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{{short description|Regular polygon that can be constructed with compass and straightedge}}
{{short description|Regular polygon that can be constructed with compass and straightedge}}
[[Image:Pentagon construct.gif|thumb|Construction of a regular pentagon]]
[[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.
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==
==Conditions for constructibility==
[[File:constructible_polygoon_set.svg|thumb|300px|Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)]]
[[File:constructible_polygoon_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 17-gon]]
[[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 polygons with ''n'' edges) are constructible and which are not?
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?


[[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]], 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''':
[[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]], 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 (that is, a polygon with ''n'' sides) 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 [[Fermat prime]]s (including none).
: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 [[Fermat prime]]s (including none).


(A Fermat prime is a [[prime number]] of the form <math>2^{(2^m)}+1.</math>)
A Fermat prime is a [[prime number]] of the form <math>2^{(2^m)} + 1.</math>


In order to reduce a 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 roots. Equivalently, a regular ''n''-gon is constructible if any [[root of a function|root]] of the ''n''th [[cyclotomic polynomial]] is constructible.
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===
===Detailed results by Gauss's theory===


Restating the Gauss-Wantzel theorem:
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 integers, and the ''p''<sub>''i''</sub>'s (when ''t'' > 0) are distinct Fermat primes.
: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:
The five known [[Fermat primes]] are:
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Since there are 31 combinations of anywhere from one to five Fermat primes, there are 31 known constructible polygons with an odd number of sides.
Since there are 31 combinations of anywhere from one to five 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.<ref>[http://www.prothsearch.com/fermat.html Prime factors k · 2n + 1 of Fermat numbers Fm
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>
and complete factoring status] by Wilfrid Keller.</ref>


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


Since there are 5 known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence 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, 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 1s 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 × 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 odd-sided regular constructible polygons.
Since there are 5 known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence 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 1s 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==
==General theory==
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| series = Pure and Applied Mathematics
| series = Pure and Applied Mathematics
| title = Galois Theory
| 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 over the base field that is a power of two.
| 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]]
In the specific case of a regular ''n''-gon, the question reduces to the question of [[constructible number|constructing a length]]


:cos {{sfrac|2{{pi}}|''n''}} ,
: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 subfield, which is a [[Totally real number field|totally real field]] and a [[rational number|rational]] [[vector space]] of [[Hamel dimension|dimension]]
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]]


:½φふぁい(''n''),
:½&thinsp;φふぁい(''n''),


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


As for the construction of Gauss, when the Galois group is 2-group it follows that it has a sequence of subgroups of orders
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, ...
:1, 2, 4, 8, ...


that are nested, each in the next (a [[composition series]], in [[group theory]] terms), something simple to prove by 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'' = 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
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 {{sfrac|2{{pi}}|17}} .
: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.
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.
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== Compass and straightedge constructions ==
== Compass and straightedge constructions ==


[[Compass and straightedge construction]]s are known for all known constructible polygons. If ''n''&nbsp;=&nbsp;''p''·''q'' with ''p''&nbsp;=&nbsp;2 or ''p'' and ''q'' [[coprime]], an ''n''-gon can be constructed from a ''p''-gon and a ''q''-gon.
[[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, 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 relatively prime, there exists integers ''a'',''b'' such that ''ap + bq = 1''. Then ''2aπぱい/q + 2bπぱい/p = 2πぱい/pq''. From this, a ''p''·''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ぱい/''q'' + 2''b''πぱい/''p'' = 2πぱい/''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.
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]].
*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 BC), and by [[Ptolemy]] (''[[Almagest]]'', ca AD 150). See [[pentagon]].
*Constructions for the regular pentagon were described both by [[Euclid]] (''[[Euclid's Elements|Elements]]'', ca 300 BC), and by [[Ptolemy]] (''[[Almagest]]'', ca AD 150); see [[Pentagon]].
*Although Gauss ''proved'' that the regular 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' work. See [[heptadecagon]].
*Although Gauss ''proved'' that the regular 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' work; see [[Heptadecagon]].
*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}}</ref>
*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}}</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>
*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>
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[[File:Regular 257-gon Using Carlyle Circle.gif]]
[[File:Regular 257-gon Using Carlyle Circle.gif]]
[[File:Regular 65537-gon First Carlyle Circle.gif|257px]]<BR>
[[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 complete.
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==
==Other constructions==


