Squaring the circle

Squaring the circle is a classic problem of geometry. The task is to construct from a given circuit in a finite number of steps, a square with the same area. It is equivalent to the so-called rectification of the circle, so the construction of a straight line that corresponds to the circumference. Confining the construction funds on ruler and compass, the problem is unsolvable. However, this could be proved only in 1882 by the German mathematician Ferdinand von Lindemann.

Squaring the circle is one of the most popular problems of mathematics. For centuries, mathematicians studied alongside again and again lay in vain for a solution. The term quadrature of the circle has become in many languages ​​a metaphor for an impossible task.

• 4.1 Tarski's problem of squaring the circle
• 4.2 lemniscate

• 6.1 Notes and references

History

→ Main article: Kreiszahl

Prehistory

Already in the ancient Near Eastern civilizations there were procedures for calculating circular areas. In a problem of the Rhind papyrus ( c. 1650 BC), for example, the area of ​​a circle is approximated by the diameter 9 a ( irregular ) Octagon - by both are inscribed in a square of side length 9, which is broken down into 9 smaller squares - and about equal to the area of ​​a square of side length 8, which a fairly accurate value for the mathematical constant π of 3 1/9 1/ 27 1/81 = 3.16 ... equivalent. Such patterns solutions were obtained from the practice and intended for practice, there was no further theoretical considerations, in particular no difference between exact solution and approximation has been made.

A deductive approach in mathematics, replace the sample tasks in the supporting evidence sets, developed from the 6th century BC in Greece. To some extent it is already at Thales of Miletus, seen more clearly in the founded by Pythagoras of Samos Pythagorean school. With the commonly ascribed to the Pythagoreans Hippasos of Metapontum discovery of incommensurable lines in the late 6th or early 5th century, it turned out that there is constructible objects ( for example, the diagonal of a square ) that are not representable as an integer ratio. As a result of this discovery, the arithmetic entered in favor of the geometry in the background, equations had to be solved geometrically now, such as Aneinanderlegung of figures and conversion of different figures into rectangles or squares. From the late 5th century, the three classical construction problems of ancient mathematics, in addition to the quadrature of the circle, the object of the trisection of an angle and the Delian problem of doubling the cube come yet.

A limitation of the design tools on ruler and compass was not generally required. During the study of the classical problems early solutions were found based on further aids. However, crystallized in the course of time, a stance, which requires the greatest possible restriction. At the latest when Pappus this maximum limitation had become the measure.

Early work

As one of the first Greek writer Plutarch, according to the philosopher Anaxagoras " written to square the circle in prison (or drawn, AltGr ἔγραφε. ) " Have details of Anaxagoras ' construction makes not Plutarch. A prison of Anaxagoras would be dated to about 430 BC, when the philosopher in Athens was accused of impiety and finally had to flee. More detailed sources to the beginnings of research are mainly late antique Comments on works of Aristotle, writings, therefore, which were taken with a temporal distance of around 900 years. Accordingly unsure chronological order and exact thoughts of the first approaches. The most important works of the 5th century BC comes from Hippocrates of Chios, Antiphon, Bryson of Heraclea and Hippias of Elis.

The transfer of triangles in rectangles of rectangles into squares or the addition of two squares had to cope with the known geometric sets even then elementary. The basic question is whether curvilinear surfaces can be precisely transferred into squares, could be answered positively by Hippocrates of Chios about 440 BC. Based on the still commonly used as an axiom in his sentence that the areas of similar segments of circles proportional to the squares of their tendons, succeeded Hippocrates, bounded by circular arcs surfaces to square the so-called " half-moon of Hippocrates ". Squaring the circle is, however, not be achieved in this way, as only certain lune - for example, on the side of the square, but not on the side of a regular hexagon - are quadrierbar.

Since triangles (and thus any polygons ) could be converted into a square, was a second approach, to construct a circle the same area polygon. Antiphon had the idea to approximate the circle by inscribed polygons. Bryson of Heraclea refined this approach by additionally approached the circle circumscribed by polygons and an intermediate value was formed.

