Non-Euclidean geometry

The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l.

Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line (in the same plane):

• In Euclidean geometry, the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.
• In hyperbolic geometry, they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
• In elliptic geometry, the lines "curve toward" each other and intersect.

Euclidean geometry can be axiomatically described in several ways. Unfortunately, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms. Hilbert's system consisting of 20 axioms[18] most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of undefined terms obtain the same geometry by different paths. All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff, for instance, uses the axiom that says that, "There exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry. As the first 28 propositions of Euclid (in The Elements) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.[19]

To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation. Negating the Playfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways:

• Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line. In the first case, replacing the parallel postulate (or its equivalent) with the statement "In a plane, given a point P and a line l not passing through P, there exist two lines through P, which do not meet l" and keeping all the other axioms, yields hyperbolic geometry.[20]
• The second case is not dealt with as easily. Simply replacing the parallel postulate with the statement, "In a plane, given a point P and a line l not passing through P, all the lines through P meet l", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry,[21] but this statement says that there are no parallel lines. This problem was known (in a different guise) to Khayyam, Saccheri and Lambert and was the basis for their rejecting what was known as the "obtuse angle case". To obtain a consistent set of axioms that includes this axiom about having no parallel lines, some other axioms must be tweaked. These adjustments depend upon the axiom system used. Among others, these tweaks have the effect of modifying Euclid's second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded. Riemann's elliptic geometry emerges as the most natural geometry satisfying this axiom.

Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensions

On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.

Models of non-Euclidean geometry are mathematical models of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawn parallel to a given line l through a point that is not on l. In hyperbolic geometric models, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometric models, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.)

Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same). The pseudosphere has the appropriate curvature to model hyperbolic geometry.

Lambert quadrilateral in hyperbolic geometry

Saccheri quadrilaterals in the three geometries

Euclidean and non-Euclidean geometries naturally have many similar properties, namely those that do not depend upon the nature of parallelism. This commonality is the subject of absolute geometry (also called neutral geometry). However, the properties that distinguish one geometry from others have historically received the most attention.

Besides the behavior of lines with respect to a common perpendicular, mentioned in the introduction, we also have the following:

• A Lambert quadrilateral is a quadrilateral with three right angles. The fourth angle of a Lambert quadrilateral is acute if the geometry is hyperbolic, a right angle if the geometry is Euclidean or obtuse if the geometry is elliptic. Consequently, rectangles exist (a statement equivalent to the parallel postulate) only in Euclidean geometry.
• A Saccheri quadrilateral is a quadrilateral with two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry is Euclidean and obtuse angles if the geometry is elliptic.
• The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180° if the geometry is elliptic. The defect of a triangle is the numerical value (180° − sum of the measures of the angles of the triangle). This result may also be stated as: the defect of triangles in hyperbolic geometry is positive, the defect of triangles in Euclidean geometry is zero, and the defect of triangles in elliptic geometry is negative.

Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the mathematical model of space. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.

The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant's treatment of human knowledge had a special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift.[23]

Non-Euclidean geometry is an example of a scientific revolution in the history of science, in which mathematicians and scientists changed the way they viewed their subjects.[24] Some geometers called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work.[25][26]

The existence of non-Euclidean geometries impacted the intellectual life of Victorian England in many ways[27] and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements. This curriculum issue was hotly debated at the time and was even the subject of a book, Euclid and his Modern Rivals, written by Charles Lutwidge Dodgson (1832–1898) better known as Lewis Carroll, the author of Alice in Wonderland.

In analytic geometry a plane is described with Cartesian coordinates: C = { (x,y) : x, y ∈ ℝ }. The points are sometimes identified with complex numbers z = x + y ε where ε² ∈ { –1, 0, 1}.

The Euclidean plane corresponds to the case ε² = −1 since the modulus of z is given by

${\displaystyle zz^{\ast }=(x+y\epsilon )(x-y\epsilon )=x^{2}+y^{2}}$

and this quantity is the square of the Euclidean distance between z and the origin. For instance, {z | z z* = 1} is the unit circle.

For planar algebra, non-Euclidean geometry arises in the other cases. When ε² = +1, then z is a split-complex number and conventionally j replaces epsilon. Then

${\displaystyle zz^{\ast }=(x+y\mathbf {j} )(x-y\mathbf {j} )=x^{2}-y^{2}\!}$

and {z | z z* = 1} is the unit hyperbola.

When ε² = 0, then z is a dual number.[28]

This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope in the dual number plane and hyperbolic angle in the split-complex plane correspond to angle in Euclidean geometry. Indeed, they each arise in polar decomposition of a complex number z.[29]

Hyperbolic geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in 1908. Minkowski introduced terms like worldline and proper time into mathematical physics. He realized that the submanifold, of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions.[30][31] Already in the 1890s Alexander Macfarlane was charting this submanifold through his Algebra of Physics and hyperbolic quaternions, though Macfarlane did not use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model of hyperbolic geometry.

