Editing Hyperbolic geometry (section)

== History ==
{{see also|Non-euclidean Geometry}}
Since the publication of [[Euclid's Elements|Euclid's ''Elements'']] circa 300BC, many [[geometers]] tried to prove the [[parallel postulate]]. Some tried to prove it by [[Proof by contradiction|assuming its negation and trying to derive a contradiction]]. Foremost among these were [[Proclus]], [[Ibn al-Haytham]] (Alhacen), [[Omar Khayyám]],<ref>See for instance, {{cite web|url=http://www.resonancepub.com/omarkhayyam.htm|title=Omar Khayyam 1048–1131|access-date=2008-01-05|archive-date=2007-09-28|archive-url=https://web.archive.org/web/20070928084550/http://www.resonancepub.com/omarkhayyam.htm|url-status=dead}}</ref> [[Nasīr al-Dīn al-Tūsī]], [[Witelo]], [[Gersonides]], [[Abner of Burgos|Alfonso]], and later [[Giovanni Gerolamo Saccheri]], [[John Wallis]], [[Johann Heinrich Lambert]], and [[Adrien-Marie Legendre|Legendre]].<ref>{{cite web|url=http://www.math.columbia.edu/~pinkham/teaching/seminars/NonEuclidean.html|title=Non-Euclidean Geometry Seminar|website=Math.columbia.edu|access-date=21 January 2018}}</ref>
Their attempts were doomed to failure (as we now know, the parallel postulate is not provable from the other postulates), but their efforts led to the discovery of hyperbolic geometry.

The theorems of Alhacen, Khayyam and al-Tūsī on [[quadrilateral]]s, including the [[Ibn al-Haytham–Lambert quadrilateral]] and [[Khayyam–Saccheri quadrilateral]], were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.<ref>Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., ''[[Encyclopedia of the History of Arabic Science]]'', Vol. 2, p. 447–494 [470], [[Routledge]], London and New York: {{blockquote|"Three scientists, Ibn al-Haytham, Khayyam and al-Tūsī, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's ''[[Book of Optics]]'' (''Kitab al-Manazir'') – was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar [[Gersonides|Levi ben Gerson]], who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that ''Pseudo-Tusi's Exposition of Euclid'' had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."}}</ref>

In the 18th century, [[Johann Heinrich Lambert]] introduced the [[hyperbolic functions]]<ref>{{citation|title=Foundations and Fundamental Concepts of Mathematics|first=Howard|last=Eves|publisher=Courier Dover Publications|year=2012|isbn=9780486132204|page=59|url=https://books.google.com/books?id=J9QcmFHj8EwC&pg=PA59|quote=We also owe to Lambert the first systematic development of the theory of hyperbolic functions and, indeed, our present notation for these functions.}}</ref> and computed the area of a [[hyperbolic triangle]].<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=Graduate Texts in Mathematics|first=John|last=Ratcliffe|publisher=Springer|year=2006|isbn=9780387331973|page=99|url=https://books.google.com/books?id=JV9m8o-ok6YC&pg=PA99|quote=That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph ''Theorie der Parallellinien'', which was published posthumously in 1786.}}</ref>

===19th-century developments===

In the 19th century, hyperbolic geometry was explored extensively by [[Nikolai Lobachevsky]], [[János Bolyai]], [[Carl Friedrich Gauss]] and [[Franz Taurinus]]. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry.<ref>{{Cite book|author=Bonola, R.|title=Non-Euclidean geometry: A critical and historical study of its development|year=1912|location=Chicago|publisher=Open Court|url=https://archive.org/details/noneuclideangeom00bono}}</ref><ref>{{cite book|author-link1=Marvin Greenberg|last1=Greenberg|first1=Marvin Jay|title=Euclidean and non-Euclidean geometries: development and history|url=https://archive.org/details/euclideannoneucl00gree_304|url-access=limited|date=2003|publisher=Freeman|location=New York|isbn=0716724464|page=[https://archive.org/details/euclideannoneucl00gree_304/page/n194 177]|edition=3rd|quote=Out of nothing I have created a strange new universe. JÁNOS BOLYAI}}</ref>

Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "[[non-Euclidean geometry]]"<ref>Felix Klein, ''Elementary Mathematics from an Advanced Standpoint: Geometry'', Dover, 1948 (reprint of English translation of 3rd Edition, 1940. First edition in German, 1908) pg. 176</ref> causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self-consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.