Hippias of Elis developed about 425 BC to the solution of angle trisection a curve that was mechanically generated by the superposition of a circular with a linear movement. More than a century later discovered Dinostratus that using this curve, the so-called quadratrix that stretch the length - and thus using another elementary constructions, a square of area π - can be constructed. However, since the quadratrix itself is a so-called transcendental curve, ie not to generate ruler and compass, the solution was not reached so in the strict sense.

Archimedes

A detailed treatise entitled circle measurement is known of Archimedes. Archimedes proves in this work, three basic sets:

On the first set, the problem of squaring the circle to the question of the constructability of the circumference of a circle of the given radius and therefore on the constructability of π is returned. In the third set Archimedes is equal to a simple and accurate approximation of this number, namely 22/7, a value (~ 3143 ), which today is used for practical purposes. The second movement is a simple corollary of the other two; that the area of ​​a circle is proportional to the square of its diameter, was known to Euclid, Archimedes gives here the value of the proportionality of.

To prove his statements Archimedes accesses the Brysonsche idea of ​​any approximation of the circle by one and circumscribed regular polygons. Based on the inscribed hexagon and circumscribed triangle reaches Archimedes by successively doubling the number of pages on each of the 96 -gon. A clever estimation of square roots occurring in the individual steps of calculation results and above the 3 barriers.

In another work On Spirals, Archimedes describes the construction of the later named after him Archimedean spiral, which is obtained similar to Hippias ' quadratrix by the superposition of a circular with a linear movement. It shows that at this spiral the circumference of a circle can be removed in a straight line by the application of the tangent. In the work done thus squaring the circle indicate later commentators, Archimedes himself did not make a statement. As with the quadratrix neither spiral nor their tangent with compass and straightedge are constructible.

Middle Ages

As a result of greater interest for the ancient mathematics in Christian Europe from about the 11th century created a number of treatises on squaring the circle, but without causing significant contributions were made to the actual solution. Is to be regarded as a step backwards, that in the Middle Ages the Archimedean approximation of 22/7 was long regarded as exact for the district number.

One of the first authors of the Middle Ages, which resumed the problem of squaring the circle, was Franco of Liege. In 1050 his work De quadratura Circuli arose. Franco is in first three quartiles, which he rejects. The first two indicate for the side length of the square 7/8, respectively, for the diagonal 10/8 of the circle diameter, which relatively poor approximations of 31 /16 and 31/8 corresponds to π. The third proposal, in turn, sets the perimeter of the square is equal to the circumference, that requires the rectification of the latter.

Franco's own solution is based on a circle with diameter 14. Its area is, in his view exactly 7 ² x 22/7 = 154 Mathematically can be no argument after Franco's surface equal square find as the square root of 22/7 is irrational, but constructively so. For this purpose he divided the circle into 44 equal sectors, which he put together to form a rectangle of side lengths 11 and 14. The necessary artifice, in which he replaced the circular sectors by right triangles with short sides of length 1 and 7, Franco, however, not explained. Another problem is not quite of successful experiment, then to convert the rectangle by a suitable decomposition into a square. Obviously Franco was the ancient Greek method not common.

Later treatises of the scholastic exhaust themselves more or less in a consideration of the arguments of the well-known classic. Only with the spread of Latin translations of Archimedes writings in the late Middle Ages, the value was 22/7 recognized as an approximation and search for new solutions to the problem, such as Nicholas of Cusa. This took up the idea to approximate the circle by a sequence of regular polygons with increasing number of sides back on, but did not seek to determine, in contrast to Archimedes the circumference, but the circle radius at a given constant perimeter of polygons. In a letter of Cusa was such a solution, which he believed to be exactly at. The resulting determined value for the wave number is also at least between the given limits of Archimedes. The actual Cusa's work on the subject provide significantly poorer approximations and were thus the target of a pamphlet of Regiomontanus, who proved the inaccuracy of the calculations and the evidence referred to " as a philosophical, not a mathematical ".

Progress of the circle measurement in the early modern period

Advances in circuit calculation applied from the 16th century, the development of the Archimedean approximation method as well as the advent of modern analytical methods.

