Editing Constructible polygon (section)

==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.