In 1868, [[Eugenio Beltrami]] provided models of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent [[if and only if]] Euclidean geometry was.

The term "hyperbolic geometry" was introduced by [[Felix Klein]] in 1871.<ref>F. Klein. "Über die sogenannte Nicht-Euklidische Geometrie". ''Math. Ann.'' 4, 573–625 (also in ''Gesammelte Mathematische Abhandlungen'' 1, 244–350).</ref> Klein followed an initiative of [[Arthur Cayley]] to use the transformations of [[projective geometry]] to produce [[isometries]]. The idea used a [[conic section]] or [[quadric]] to define a region, and used [[cross ratio]] to define a [[metric (mathematics)|metric]]. The projective transformations that leave the conic section or quadric [[invariant (mathematics)#Invariant set|stable]] are the isometries. "Klein showed that if the [[Cayley absolute]] is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane..."<ref>Rosenfeld, B.A. (1988) ''A History of Non-Euclidean Geometry'', page 236, Springer-Verlag {{ISBN|0-387-96458-4}}</ref>

=== Philosophical consequences ===

The discovery of hyperbolic geometry had important [[philosophical]] consequences. Before its discovery many philosophers (such as [[Hobbes]] and [[Spinoza]]) viewed philosophical rigor in terms of the "geometrical method", referring to the method of reasoning used in [[Euclid's Elements|Euclid's ''Elements'']].

[[Kant]] in [[Critique of Pure Reason#Space and time|''Critique of Pure Reason'']] concluded that space (in [[Euclidean geometry]]) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.<ref>{{cite book |last=Lucas |first=John Randolph |author-link=John Lucas (philosopher) |title= Space, Time and Causality |year=1984 |page=149 |publisher=Clarendon Press |isbn=0-19-875057-9}}</ref>

It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the [[Boeotia]]ns" (stereotyped as dullards by the ancient Athenians<ref>{{cite journal
 | last = Wood | first = Donald
 | date = April 1959
 | doi = 10.1177/030639685900100207
 | issue = 2
 | journal = Race
 | pages = 65–71
 | title = Some Greek stereotypes of other peoples
 | volume = 1}}</ref>), which would ruin his status as ''princeps mathematicorum'' (Latin, "the Prince of Mathematicians").<ref>{{cite book|last1=Torretti|first1=Roberto|title=Philosophy of Geometry from Riemann to Poincare|date=1978|publisher=Reidel|location=Dordrecht Holland|page=255}}</ref>
The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in [[mathematical rigour]], [[analytical philosophy]] and [[logic]]. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.

=== Geometry of the universe (spatial dimensions only) ===
{{main article|Philosophy of space and time}}
{{see also|Shape of the universe#Curvature of the universe}}

Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature?

Lobachevsky had already tried to measure the curvature of the universe by measuring the [[parallax]] of [[Sirius]] and treating Sirius as the ideal point of an [[angle of parallelism]]. He realized that his measurements were [[margin of error|not precise enough]] to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then the [[#Standardized Gaussian curvature|absolute length]] is at least one million times the diameter of [[Earth's orbit]] ({{val|2000000|ul=AU}}, 10 [[parsec]]).<ref>{{cite book|last1=Bonola|first1=Roberto|title=Non-Euclidean geometry : a critical and historical study of its developments|url=https://archive.org/details/noneuclideangeom0000bono|url-access=registration|date=1955|publisher=Dover|location=New York, NY|isbn=0486600270|page=[https://archive.org/details/noneuclideangeom0000bono/page/95 95]|edition=Unabridged and unaltered republ. of the 1. English translation 1912.}}</ref>
Some argue that his measurements were methodologically flawed.<ref>{{cite book|last1=Richtmyer|first1=Arlan Ramsay, Robert D.|title=Introduction to hyperbolic geometry|date=1995|publisher=Springer-Verlag|location=New York|isbn=0387943390|pages=[https://archive.org/details/introductiontohy0000rams/page/118 118–120]|url=https://archive.org/details/introductiontohy0000rams/page/118}}</ref>