In the original method of Archimedes the circumference of a circle inscribed by the scope of the circle and the one estimated the circle circumscribed polygon. Exact bounds are obtained by an increase in the number of corners. The Dutch mathematician Willibrord van Roijen Snell ( Snell ) found that even without the page number to enlarge finer bounds on the length of an arc segment than the tendons of the polygons can be specified. He could not strictly prove this result, however. The development and improvement of Snell's approach made ​​Christiaan Huygens in his work De Circuli magnitudine inventa, in which he also yielded evidence of established Snell sets. On a purely elementargeometrischem way Huygens such a good containment of the area lying between polygon and circle, that he with a corresponding page number of polygons, the circle number at least three times as many significant digits received as Archimedes with his method.

The purely geometric approach to determine the circuit constants was exhausted working mainly with Huygens. Better approximations resulted by using infinite series, especially the expansion of trigonometric functions. Although François Viète had found at the end of the 16th century through the study of particular track conditions of successive polygons a first exact representation of π by an infinite product, but proved this formula as unwieldy. A simpler series that gets beyond just rational operations, comes from John Wallis, a further illustration of circle number as a continued fraction of William Brouncker. More important for the practice was the number found by James Gregory and independent of Gottfried Leibniz for the arc tangent. Although this series converges even slowly, one can deduce from her other series, in turn, are very well suited for the calculation of the circle number. Beginning of the 18th century such series were calculated using over 100 points of π, new insights into the problem of squaring the circle could thus not, however, be won.

Algebraic problem and irrationality of π

To solve the problem, it was necessary for one of the way to give the geometric term " constructible " an algebraic meaning, on the other hand a closer inspection of the properties of the circle number.

A geometric design with ruler and compass is based on a finite number of predetermined points determined in a finite number of steps new points by cutting two lines, two circles or a line and a circle. The translation of this approach into the language of algebra was achieved by the introduction of coordinate systems in the context of the 17th century, mainly developed by Pierre de Fermat and René Descartes analytical geometry. Straight lines and circles could be described with the new means by equations intersections are determined by the solution of equations. It turned out that the constructible with compass and straightedge path lengths are precisely those which can be by a finite number of rational operations (addition, subtraction, multiplication and division ) and a finite number of square roots of a given length are derived. In particular, these lengths are algebraic numbers, so a portion of the figures, which is a solution of an algebraic equation of any degree with rational coefficients. Numbers that are not algebraic are called transcendent and are not constructible.

Starting point for further investigations of the circle number were some basic knowledge Leonhard Euler, which he had published in 1748 in his book Introductio in analysin infinitorum. Euler introduced, among others, the well-known formula

The first time a relationship between the trigonometric functions and the exponential function here, and also generated some continued fraction and series representations of π and later named after him, Euler's number e

This preliminary work went Johann Heinrich Lambert advantage that could show using one of Euler's continued fraction developments in 1766 for the first time that e and π irrational, thus not representable by an integer fraction numbers. A small gap in Lambert's evidence was closed in 1806 by Adrien -Marie Legendre, who provided the Irrationalitätsbeweis for the same time.

The assumption that π can not be algebraic, now stood in the room was at least enunciated by Euler, Lambert and Legendre. In this case, up to the middle of the 19th century was not yet clear that there had to be at all transcendental numbers. Evidence for this comes 1844/1851 Joseph Liouville by explicit construction of Liouville transcendental numbers.

Proof of the impossibility

Ferdinand von Lindemann proved that π is not algebraic but transcendental in 1882 finally. Therefore, π can not be constructed in a straight line and squaring the circle is impossible.

Lindemann reached into his work back to an outcome of the French mathematician Charles Hermite. This had shown in 1873 that the Euler number e is transcendental. Could Based on Lindemann prove the so-called set of Lindemann - Weierstrass which states that for any distinct algebraic numbers and for any algebraic numbers the equation

Can only apply if all have the value zero. In particular, may result for any of the non-zero algebraic number z of the expression is a rational number. After this preparation, Lindemann was the assumption that π is algebraically using Euler's identity lead to a contradiction; π therefore had to be transcendent.

Lindemann's proof of the transcendence of π has been simplified considerably in the following years and decades, such as by David Hilbert in 1893.

Popularity squaring the circle

Squaring the circle reached like few other issues also outside mathematics a lot of popularity. As a result, many mathematical laymen tried to resolve the apparently simple problem, and many believed they had found them.

