power of 2]] and any number of distinct (unequal) [[Fermat prime]]s. Here, a power of 2 is a number of the form <math>2^m</math>, where ''m'' &ge; 0 is an integer. A Fermat prime is a [[prime number]] of the form <math>2^{(2^m)} + 1</math>, where ''m'' &ge; 0 is an integer. The number of Fermat primes involved can be 0, in which case ''n'' is a power of 2.

In order to reduce a [[geometry|geometric]] problem to a problem of pure [[number theory]], the proof uses the fact that a regular ''n''-gon is constructible if and only if the [[cosine]] <math>\cos(2\pi/n)</math> is a [[constructible number]]—that is, can be written in terms of the four basic arithmetic operations and the extraction of [[square root]]s. Equivalently, a regular ''n''-gon is constructible if any [[root of a function|root]] of the ''n''th  [[cyclotomic polynomial]] is constructible.

===Detailed results by Gauss's theory===

Restating the Gauss–Wantzel theorem:
:A regular ''n''-gon is constructible with straightedge and compass if and only if ''n'' = 2<sup>''k''</sup>''p''<sub>1</sub>''p''<sub>2</sub>...''p''<sub>''t''</sub> where ''k'' and ''t'' are non-negative [[integer]]s, and the ''p''<sub>''i''</sub>'s (when ''t'' > 0) are distinct Fermat primes.

The five known [[Fermat primes]] are:
:''F''<sub>0</sub> = 3, ''F''<sub>1</sub> = 5, ''F''<sub>2</sub> = 17, ''F''<sub>3</sub> = 257, and ''F''<sub>4</sub> = 65537 {{OEIS|id=A019434}}.

Since there are 31 nonempty subsets of the five known Fermat primes, there are 31 known constructible polygons with an odd number of sides.

The next twenty-eight Fermat numbers, ''F''<sub>5</sub> through ''F''<sub>32</sub>, are known to be [[composite number|composite]].<ref>[http://www.prothsearch.com/fermat.html Prime factors  k · 2n + 1  of Fermat numbers  Fm and complete factoring status] by Wilfrid Keller.</ref>

Thus a regular ''n''-gon is constructible if
:''n'' = [[Equilateral triangle|3]], [[Square|4]], [[Pentagon|5]], [[Hexagon|6]], [[Octagon|8]], [[Decagon|10]], [[Dodecagon|12]], [[Pentadecagon|15]], [[Hexadecagon|16]], [[Heptadecagon|17]], [[Icosagon|20]], [[Icositetragon|24]], [[Triacontagon|30]], 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, [[257-gon|257]], 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, ... {{OEIS|id=A003401}},

while a regular ''n''-gon is not constructible with compass and straightedge if
:''n'' = [[Heptagon|7]], [[Enneagon|9]], [[Hendecagon|11]], [[Tridecagon|13]], [[Tetradecagon|14]], [[Octadecagon|18]], 19, 21, 22, [[Icositrigon|23]], 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, ... {{OEIS|id=A004169}}.

===Connection to Pascal's triangle===

Since there are five known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence we know of 31 constructible odd-sided regular polygons. These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, [[65537-gon|65537]], 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295 {{OEIS|id=A045544}}. As [[John Conway]] commented in ''The Book of Numbers'', these numbers, when written in [[binary number|binary]], are equal to the first 32 rows of the [[Modular arithmetic|modulo]]-2 [[Pascal's triangle]], minus the top row, which corresponds to a [[monogon]]. (Because of this, the ''1''s in such a list form an approximation to the [[Sierpiński triangle]].) This pattern breaks down after this, as the next Fermat number is composite (4294967297 = 641&thinsp;×&nbsp;6700417), so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, and it is therefore unknown how many odd-sided constructible regular polygons exist. In general, if there are ''q'' Fermat primes, then there are 2<sup>''q''</sup>−1 {{nowrap|odd-sided}} regular constructible polygons.

==General theory==

In the light of later work on [[Galois theory]], the principles of these proofs have been clarified. It is straightforward to show from [[analytic geometry]] that constructible lengths must come from base lengths by the solution of some sequence of [[quadratic equation]]s.<ref>{{citation
 | last = Cox | first = David A. | authorlink = David A. Cox
 | contribution = Theorem 10.1.6
 | doi = 10.1002/9781118218457
 | edition = 2nd
 | isbn = 978-1-118-07205-9
 | page = 259
 | publisher = John Wiley & Sons
 | series = Pure and Applied Mathematics
 | title = Galois Theory
 | year = 2012}}.</ref> In terms of [[field theory (mathematics)|field theory]], such lengths must be contained in a [[field extension]] generated by a tower of [[quadratic extension]]s. It follows that a field generated by constructions will always have [[degree of a field extension|degree]] over the base field that is a power of two.