[[Henri Poincaré]], with his [[sphere-world]] [[thought experiment]], came to the conclusion that everyday experience does not necessarily rule out other geometries.

The [[geometrization conjecture]] gives a complete list of eight possibilities for the fundamental geometry of our space. The problem in determining which one applies is that, to reach a definitive answer, we need to be able to look at extremely large shapes – much larger than anything on Earth or perhaps even in our galaxy.<ref>{{cite web|url=http://www.learner.org/courses/mathilluminated/units/8/textbook/08.php|title=Mathematics Illuminated - Unit 8 - 8.8 Geometrization Conjecture|website=Learner.org|access-date=21 January 2018}}</ref>

=== Geometry of the universe (special relativity) ===
[[Special relativity]] places space and time on equal footing, so that one considers the geometry of a unified [[spacetime]] instead of considering space and time separately.<ref>{{cite book|title=Classical Theory of Fields|author1=L. D. Landau |author2=E. M. Lifshitz |series=[[Course of Theoretical Physics]]|edition=4th|volume=2|publisher=Butterworth Heinemann|isbn=978-0-7506-2768-9|year=1973|pages=1–4}}</ref><ref>{{cite book|title=[[Feynman Lectures on Physics]]|author1=R. P. Feynman |author2=R. B. Leighton |author3=M. Sands |volume=1|isbn=0-201-02116-1|year=1963|publisher=Addison Wesley|page=(17-1)–(17-3)}}</ref> [[Minkowski space|Minkowski geometry]] replaces [[Galilean geometry]] (which is the 3-dimensional Euclidean space with time of [[Galilean relativity]]).<ref>{{cite book|title=Dynamics and Relativity|url=https://archive.org/details/dynamicsrelativi00fors|url-access=limited|author1=J. R. Forshaw |author-link1=Jeff Forshaw|author2=A. G. Smith |series=Manchester physics series|isbn=978-0-470-01460-8|year=2008|publisher=Wiley|pages=[https://archive.org/details/dynamicsrelativi00fors/page/n260 246]–248}}</ref>

In relativity, rather than Euclidean, elliptic and hyperbolic geometry, the appropriate geometries to consider are [[Minkowski space]], [[de Sitter space]] and [[anti-de Sitter space]],<ref>{{cite book |author1=Misner |author2=Thorne |author3=Wheeler |date=1973 |title=Gravitation |url=https://archive.org/details/gravitation00cwmi |url-access=limited |pages=[https://archive.org/details/gravitation00cwmi/page/n53 21], 758 }}</ref><ref>{{cite book |author1=John K. Beem|author2=Paul Ehrlich |author3=Kevin Easley |date=1996 |title=Global Lorentzian Geometry |edition=Second}}</ref> corresponding to zero, positive and negative curvature respectively.

Hyperbolic geometry enters special relativity through [[rapidity]], which stands in for [[velocity]], and is expressed by a [[hyperbolic angle]]. The study of this velocity geometry has been called [[Non-Euclidean geometry#Kinematic geometries|kinematic geometry]]. The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points (velocities).<ref>{{cite book|title=Classical Theory of Fields|author1=L. D. Landau |author2=E. M. Lifshitz |series=[[Course of Theoretical Physics]]|edition=4th|volume=2|publisher=Butterworth Heinemann|isbn=978-0-7506-2768-9|year=1973|page=38}}</ref>