As the earliest evidence of the emergence of a so-called " Kreisquadrierers " or " Quadrators " means an entity in Aristophanes ' comedy is occasionally the birds cited, occurs as a surveyor in the Meton and wants to set the floor plan of a new city with geometric tools so that "the circle a quadrilateral will ". However, this is meant not the squaring of a circle, but the creation of two perpendicular streets of juxtaposed, even if the term conveys an allusion to the squaring the circle.

Reports on a growing accumulation of amateur works from the 18th and 19th centuries, examples of topic can be found in Jean- Étienne Montucla, Johann Heinrich Lambert and Augustus de Morgan. Usually it was process in which the problem has been mechanically, numeric or "exact" achieved by a geometric proximity construction. Such work brought to my attention in such a large number of mathematicians and scientific institutions that saw 1775 compelled, for example, the Paris Academy of Sciences officially refuse further investigation of alleged solutions squaring the circle:

" L' Académie a pris, cette année, la résolution de ne plus examiner aucune solution of problèmes de la duplication du cube, trisection de l'angle de la ou de la quadrature du cercle, ni aucune machine anoncée comme un mouvement perpétuel. "

" The academy has this year made ​​the decision in the future, neither the solutions of mathematical problems concerning the duplication of the cube, the trisection of an angle and squaring the circle, nor any machine with the claim of a " perpetual motion " to investigate. "

Even after the lindemann between impossibility proof alleged quadratures were still published. More recently, the futile attempts of the amateur mathematician fabric of recreational mathematics have become.

A major reason for the high just for mathematical lay attractiveness is probably the very basic problem that can be understood without deeper mathematical knowledge appears or at least understandable. In conjunction with the many futile attempts to solve established scientists a veritable nimbus around the squaring the circle was formed.

Another not to be underestimated as a reason for the numerous efforts to square the circle was the common opinion on the solution of the problem was a high price exposed - a mistaken belief that possibly goes back to the false assumption that squaring the circle would be in direct communication with the also long open problem of the exact determination of longitude at sea, on whose solution, in fact, were exposed to prices. The saga of the contest remained so persistent that even 1891 was in Meyers Lexicon to read yet, that " Charles V 100,000 Thaler and the Dutch States-General an even higher sum " would have exposed.

A prominent example for an amateur mathematician who believed he had found the quadrature of the circle, the English philosopher Thomas Hobbes was. His published in 1665 in his De corpore solution - in fact, an approximate construction - was rejected by John Wallis in the same year. In the following period ensued between the two an argument advanced in sharp tone confrontation that ended only with Hobbes ' death in 1679.

Lambert reported by three circular quadratures by means of a specific rational value. The appeared in the mid-18th century work based on the approximation 35/31 for the ratio of diameter to the side of the equivalent square. For the circle number one obtains the approximation

One of the three authors, the preacher Merkel from Ravensburg, Gotthold Ephraim Lessing dedicated the poem " On the Lord M ** the inventor of the squaring of the circle ".

The squaring of the circle American physician Edward J. Goodwin appeared in 1894 even in the first volume of the American Mathematical Monthly, if only as a listing of the author. The work itself is contradictory and can, depending on reading multiple values ​​for π to. It formed the basis for a 1897 presented to Parliament by Indiana bill, called the Indiana Pi Bill, by which the findings Goodwin should be made ​​into law.

Proximity structures

Although an exact solution by ruler and compass is not possible, there are approximate constructions for squaring the circle, which are accurate enough for many purposes. Simple, well-known even in ancient method return an integer ratio of the diameter or radius of the circle to the side or diagonal of the square. In addition to the mentioned in the papyrus Rhind equation of the circle the diameter of 9 with the square of side 8, the diameter of the circle 8 with the square of the diagonal 10 has been known. This construction can be found among the Babylonians, and possibly the Roman surveyor Vitruvius. It returns the value 31 /8 for π. To specify a convenient method for drawing, Albrecht Dürer takes this construction in 1525 in his work Vnderweysung the measurement with the zirckel and richtscheyt again. Dürer is aware of this is that it is a purely approximate solution, he writes explicitly that an exact solution is not found:

