Constructible polygon

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Constructible polygon

In mathematics, a constructible polygon is a regular polygon

Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
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 internal angles in a simple pentagon total 540?....
is constructible with compass and straightedge while a regular heptagon

Heptagon

In geometry, a heptagon is a polygon with seven sides and seven angles. In a regular polygon heptagon, in which all sides and all angles are equal, the sides meet at an angle of 5p/7 radians, 128.5714286 degree s....
is not.

Conditions for constructibility
Some regular polygons are easy to construct with compass and straightedge; others are not.

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Encyclopedia

In mathematics, a constructible polygon is a regular polygon

Regular polygon

Pentagon

Heptagon

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 Germans mathematician and scientist who contributed significantly to many fields, including number theory, statistics, mathematical analysis, Differential geometry and topology, geodesy, electrostatics, astronomy and optics....
proved the constructibility of the regular 17-gon

Heptadecagon

In geometry, a heptadecagon is a seventeen-sided polygon....
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 root of unity. They permit explicit calculations in cyclotomic fields, in relation both with Galois theory and with harmonic analysis ....
s in his Disquisitiones Arithmeticae
Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae is a textbook of number theory written by Germany mathematician 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 conjecture

Conjecture

In mathematics, a conjecture is a mathematical statement which appears resourceful, but has not been formally proven to be true under the rules of mathematical logic....
d that this condition was also necessary, but he offered no proof of this fact, which was proved by Pierre Wantzel

Pierre Wantzel

Pierre Laurent Wantzel was a France mathematician who proved that several ancient geometric problems were impossible to solve.In a paper from 1837, Wantzel proved that the problems of...
in 1837. It seems very unlikely that Gauss had a correct proof, because by taking n = 9, one can immediately deduce the impossibility of trisecting an angle of 120°, a fact of which Gauss was certainly aware.

Detailed results by Gauss' theory

Only five Fermat primes are known:

F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537

The next seven Fermat numbers, F₅ through F₁₁, 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,...

General theory

In the light of later work on Galois theory

Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after ?variste 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, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra; the modern development of analytic geometry is thus suggestively called algebraic geometry....
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 polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
s. 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(2p/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 root of unity to Q, the field of rational numbers. The n-th cyclotomic field Q is obtained by adjoining a primitive n-th root of unity ?n to the rational numbers....
— and in fact in its real subfield, which is a totally real field and a rational vector space

Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
of dimension

½f(n),

where f(n) is Euler's totient function

Euler's totient function

In number theory, the totient 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....
. Wantzel's result comes down to a calculation showing that f(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 module, hence cannot be decomposed into a direct sum...
, in group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
terms), something simple to prove by induction in this case of an abelian group

Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
. 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 root of unity. They permit explicit calculations in cyclotomic fields, in relation both with Galois theory and with harmonic analysis ....
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(2p/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 coprime

Coprime

In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common divisor other than 1 or, equivalently, if their greatest common divisor is 1....
, an n-gon can be constructed from a p-gon and a q-gon.

If p = 2, draw a q-gon and bisect
Bisection

In 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 segment bisectors and angle bisectors....
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 are two vertices a central angle 360°/(p·q) apart. 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 Antiquity
Ancient history

Ancient history is the history from the History of writing until the Early Middle Ages in Europe, the Qin Dynasty in China, the Chola Empire in India, and some less defined point in the rest of the world ....
. See equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
.
Constructions for the regular pentagon were described both by Euclid
Euclid

Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
(Elements
Euclid's Elements

Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC....
, ca 300 BC), and by Ptolemy
Ptolemy

Claudius Ptolemaeus , known in English as Ptolemy , was a Roman Greek mathematics, Greek astronomy, geographer and astrologer. He lived in History of Roman Egypt, and was probably born there in a town in the Thebaid called Ptolemais Hermiou; he died in Alexandria around 168 AD....
(Almagest
Almagest

Almagest is the Latin form of the Arabic language name of a mathematical and astronomical treatise proposing the complex motions of the stars and planetary paths, originally written in Greek language as by Ptolemy of Alexandria, Egypt, written in the 2nd century....
, ca AD 150). See pentagon
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
.
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 heptadecagon
Heptadecagon

In geometry, a heptadecagon is a seventeen-sided polygon....
.
The first explicit construction of a regular 257-gon was given by Friedrich Julius Richelot
Friedrich Julius Richelot

Friedrich Julius Richelot was a Germany Mathematics, born in K?nigsberg. He was a student of Carl Gustav Jacob Jacobi. He promoted in 1831 at the Philosophical Faculty of the University of K?nigsberg with a dissertation on the division of the circle into 257 equal parts and was a professor there....
(1832).
A construction for a regular 65537-gon was first given by Johann Gustav Hermes
Johann Gustav Hermes

Johann Gustav Hermes was a German mathematician....
(1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript. (Conway
John Horton Conway

John Horton Conway is a prolific mathematician active in the theory of finite group , knot theory, number theory, combinatorial game theory and coding theory....
has cast doubt on the validity of Hermes' construction, however.)

Other constructions

It should be stressed that the concept of constructibility as discussed in this article applies specifically to compass and straightedge

Compass and straightedge

Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or angles using only an Idealization 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....
s, for example, make use of a marked ruler. The construction of a regular heptagon

Heptagon

Almost all

In mathematics, the phrase almost all has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finite setly many" or "all but a countable set" ; see almost....
polygons remain inconstructible.

External links

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, Ask Dr. Math FAQ.

Constructible polygon

Conditions for constructibility

Detailed results by Gauss' theory

General theory

Compass and straightedge constructions

Other constructions

See also

External links