Constructible polygon
Encyclopedia
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
that can be constructed with compass and straightedge. For example, a regular pentagon
Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
is constructible with compass and straightedge while a regular heptagon is not.
Conditions for constructibility
Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not?Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
proved the constructibility of the regular 17-gon
Heptadecagon
In geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....
in 1796. Five years later, he developed the theory of Gaussian period
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
s in his Disquisitiones Arithmeticae
Disquisitiones Arithmeticae
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...
. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:
- A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes.
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
Pierre Wantzel
Pierre Laurent Wantzel was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge....
in 1837. The result is known as the Gauss–Wantzel theorem.
Detailed results by Gauss' theory
Only five Fermat primes are known:- F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537
The next twenty-eight Fermat numbers, F5 through F32, are known to be composite.
Thus an n-gon is constructible if
- n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, … ,
while an n-gon is not constructible with compass and straightedge if
- n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, … .
Connection to Pascal's triangle
There are 31 known numbers that are multiples of distinct Fermat primes, which correspond to the 31 odd-sided regular polygons that are known to be constructible. These are 3, 5, 15, 17, 51, 85, 255, 257, …, 4294967295 . As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the moduloModular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
-2 Pascal's triangle
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
, minus the top row. This pattern breaks down after there, as the 6th Fermat number is composite, so the following rows do not correspond to constructible polygons. It is unknown whether any more Fermat primes exist, and is therefore unknown how many odd-sided constructible polygons exist. In general, if there are x Fermat primes, then there are 2x−1 odd-sided constructible polygons.
General theory
In the light of later work on Galois theoryGalois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, the principles of these proofs have been clarified. It is straightforward to show from analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
that constructible lengths must come from base lengths by the solution of some sequence of quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
s. In terms of field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, 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/n).
This number lies in the n-th cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
— and in fact in its real subfield, which is a totally real field and a rational vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of dimension
- ½φ(n),
where φ(n) is Euler's totient function
Euler's totient function
In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...
. 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 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
Composition series
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence...
, in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
terms), something simple to prove by induction in this case of an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
. 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
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
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/17).
Each of those is a root of a quadratic equation in terms of the one before. Moreover these equations have real rather than imaginary roots, so in principle can be solved by geometric construction: this 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 constructible polygons. If n = p·q with p = 2 or p and q coprimeCoprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
, an n-gon can be constructed from a p-gon and a q-gon.
- If p = 2, draw a q-gon and bisectBisectionIn geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector. The most often considered types of bisectors are the segment bisector and the angle bisector In geometry, bisection is the division of something into two equal...
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 relatively prime, there exists integers a,b such that ap + bq = 1. Then 2aπ/q + 2bπ/p = 2π/pq. From this, a p·q-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 AntiquityAncient historyAncient history is the study of the written past from the beginning of recorded human history to the Early Middle Ages. The span of recorded history is roughly 5,000 years, with Cuneiform script, the oldest discovered form of coherent writing, from the protoliterate period around the 30th century BC...
. See equilateral triangle. - Constructions for the regular pentagon were described both by EuclidEuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
(ElementsEuclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, ca 300 BC), and by PtolemyPtolemyClaudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
(AlmagestAlmagestThe Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
, ca AD 150). See pentagonPentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
. - Although Gauss proved that the regular 17-gon is constructible, he didn't actually show how to do it. The first construction is due to Erchinger, a few years after Gauss' work. See heptadecagonHeptadecagonIn geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....
. - The first explicit construction of a regular 257-gon was given by Friedrich Julius RichelotFriedrich Julius RichelotFriedrich Julius Richelot was a German mathematician, born in Königsberg. He was a student of Carl Gustav Jacob Jacobi....
(1832). - A construction for a regular 65537-gon was first given by Johann Gustav HermesJohann Gustav HermesJohann Gustav Hermes was a German mathematician.-Life:Hermes was born in Königsberg and was educated at the Kneiphöfischen Gymnasium. He undertook his Abitur at the school in 1866. After completing his secondary education, he studied mathematics from 1866 to 1870, mostly in Königsberg...
(1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript. (ConwayJohn Horton ConwayJohn Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
has cast doubt on the validity of Hermes' construction, however.)
Gallery
From left to right, constructions of a 17-gon
Heptadecagon
In geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....
, 257-gon and 65537-gon.
Other constructions
It should be stressed that the concept of constructable as discussed in this article applies specifically to compass and straightedgeCompass and straightedge
Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
construction. More constructions become possible if other tools are allowed. The so-called neusis construction
Neusis construction
The neusis is a geometric construction method that was used in antiquity by Greek mathematicians.- Geometric construction :The neusis construction consists of fitting a line element of given length in between two given lines , in such a way that the line element, or its...
s, for example, make use of a marked ruler. The constructions are a mathematical idealization and are assumed to be done exactly.