In the specific case of a regular ''n''-gon, the question reduces to the question of [[constructible number|constructing a length]]

:cos&thinsp;{{sfrac|2{{pi}}|''n''}} ,

which is a [[trigonometric number]] and hence an [[algebraic number]]. This number lies in the ''n''-th [[cyclotomic field]] &mdash; and in fact in its [[real number|real]] [[field extension|subfield]], which is a [[Totally real number field|totally real field]] and a [[rational number|rational]] [[vector space]] of [[Hamel dimension|dimension]]

:½&thinsp;φ(''n''),

where φ(''n'') is [[Euler's totient function]]. Wantzel's result comes down to a calculation showing that φ(''n'') is a power of 2 precisely in the cases specified.

As for the construction of Gauss, when the [[Galois group]] is a 2-group it follows that it has a sequence of [[subgroup]]s of orders

:1, 2, 4, 8, ...

that are nested, each in the next (a [[composition series]], in [[group theory]] terminology), something simple to prove by [[mathematical induction|induction]] in this case of an [[abelian group]]. Therefore, there are subfields nested inside the cyclotomic field, each of degree 2 over the one before. Generators for each such field can be written down by [[Gaussian period]] theory. For example, for [[heptadecagon|''n''&nbsp;=&thinsp;17]] there is a period that is a sum of eight [[roots of unity]], one that is a sum of four roots of unity, and one that is the sum of two, which is

:cos&thinsp;{{sfrac|2{{pi}}|17}} .

Each of those is a root of a [[quadratic equation]] in terms of the one before. Moreover, these equations have [[real number|real]] rather than [[complex number|complex]] roots, so in principle can be solved by geometric construction: this is because the work all goes on inside a totally real field.

In this way the result of Gauss can be understood in current terms; for actual calculation of the equations to be solved, the periods can be squared and compared with the 'lower' periods, in a quite feasible algorithm.

== 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.

==Other constructions==

The concept of constructibility as discussed in this article applies specifically to [[compass and straightedge]] constructions. More constructions become possible if other tools are allowed. The so-called [[neusis construction]]s, for example, make use of a ''marked'' ruler. The constructions are a mathematical idealization and are assumed to be done exactly.

A regular polygon with ''n'' sides can be constructed with ruler, compass, and [[angle trisection|angle trisector]] if and only if <math>n=2^r3^sp_1p_2\cdots p_k,</math> where ''r, s, k'' ≥ 0 and where the ''p''<sub>''i''</sub> are distinct [[Pierpont prime]]s greater than 3 (primes of the form <math>2^t3^u +1).</math><ref name="Gleason">{{cite journal|last=Gleason|first=Andrew M.|authorlink=Andrew M. Gleason|title=Angle trisection, the heptagon, and the triskaidecagon |journal=[[American Mathematical Monthly]]|date=March 1988|volume=95|issue=3 |pages=185–194 |doi= 10.2307/2323624|jstor=2323624 }}</ref>{{rp|Thm. 2}} These polygons are exactly the regular polygons that can be constructed with [[conic section]]s, and the regular polygons that can be constructed with [[Mathematics of paper folding|paper folding]]. The first numbers of sides of these polygons are:
:3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, 111, 112, 114, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 144, 146, 148, 152, 153, 156, 160, 162, 163, 168, 170, 171, 180, 182, 185, 189, 190, 192, 193, 194, 195, 204, 208, 210, 216, 218, 219, 221, 222, 224, 228, 234, 238, 240, 243, 247, 252, 255, 256, 257, 259, 260, 266, 270, 272, 273, 280, 285, 288, 291, 292, 296, ... {{OEIS|A122254}}

==See also==
*[[Polygon]]
*[[Carlyle circle]]

==References==

<references/>

==External links==
* {{cite journal | author=Duane W. DeTemple |title=Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions | journal=[[The American Mathematical Monthly]] |volume=98 |issue=2 |year=1991 |pages=97–108 |doi=10.2307/2323939 |mr=1089454 |jstor=2323939}}
* {{cite journal | author=Christian Gottlieb |title=The Simple and Straightforward Construction of the Regular 257-gon |journal=[[Mathematical Intelligencer]] |volume=21 |issue=1 |year=1999 |pages=31–37 |mr=1665155 | doi=10.1007/BF03024829|s2cid=123567824 }}
* [https://web.archive.org/web/20080412084431/http://mathforum.org/dr.math/faq/formulas/faq.regpoly.html Regular Polygon Formulas], Ask Dr. Math FAQ.
* Carl Schick: Weiche Primzahlen und das 257-Eck : eine analytische Lösung des 257-Ecks. Zürich : C. Schick, 2008. {{ISBN|978-3-9522917-1-9}}.
*[https://commons.wikimedia.org/wiki/File:01-65.537-Eck-Quadratrix.svg 65537-gon, exact construction for the 1st side], using the [[Quadratrix of Hippias]] and [[GeoGebra]] as additional aids, with brief description (German)

[[Category:Constructible polygons| ]]
[[Category:Euclidean plane geometry]]
[[Category:Carl Friedrich Gauss]]
[[Category:Greek mathematics]]