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. A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ... A number of ancient problems in geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... In geometry, a pentagon is any five-sided polygon. ... A heptagon is a plane figure with seven sides and seven angles. ...

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 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: Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... In geometry, a heptadecagon is a seventeen-sided polygon. ... In mathematics, a Gaussian period is a certain kind of sum of roots of unity. ... The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ...

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 conjectured that this condition was also necessary, but he offered no proof of this fact, which was proved by Pierre Wantzel in (1836). 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. In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... Pierre Wantzel (1814–1848) was a French mathematician. ... 1836 was a leap year starting on Friday (see link for calendar). ...

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. 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 mathematics, Galois theory is a branch of abstract algebra. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... Graph of a quadratic function: y = x2 - x - 2 = (x+1)(x-2) The x-coordinates of the points where the graph crosses the x-axis, x = -1 and x = 2, are the roots of the quadratic equation: x2 - x - 2 = 0. ... Field theory is a branch of mathematics which studies the properties of fields. ... In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have [L:K] = n and L is generated over K by a root of a polynomial Xn − a with a in K, and K contains n distinct roots of...

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 — and in fact in its real subfield, which is a totally real field of degree over the rational numbers In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In number theory , a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

½φ(n)

where φ(n) is Euler's toilet 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 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, 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 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 In mathematics, a composition series of a group G is a chain of subgroups of G satisfying where stands for normal subgroup, such that each quotient group Hi+1/Hi is a simple group. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

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.

Detailed results in terms of Fermat primes

Only five Fermat primes are known:

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

(sequence A019434 in OEIS).

Thus an n-gon is constructible if The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, ...

(sequence A003401 in OEIS),

while and an n-gon is not constructible with compass and straightedge if The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25,...

(sequence A004169 in OEIS).

The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

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, an n-gon can be constructed from a p-gon and a q-gon. Coprime - Wikipedia /**/ @import /skins-1. ...

• 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 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. For the numerical analysis algorithm, see bisection method. ...

• The construction for an equilateral triangle is simple and has been known since Antiquity. See equilateral triangle.
• Constructions for the regular pentagon were described both by Euclid (Elements, ca 300 BC), and by Ptolemy (Almagest, ca AD 150). See pentagon.
• 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.
• The first explicit construction of a regular 257-gon was given by F.J. Richelot (1832).
• A construction for a regular 65537-gon was first given by J. Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript. (John Conway has cast doubt on the validity of Hermes' construction, however.)

Antiquity means ancient times, and may be used of any period before the Middle Ages. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Euclid of Alexandria (Greek: ) (ca. ... Euclids Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. ... Claudius Ptolemaeus, given contemporary German styling, in a 16th century engraved book frontispiece . Claudius Ptolemaeus (Greek: Κλαύδιος Πτολεμαῖος; ca. ... Almagest is Latin form of the Arabic name (al-kitabu-l-mijisti, i. ... In geometry, a pentagon is any five-sided polygon. ... In geometry, a heptadecagon is a seventeen-sided polygon. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ...

Other constructions

It should be stressed that 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 construction of a regular heptagon is then easy. A number of ancient problems in plane geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ... A heptagon is a plane figure with seven sides and seven angles. ...

See also

• Fermat prime.
• Polygon.
• Compass-and-straightedge construction.

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... Look up Polygon on Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... A number of ancient problems in plane geometry involve the construction of lengths or angles using only an idealized ruler and compass, or more properly a straightedge and compass. ...

External links

• Regular Polygon Formulas, Ask Dr. Math FAQ.
• Why Gauss could not have proved necessity of constructible regular polygons