The concept of constructibility as discussed in this article applies specifically to [[compass and straightedge]] construction. 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.
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 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|archivedate=2015-12-19 }}</ref>{{rp|Thm. 2}}
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|archivedate=2015-12-19 }}</ref>{{rp|Thm. 2}}


==See also==
==See also==

Revision as of 23:00, 7 August 2021

Construction of a regular pentagon

In mathematics, a constructible polygon is a regular polygon that can be 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

Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)
Construction of the regular 17-gon

Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides,[1]: p. xi  and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.[1]: 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, polygons with n edges) are constructible and which are not?

Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods 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, 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 Fermat primes (including none).

A Fermat prime is a prime number of the form

In order to reduce a 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 is a constructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. Equivalently, a regular n-gon is constructible if any root of the nth 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 = 2kp1p2...pt where k and t are non-negative integers, and the pi's (when t > 0) are distinct Fermat primes.

The five known Fermat primes are:

F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in the OEIS).

Since there are 31 combinations of anywhere from one to five Fermat primes, there are 31 known constructible polygons with an odd number of sides.

The next twenty-eight Fermat numbers, F5 through F32, are known to be composite.[2]

Thus a regular n-gon is constructible if

n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 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, ... (sequence A003401 in the OEIS),

while a regular n-gon is not constructible with compass and straightedge if

n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 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, ... (sequence A004169 in the OEIS).

Connection to Pascal's triangle

Since there are 5 known Fermat primes, we know of 31 numbers that are products of distinct Fermat primes, and hence 31 constructible odd-sided regular polygons. These are 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295 (sequence A045544 in the OEIS). As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the modulo-2 Pascal's triangle, minus the top row, which corresponds to a monogon. (Because of this, the 1s 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 × 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 2q−1 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 equations.[3] In terms of field theory, such lengths must be contained in a field extension generated by a tower of quadratic extensions. It follows that a field generated by constructions will always have 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 constructing a length

cos 2πぱい/n ,

which is a trigonometric number and hence an algebraic number. This number lies in the n-th cyclotomic field — and in fact in its real subfield, which is a totally real field and a rational vector space of dimension

½ φふぁい(n),

where φふぁい(n) is Euler's totient function. Wantzel's result comes down to a calculation showing that φふぁい(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 subgroups 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 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 n = 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 2πぱい/17 .

Each of those is a root of a quadratic equation in terms of the one before. Moreover, these equations have real rather than 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 constructions are known for all known constructible polygons. If n = pq with p = 2 or p and q coprime, an n-gon can be constructed from a p-gon and a q-gon.

  • If p = 2, draw a q-gon and bisect one of its central angles. From this, a 2q-gon can be constructed.
  • If p > 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 2aπぱい/q + 2bπぱい/p = 2πぱい/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.

Gallery


From left to right, constructions of a 15-gon, 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 constructions, 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 trisector if and only if where r, s, k ≥ 0 and where the pi are distinct Pierpont primes greater than 3 (primes of the form [7]: Thm. 2 

See also

References

  1. ^ a b Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969).
  2. ^ [http://www.prothsearch.com/fermat.html Prime factors k · 2n + 1 of Fermat numbers Fm and complete factoring status] by Wilfrid Keller.
  3. ^ Cox, David A. (2012), "Theorem 10.1.6", Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, p. 259, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9.
  4. ^ Magnus Georg Paucker (1822). "Geometrische Verzeichnung des regelmäßigen Siebzehn-Ecks und Zweyhundersiebenundfünfzig-Ecks in den Kreis". Jahresverhandlungen der Kurländischen Gesellschaft für Literatur und Kunst (in German). 2: 160–219.
  5. ^ Friedrich Julius Richelot (1832). "De resolutione algebraica aequationis x257 = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata". Journal für die reine und angewandte Mathematik (in Latin). 9: 1–26, 146–161, 209–230, 337–358. doi:10.1515/crll.1832.9.337.
  6. ^ Johann Gustav Hermes (1894). "Über die Teilung des Kreises in 65537 gleiche Teile". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (in German). 3. Göttingen: 170–186.
  7. ^ Gleason, Andrew M. (March 1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. {{cite journal}}: |archive-date= requires |archive-url= (help)

External links

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