Editing Constructible number (section)

==History==

The birth of the concept of constructible numbers is inextricably linked with the history of the three impossible compass and straightedge constructions: doubling 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]] due to a passage in [[Plutarch]]. According to Plutarch, Plato gave the duplication of the cube (Delian) problem to [[Eudoxus of Cnidus|Eudoxus]] and [[Archytas]] and [[Menaechmus]], who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using [[pure geometry]].<ref>Plutarch, ''Quaestiones convivales'' [https://web.archive.org/web/20080815001514/http://ebooks.adelaide.edu.au/p/plutarch/symposiacs/chapter8.html#section80 VIII.ii], 718ef.</ref> However, this attribution is challenged,{{sfnp|Kazarinoff|2003|p=28}} due, in part, to the existence of another version of the story (attributed to [[Eratosthenes]] by [[Eutocius of Ascalon]]) that says that all three found solutions but they were too abstract to be of practical value.{{sfnp|Knorr|1986|p=4}} [[Proclus]], citing [[Eudemus of Rhodes]], credited [[Oenopides]] ({{circa}} 450 BCE) with two ruler and compass constructions, leading some authors to hypothesize that Oenopides originated the restriction.{{sfnp|Knorr|1986|pp=15–17}} 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 of Hippias|Quadratrix]] of [[Hippias of Elis]], the [[conic section|conics]] of Menaechmus, or the marked straightedge ([[Neusis construction|neusis]]) construction of [[Archimedes]] have all been used, as has a more modern approach via [[mathematics of paper folding|paper folding]].{{sfnp|Friedman|2018|pp=1–3}}
 
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 {{nowrap|<math>n</math>-gons}} with <math>n=2^h</math> (for any integer <math>h\ge 2</math>), 3, 5, or the product of any two or three of these numbers, but other regular {{nowrap|<math>n</math>-gons}} eluded them. In 1796 [[Carl Friedrich Gauss]], then an eighteen-year-old student, announced in a newspaper that he had constructed a [[Heptadecagon|regular 17-gon]] with straightedge and compass.{{sfnp|Kazarinoff|2003|p=29}} 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]]'' giving the {{em|sufficient}} condition for the construction of a regular {{nowrap|<math>n</math>-gon.}} Gauss claimed, but did not prove, that the condition was also necessary and several authors, notably [[Felix Klein]],{{sfnp|Klein|1897|p=16}} attributed this part of the proof to him as well.{{sfnp|Kazarinoff|2003|p=30}} Alhazen's problem is also not one of the classic three problems, but despite being named after [[Ibn al-Haytham]] (Alhazen), a [[Mathematics in medieval Islam|medieval Islamic mathematician]], it already appears in [[Ptolemy]]'s [[Optics (Ptolemy)|work on optics]] from the second century.{{sfnp|Neumann|1998}}

[[Pierre Wantzel]] proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve using 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]] the number of its sides is the product of a [[power of two]] and any number of distinct [[Fermat prime]]s (i.e., the sufficient conditions given by Gauss are also necessary).{{sfnmp|Wantzel|1837|Martin|1998|2p=46}} An attempted proof of the impossibility of squaring the circle was given by [[James Gregory (astronomer and mathematician)|James Gregory]] in ''{{lang|la|Vera Circuli et Hyperbolae Quadratura|italic=unset}}'' (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 {{pi}}. It was not until 1882 that [[Ferdinand von Lindemann]] rigorously proved its impossibility, by extending the work of [[Charles Hermite]] and proving that {{pi}} is a [[transcendental number]].{{sfnp|Martin|1998|p=44}}<ref>{{harvp|Klein|1897|pp=68–77|loc=Chapter IV}}: The transcendence of the number {{pi}}.</ref> Alhazen's problem was not proved impossible to solve by compass and straightedge until the work of Jack Elkin.<ref>{{harvp|Elkin|1965}}; see also {{harvp|Neumann|1998}} for an independent solution with more of the history of the problem.</ref>

The study of constructible numbers, per se, was initiated by [[René Descartes]] in [[La Géométrie]], an appendix to his book ''[[Discourse on the Method]]'' 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 of Alexandria|Pappus]].{{sfnp|Boyer|2004|pp=83–88}}