" Would be necessary to know Quadratura Circuli, which is the likeness of a circle and a square, so that one would have as much content as the other. But such is not yet demonstrirt by scholars. Mechanice that is casual, so that the plant is missing or only a little while, this equality may thus be made. Tear a transept and share the local bar into ten parts and tear after a Zirkelriß whose diameter is said to have eight parts, like squaring their 10; as I have torn down. "

A classical approximate solution is the approximate construction of Kochański, the Polish mathematician Adam Adam Andy Kochański discovered in 1685. It comes from having only one circle opening. The actual structure consists of a rectification of the half circle, Kochanski constructed from the predetermined radius r of approximately a straight line of length. Squaring follows from elementary using the Kathetensatzes. The county number is accurately approximated with Kochański to four decimal places:

In 1913, a construction of the Indian mathematician Ramanujan SA, which appeared on the approximation

Based, a value that is accurate even at six decimal places and in China was in Europe at least since the 17th century since the 5th century known. Ramanujan noted with respect to the accuracy of his method is that in a circular area of 140,000 square miles of the constructed square side differed by only about one inch from the true value. In a paper from the following year, Ramanujan supplied in addition to other approximation methods, a further quadrature with ruler and compass. This is the value

Basis, which even comes close to π to eight digits.

A simpler method published Louis Loynes 1961. It is based on the finding that the area of ​​the circumference of a right triangle is equal to the square on the larger cathetus, if the tangent of the smaller angle, that is, the ratio of smaller to larger cathetus

Is a value that is very close to the fracture

Lies. This gives a simple approximation by the ( constructible ) used one right triangle with the catheti ratio 23:44 for quadrature. The approximate value for the number of circuit

Is slightly better than Kochanski construction.

Variants

Tarski's problem of squaring the circle

Alfred Tarski in 1925 presented the task to piece a circle into any number of parts and to then move through pure movement (without extension) so that a square is formed.

Miklós Laczkovich came in 1989 the solution: He proved that it is possible to divide a circle into finitely many parts and to move only through exercise so that a square is formed. He divided the circle into 1050 pieces. However, for the proof he needs the axiom of choice, which is accepted by most scientists today, but is not self-evident. The proof is very similar to the Banach - Tarski paradox.

Laczkovich has indeed proved that (assuming the axiom of choice ) then there exists a decomposition, but this decomposition can not be specified explicitly.

Lemniscate

In contrast to the circuit, it is possible (∞) represent a lemniscate by two squares whose side length corresponds to the largest Lemniskatenradius a.

Literature and sources

Generally

• Eugen bag: Squaring the Circle. 2nd edition. Teubner, Leipzig, 1920. ( Digitized )
• Moritz Cantor Lectures on the History of Mathematics. Teubner, Leipzig, 1880-1908 (4 vols ).
• Helmuth Gericke: Mathematics in Antiquity and the Middle East. Springer, Berlin 1984, ISBN 3-540-11647-8.
• Helmuth Gericke: Mathematics in the West. Springer, Berlin, 1990, ISBN 3-540-51206-3.
• Thomas Little Heath: A History of Greek Mathematics. Volume 1, Clarendon Press, Oxford, 1921 (reprint: Dover, New York 1981, ISBN 0-486-24073-8. ).
• Klaus Mainzer: History of Geometry. Bibliographical Institute, Mannheim, inter alia, 1980, ISBN 3-411-01575-6.
• Ferdinand Rudio: Archimedes, Huygens, Lambert, Legendre. Four essays on the circle measurement. Teubner, Leipzig, 1892. ( Digitized )

To the transcendence of π

• Ferdinand Lindemann: π about the number. In: Mathematische Annalen 20 (1882 ), pp. 213-225.
• David Hilbert: About the transcendence of e and π. In: Mathematische Annalen 43 (1893 ), pp. 216-219.
• Paul Albert Gordan: transcendence of e and π. In: Mathematische Annalen 43 (1893 ), pp. 222-224.
• Theodor Vahlen: proof of the theorem on the exponential function Lindemann'schen. In: Mathematische Annalen 53 (1900 ), pp. 457-460.

Recreational mathematics

• Underwood Dudley: Mathematics between delusion and wit. Fallacies, false evidence and the significance of the number 57 for the American history, Birkhäuser, Basel 1995, ISBN 3-7643-5145-4. ( original English title: Mathematical cranks )