Inmathematics, aconstructible polygon is aregular polygon that can beconstructed with compass and straightedge. For example, a regularpentagon is constructible with compass and straightedge while a regularheptagon is not. There are infinitely many constructible polygons, but only 31 with anodd number of sides are known.
Some regular polygons are easy to construct with compass and straightedge; others are not. Theancient 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 constructall regular polygons with compass and straightedge? If not, whichn-gons (that is,polygons withn edges) are constructible and which are not?
Carl Friedrich Gauss proved the constructibility of the regular17-gon in 1796. Five years later, he developed the theory ofGaussian periods in hisDisquisitiones Arithmeticae. This theory allowed him to formulate asufficient condition for the constructibility of regular polygons. Gauss stated without proof that this condition was alsonecessary,[2] but never published his proof.
A full proof of necessity was given byPierre Wantzel in 1837. The result is known as theGauss–Wantzel theorem: A regularn-gon can be constructed with compass and straightedgeif and only ifn is the product of apower of 2 and any number of distinct (unequal)Fermat primes. Here, a power of 2 is a number of the form, wherem ≥ 0 is an integer. A Fermat prime is aprime number of the form, wherem ≥ 0 is an integer. The number of Fermat primes involved can be 0, in which casen is a power of 2.
In order to reduce ageometric problem to a problem of purenumber theory, the proof uses the fact that a regularn-gon is constructible if and only if thecosine is aconstructible number—that is, can be written in terms of the four basic arithmetic operations and the extraction ofsquare roots. Equivalently, a regularn-gon is constructible if anyroot of thenthcyclotomic polynomial is constructible.
Restating the Gauss–Wantzel theorem:
The five knownFermat primes are:
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,F5 throughF32, are known to becomposite.[3]
Thus a regularn-gon is constructible if
while a regularn-gon is not constructible with compass and straightedge if
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, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295 (sequenceA045544 in theOEIS). AsJohn Conway commented inThe Book of Numbers, these numbers, when written inbinary, are equal to the first 32 rows of themodulo-2Pascal's triangle, minus the top row, which corresponds to amonogon. (Because of this, the1s in such a list form an approximation to theSierpiński 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 areq Fermat primes, then there are 2q−1odd-sided regular constructible polygons.
In the light of later work onGalois theory, the principles of these proofs have been clarified. It is straightforward to show fromanalytic geometry that constructible lengths must come from base lengths by the solution of some sequence ofquadratic equations.[4] In terms offield theory, such lengths must be contained in afield extension generated by a tower ofquadratic extensions. It follows that a field generated by constructions will always havedegree over the base field that is a power of two.
In the specific case of a regularn-gon, the question reduces to the question ofconstructing a length
which is atrigonometric number and hence analgebraic number. This number lies in then-thcyclotomic field — and in fact in itsrealsubfield, which is atotally real field and arationalvector space ofdimension
where φ(n) isEuler'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 theGalois group is a 2-group it follows that it has a sequence ofsubgroups of orders
that are nested, each in the next (acomposition series, ingroup theory terminology), something simple to prove byinduction in this case of anabelian 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 byGaussian period theory. For example, forn = 17 there is a period that is a sum of eightroots of unity, one that is a sum of four roots of unity, and one that is the sum of two, which is
Each of those is a root of aquadratic equation in terms of the one before. Moreover, these equations havereal rather thancomplex 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 are known for all known constructible polygons. Ifn = pq withp = 2 orp andqcoprime, ann-gon can be constructed from ap-gon and aq-gon.
Thus one only has to find a compass and straightedge construction forn-gons wheren is a Fermat prime.
From left to right, constructions of a15-gon,17-gon,257-gon and65537-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.
The concept of constructibility as discussed in this article applies specifically tocompass and straightedge constructions. More constructions become possible if other tools are allowed. The so-calledneusis constructions, for example, make use of amarked ruler. The constructions are a mathematical idealization and are assumed to be done exactly.
A regular polygon withn sides can be constructed with ruler, compass, andangle trisector if and only if wherer, s, k ≥ 0 and where thepi are distinctPierpont primes greater than 3 (primes of the form[8]: Thm. 2 These polygons are exactly the regular polygons that can be constructed withconic sections, and the regular polygons that can be constructed withpaper folding. The first numbers of sides of these polygons are:
