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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is constructible if and only if there is a
closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
for r using only integers and the operations for addition, subtraction, multiplication, division, and square roots. The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it can be produced as one of the points of a compass and straight edge construction (an endpoint of a line segment or crossing point of two lines or circles), starting from a given unit length segment. Alternatively and equivalently, taking the two endpoints of the given segment to be the points (0, 0) and (1, 0) of a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, a point is constructible if and only if its Cartesian coordinates are both constructible numbers. Constructible numbers and points have also been called ruler and compass numbers and ruler and compass points, to distinguish them from numbers and points that may be constructed using other processes. The set of constructible numbers forms a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
: applying any of the four basic arithmetic operations to members of this set produces another constructible number. This field is a field extension of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s and in turn is contained in the field of
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. It is the
Euclidean closure In mathematics, a Euclidean field is an ordered field for which every non-negative element is a square: that is, in implies that for some in . The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Eu ...
of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, the smallest field extension of the rationals that includes the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
s of all of its positive numbers. The proof of the equivalence between the algebraic and geometric definitions of constructible numbers has the effect of transforming geometric questions about compass and straightedge constructions into
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, including several famous problems from ancient Greek mathematics. The algebraic formulation of these questions led to proofs that their solutions are not constructible, after the geometric formulation of the same problems previously defied centuries of attack.


Geometric definitions


Geometrically constructible points

Let O and A be two given distinct points in the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, and define S to be the set of points that can be constructed with compass and straightedge starting with O and A. Then the points of S are called constructible points. O and A are, by definition, elements of S. To more precisely describe the remaining elements of S, make the following two definitions: * a line segment whose endpoints are in S is called a constructed segment, and * a circle whose center is in S and which passes through a point of S (alternatively, whose radius is the distance between some pair of distinct points of S) is called a constructed circle. Then, the points of S, besides O and A are: * the intersection of two non-parallel constructed segments, or lines through constructed segments, * the intersection points of a constructed circle and a constructed segment, or line through a constructed segment, or * the intersection points of two distinct constructed circles. As an example, the midpoint of constructed segment OA is a constructible point. One construction for it is to construct two circles with OA as radius, and the line through the two crossing points of these two circles. Then the midpoint of segment OA is the point where this segment is crossed by the constructed line.


Geometrically constructible numbers

The starting information for the geometric formulation can be used to define a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
in which the point O is associated to the origin having coordinates (0,0) and in which the point A is associated with the coordinates (1, 0). The points of S may now be used to link the geometry and algebra by defining a constructible number to be a coordinate of a constructible point. Equivalent definitions are that a constructible number is the x-coordinate of a constructible point (x,0) or the length of a constructible line segment. In one direction of this equivalence, if a constructible point has coordinates (x,y), then the point (x,0) can be constructed as its perpendicular projection onto the x-axis, and the segment from the origin to this point has length x. In the reverse direction, if x is the length of a constructible line segment, then intersecting the x-axis with a circle centered at O with radius x gives the point (x,0). It follows from this equivalence that every point whose Cartesian coordinates are geometrically constructible numbers is itself a geometrically constructible point. For, when x and y are geometrically constructible numbers, point (x,y) can be constructed as the intersection of lines through (x,0) and (0,y), perpendicular to the coordinate axes.


Algebraic definitions


Algebraically constructible numbers

The algebraically constructible real numbers are the subset of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers. Even more simply, at the expense of making these formulas longer, the integers in these formulas can be restricted to be only 0 and 1. For instance, the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...
is constructible, because it can be described by the formulas \sqrt2 or \sqrt. Analogously, the algebraically constructible
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s are the subset of complex numbers that have formulas of the same type, using a more general version of the square root that is not restricted to positive numbers but can instead take arbitrary complex numbers as its argument, and produces the principal square root of its argument. Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers. For instance, the complex number i has the formulas \sqrt or \sqrt, and its real and imaginary parts are the constructible numbers 0 and 1 respectively. These two definitions of the constructible complex numbers are equivalent.
p. 440
In one direction, if q=x+iy is a complex number whose real part x and imaginary part y are both constructible real numbers, then replacing x and y by their formulas within the larger formula x+y\sqrt produces a formula for q as a complex number. In the other direction, any formula for an algebraically constructible complex number can be transformed into formulas for its real and imaginary parts, by recursively expanding each operation in the formula into operations on the real and imaginary parts of its arguments, using the expansions *(a+ib)\pm (c+id)=(a \pm c)+i(b \pm d) *(a+ib)(c+id)=(ac-bd) + i(ad+bc) *\frac=\frac + i \frac *\sqrt = \frac + i\frac, where r=\sqrt and s=\sqrt.


Algebraically constructible points

The algebraically constructible points may be defined as the points whose two real Cartesian coordinates are both algebraically constructible real numbers. Alternatively, they may be defined as the points in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
given by algebraically constructible complex numbers. By the equivalence between the two definitions for algebraically constructible complex numbers, these two definitions of algebraically constructible points are also equivalent.


Equivalence of algebraic and geometric definitions

If a and b are the non-zero lengths of geometrically constructed segments then elementary compass and straightedge constructions can be used to obtain constructed segments of lengths a+b, , a-b, , ab, and a/b. The latter two can be done with a construction based on the
intercept theorem The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines ar ...
. A slightly less elementary construction using these tools is based on the
geometric mean theorem The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states ...
and will construct a segment of length \sqrt from a constructed segment of length a. It follows that every algebraically constructible number is geometrically constructible, by using these techniques to translate a formula for the number into a construction for the number. In the other direction, a set of geometric objects may be specified by algebraically constructible real numbers: coordinates for points, slope and y-intercept for lines, and center and radius for circles. It is possible (but tedious) to develop formulas in terms of these values, using only arithmetic and square roots, for each additional object that might be added in a single step of a compass-and-straightedge construction. It follows from these formulas that every geometrically constructible number is algebraically constructible.


Algebraic properties

The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
. Thus, the constructible numbers (defined in any of the above ways) form a field. More specifically, the constructible real numbers form a
Euclidean field In mathematics, a Euclidean field is an ordered field for which every non-negative element is a square: that is, in implies that for some in . The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Eu ...
, an ordered field containing a square root of each of its positive elements. Examining the properties of this field and its subfields leads to necessary conditions on a number to be constructible, that can be used to show that specific numbers arising in classical geometric construction problems are not constructible. It is convenient to consider, in place of the whole field of constructible numbers, the subfield \mathbb(\gamma) generated by any given constructible number \gamma, and to use the algebraic construction of \gamma to decompose this field. If \gamma is a constructible real number, then the values occurring within a formula constructing it can be used to produce a finite sequence of real numbers \alpha_1,\dots, a_n=\gamma such that, for each i, \mathbb(\alpha_1,\dots,a_i) is an
extension Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
of \mathbb(\alpha_1,\dots,a_) of degree 2. Using slightly different terminology, a real number is constructible if and only if it lies in a field at the top of a finite
tower A tower is a tall Nonbuilding structure, structure, taller than it is wide, often by a significant factor. Towers are distinguished from guyed mast, masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting ...
of real
quadratic extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s, \mathbb = K_0 \subseteq K_1 \subseteq \dots \subseteq K_n, starting with the rational field \mathbb where \gamma is in K_n and for all 0< j\le n, _j:K_2. It follows from this decomposition that the degree of the field extension mathbb(\gamma):\mathbb/math> is 2^r, where r counts the number of quadratic extension steps. Analogously to the real case, a complex number is constructible if and only if it lies in a field at the top of a finite tower of complex quadratic extensions. More precisely, \gamma is constructible if and only if there exists a tower of fields \mathbb = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n, where \gamma is in F_n, and for all 0, _j:F_ 2. The difference between this characterization and that of the real constructible numbers is only that the fields in this tower are not restricted to being real. Consequently, if a complex number \gamma is constructible, then mathbb(\gamma):\mathbb/math> is a power of two. However, this necessary condition is not sufficient: there exist field extensions whose degree is a power of two that cannot be factored into a sequence of quadratic extensions. The fields that can be generated in this way from towers of quadratic extensions of \mathbb are called ''iterated quadratic extensions'' of \mathbb. The fields of real and complex constructible numbers are the unions of all real or complex iterated quadratic extensions of \mathbb.


Trigonometric numbers

Trigonometric number In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values ...
s are the cosines or sines of angles that are rational multiples of \pi. These numbers are always algebraic, but they may not be constructible. The cosine or sine of the angle 2\pi/n is constructible only for certain special numbers n: *The
powers of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
*The
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
s, prime numbers that are one plus a power of two *The products of powers of two and distinct Fermat primes. Thus, for example, \cos(\pi/15) is constructible because 15 is the product of two Fermat primes, 3 and 5.


Impossible constructions

The
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cult ...
thought that certain problems of
straightedge and compass construction In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
they could not solve were simply obstinate, not unsolvable. However, the non-constructibility of certain numbers proves that these constructions are logically impossible to perform. (The problems themselves, however, are solvable using methods that go beyond the constraint of working only with straightedge and compass, and the Greeks knew how to solve them in this way. One such example is Archimedes'
Neusis construction In geometry, the neusis (; ; plural: grc, νεύσεις, neuseis, label=none) is a geometric construction method that was used in antiquity by Greek mathematicians. Geometric construction The neusis construction consists of fitting a line e ...
solution of the problem of Angle trisection.) In particular, the algebraic formulation of constructible numbers leads to a proof of the impossibility of the following construction problems: ;
Doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related pro ...
:The problem of doubling the unit square is solved by the construction of another square on the diagonal of the first one, with side length \sqrt2 and area 2. Analogously, the problem of doubling the cube asks for the construction of the length \sqrt /math> of the side of a cube with volume 2. It is not constructible, because the minimal polynomial of this length, x^3-2, has degree 3 over \Q. As a cubic polynomial whose only real root is irrational, this polynomial must be irreducible, because if it had a quadratic real root then the quadratic conjugate would provide a second real root. ; Angle trisection :In this problem, from a given angle \theta, one should construct an angle \theta/3. Algebraically, angles can be represented by their
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
s, such as their
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
s or cosines, which give the Cartesian coordinates of the endpoint of a line segment forming the given angle with the initial segment. Thus, an angle \theta is constructible when x=\cos\theta is a constructible number, and the problem of trisecting the angle can be formulated as one of constructing \cos(\tfrac\arccos x). For example, the angle \theta=\pi/3=60^\circ of an equilateral triangle can be constructed by compass and straightedge, with x=\cos\theta=\tfrac12. However, its trisection \theta/3=\pi/9=20^\circ cannot be constructed, because \cos\pi/9 has minimal polynomial 8x^3-6x-1 of degree 3 over \Q. Because this specific instance of the trisection problem cannot be solved by compass and straightedge, the general problem also cannot be solved. ; Squaring the circle :A square with area \pi, the same area as a
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
, would have side length \sqrt\pi, a
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
. Therefore, this square and its side length are not constructible, because it is not algebraic over \Q. ;
Regular polygons In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
:If a regular n-gon is constructed with its center at the origin, the angles between the segments from the center to consecutive vertices are 2\pi/n. The polygon can be constructed only when the cosine of this angle is a trigonometric number. Thus, for instance, a 15-gon is constructible, but the regular
heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived num ...
is not constructible, because 7 is prime but not a Fermat prime. For a more direct proof of its non-constructibility, represent the vertices of a regular heptagon as the complex roots of the polynomial x^7-1. Removing the factor x-1, dividing by x^3, and substituting y=x+1/x gives the simpler polynomial y^3+y^2-2y-1, an irreducible cubic with three real roots, each two times the real part of a complex-number vertex. Its roots are not constructible, so the heptagon is also not constructible. ;
Alhazen's problem Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD. It is named for the 11th-century Arab mathematician Alhazen (''Ibn al-Haytham'') who presented a g ...
:If two points and a circular mirror are given, where on the circle does one of the given points see the reflected image of the other? Geometrically, the lines from each given point to the point of reflection meet the circle at equal angles and in equal-length chords. However, it is impossible to construct a point of reflection using a compass and straightedge. In particular, for a unit circle with the two points (\tfrac16,\tfrac16) and (-\tfrac12,\tfrac12) inside it, the solution has coordinates forming roots of an irreducible degree-four polynomial x^4-2x^3+4x^2+2x-1. Although its degree is a power of two, the
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
of this polynomial has degree divisible by three, so it does not come from an iterated quadratic extension and Alhazen's problem has no compass and straightedge solution.


History

The birth of the concept of constructible numbers is inextricably linked with the history of the three impossible compass and straightedge constructions: duplicating the cube, trisecting an angle, and squaring the circle. The restriction of using only compass and straightedge in geometric constructions is often credited to
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
due to a passage in
Plutarch Plutarch (; grc-gre, Πλούταρχος, ''Ploútarchos''; ; – after AD 119) was a Greek Middle Platonist philosopher, historian, biographer, essayist, and priest at the Temple of Apollo in Delphi. He is known primarily for his ''P ...
. According to Plutarch, Plato gave the duplication of the cube (Delian) problem to Eudoxus and Archytas and
Menaechmus :''There is also a Menaechmus in Plautus' play, ''The Menaechmi''.'' Menaechmus ( el, Μέναιχμος, 380–320 BC) was an ancient Greek mathematician, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, w ...
, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry. However, this attribution is challenged, due, in part, to the existence of another version of the story (attributed to
Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ;  – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...
by
Eutocius of Ascalon Eutocius of Ascalon (; el, Εὐτόκιος ὁ Ἀσκαλωνίτης; 480s – 520s) was a Palestinian-Greek mathematician who wrote commentaries on several Archimedean treatises and on the Apollonian ''Conics''. Life and work Little is ...
) that says that all three found solutions but they were too abstract to be of practical value.
Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...
, citing
Eudemus of Rhodes Eudemus of Rhodes ( grc-gre, Εὔδημος) was an ancient Greek philosopher, considered the first historian of science, who lived from c. 370 BCE until c. 300 BCE. He was one of Aristotle's most important pupils, editing his teacher's work and m ...
, credited
Oenopides Oenopides of Chios ( el, Οἰνοπίδης ὁ Χῖος; born c. 490 BCE) was an ancient Greek geometer and astronomer, who lived around 450 BCE. Biography Only limited information are known about the early life of Oenopides except his bir ...
(circa 450 BCE) with two ruler and compass constructions, leading some authors to hypothesize that Oenopides originated the restriction. The restriction to compass and straightedge is essential to the impossibility of the classic construction problems. Angle trisection, for instance, can be done in many ways, several known to the ancient Greeks. The
Quadratrix In geometry, a quadratrix () is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and Ehrenfried Walther von Tschirnhaus, E. W. Tschirnhaus, ...
of
Hippias of Elis Hippias of Elis (; el, Ἱππίας ὁ Ἠλεῖος; late 5th century BC) was a Greek sophist, and a contemporary of Socrates. With an assurance characteristic of the later sophists, he claimed to be regarded as an authority on all subjects, ...
, the
conics In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
of Menaechmus, or the marked straightedge (
neusis In geometry, the neusis (; ; plural: grc, νεύσεις, neuseis, label=none) is a geometric construction method that was used in antiquity by Greek mathematics, Greek mathematicians. Geometric construction The neusis construction consists ...
) construction of
Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
have all been used, as has a more modern approach via
paper folding ) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a f ...
. Although not one of the classic three construction problems, the problem of constructing regular polygons with straightedge and compass is often treated alongside them. The Greeks knew how to construct regular with n=2^h (for any integer h\ge 2), 3, 5, or the product of any two or three of these numbers, but other regular eluded them. In 1796
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, then an eighteen-year-old student, announced in a newspaper that he had constructed a regular 17-gon with straightedge and compass. Gauss's treatment was algebraic rather than geometric; in fact, he did not actually construct the polygon, but rather showed that the cosine of a central angle was a constructible number. The argument was generalized in his 1801 book ''
Disquisitiones Arithmeticae The (Latin for "Arithmetical Investigations") 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. It is notable for having had a revolutionary impact on th ...
'' giving the ''sufficient'' condition for the construction of a regular Gauss claimed, but did not prove, that the condition was also necessary and several authors, notably
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
, attributed this part of the proof to him as well. Alhazen's problem is also not one of the classic three problems, but despite being named after
Ibn al-Haytham Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the pri ...
(Alhazen), a medieval Islamic mathematician, it already appear's in
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
's work on optics from the second century. proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve if one uses only compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible: a regular polygon is constructible
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
the number of its sides is the product of a
power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negative ...
and any number of distinct
Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
s (i.e., the sufficient conditions given by Gauss are also necessary). An attempted proof of the impossibility of squaring the circle was given by James Gregory in ''Vera Circuli et Hyperbolae Quadratura'' (The True Squaring of the Circle and of the Hyperbola) in 1667. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of . It was not until 1882 that
Ferdinand von Lindemann Carl Louis Ferdinand von Lindemann (12 April 1852 – 6 March 1939) was a German mathematician, noted for his proof, published in 1882, that (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficien ...
rigorously proved its impossibility, by extending the work of
Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermi ...
and proving that is a
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
. Alhazen's problem was not proved impossible to solve by compass and straightedge until the work of .; see also for an independent solution with more of the history of the problem. The study of constructible numbers, per se, was initiated by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
in
La Géométrie ''La Géométrie'' was published in 1637 as an appendix to ''Discours de la méthode'' (''Discourse on the Method''), written by René Descartes. In the ''Discourse'', he presents his method for obtaining clarity on any subject. ''La Géométrie ...
, an appendix to his book ''
Discourse on the Method ''Discourse on the Method of Rightly Conducting One's Reason and of Seeking Truth in the Sciences'' (french: Discours de la Méthode Pour bien conduire sa raison, et chercher la vérité dans les sciences) is a philosophical and autobiographical ...
'' published in 1637. Descartes associated numbers to geometrical line segments in order to display the power of his philosophical method by solving an ancient straightedge and compass construction problem put forth by Pappus.


See also

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Computable number In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive ...
*
Definable real number Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, \sqrt, ca ...


Notes


References

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External links

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Constructible Numbers
at
Cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
{{Number systems Euclidean plane geometry Algebraic numbers