The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the split-complex number z = e^aj can represent a spacetime event one moment into the future of a frame of reference of rapidity a. Furthermore, multiplication by z amounts to a Lorentz boost mapping the frame with rapidity zero to that with rapidity a.

Kinematic study makes use of the dual numbers ${\displaystyle z=x+y\epsilon ,\quad \epsilon ^{2}=0,}$ to represent the classical description of motion in absolute time and space: The equations ${\displaystyle x^{\prime }=x+vt,\quad t^{\prime }=t}$ are equivalent to a shear mapping in linear algebra:

${\displaystyle {\begin{pmatrix}x'\\t'\end{pmatrix}}={\begin{pmatrix}1&v\\0&1\end{pmatrix}}{\begin{pmatrix}x\\t\end{pmatrix}}.}$

With dual numbers the mapping is ${\displaystyle t^{\prime }+x^{\prime }\epsilon =(1+v\epsilon )(t+x\epsilon )=t+(x+vt)\epsilon .}$[32]

Another view of special relativity as a non-Euclidean geometry was advanced by E. B. Wilson and Gilbert Lewis in Proceedings of the American Academy of Arts and Sciences in 1912. They revamped the analytic geometry implicit in the split-complex number algebra into synthetic geometry of premises and deductions.[33][34]

Non-Euclidean geometry often makes appearances in works of science fiction and fantasy.

• In 1895, H. G. Wells published the short story "The Remarkable Case of Davidson's Eyes". To appreciate this story one should know how antipodal points on a sphere are identified in a model of the elliptic plane. In the story, in the midst of a thunderstorm, Sidney Davidson sees "Waves and a remarkably neat schooner" while working in an electrical laboratory at Harlow Technical College. At the story's close, Davidson proves to have witnessed H.M.S. Fulmar off Antipodes Island.
• Non-Euclidean geometry is sometimes connected with the influence of the 20th-century horror fiction writer H. P. Lovecraft. In his works, many unnatural things follow their own unique laws of geometry: in Lovecraft's Cthulhu Mythos, the sunken city of R'lyeh is characterized by its non-Euclidean geometry. It is heavily implied this is achieved as a side effect of not following the natural laws of this universe rather than simply using an alternate geometric model, as the sheer innate wrongness of it is said to be capable of driving those who look upon it insane.[35]
• The main character in Robert Pirsig's Zen and the Art of Motorcycle Maintenance mentioned Riemannian geometry on multiple occasions.
• In The Brothers Karamazov, Dostoevsky discusses non-Euclidean geometry through his character Ivan.
• Christopher Priest's novel Inverted World describes the struggle of living on a planet with the form of a rotating pseudosphere.
• Robert Heinlein's The Number of the Beast utilizes non-Euclidean geometry to explain instantaneous transport through space and time and between parallel and fictional universes.
• Zeno Rogue's HyperRogue is a roguelike game set on the hyperbolic plane, allowing the player to experience many properties of this geometry. Many mechanics, quests, and locations are strongly dependent on the features of hyperbolic geometry.[36]
• In the Renegade Legion science fiction setting for FASA's wargame, role-playing-game and fiction, faster-than-light travel and communications is possible through the use of Hsieh Ho's Polydimensional Non-Euclidean Geometry, published sometime in the middle of the 22nd century.
• In Ian Stewart's Flatterland the protagonist Victoria Line visits all kinds of non-Euclidean worlds.

• Hyperbolic space
• Lénárt sphere
• Projective geometry
• Non-Euclidean surface growth
• Parallel (geometry)#In non-Euclidean geometry
• Spherical geometry#Relation to similar geometries

• A'Campo, Norbert and Papadopoulos, Athanase, (2012) Notes on hyperbolic geometry, in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, ISBN 978-3-03719-105-7, DOI:10.4171/105.
• Anderson, James W. Hyperbolic Geometry, second edition, Springer, 2005
• Beltrami, Eugenio Teoria fondamentale degli spazî di curvatura costante, Annali. di Mat., ser II 2 (1868), 232–255
• Blumenthal, Leonard M. (1980), A Modern View of Geometry, New York: Dover, ISBN 0-486-63962-2
• Carroll, Lewis Euclid and His Modern Rivals, New York: Barnes and Noble, 2009 (reprint) ISBN 978-1-4351-2348-9
• H. S. M. Coxeter (1942) Non-Euclidean Geometry, University of Toronto Press, reissued 1998 by Mathematical Association of America, ISBN 0-88385-522-4.
• Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker, ISBN 0-8247-1748-1
• Jeremy Gray (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, Clarendon Press.
• Greenberg, Marvin Jay Euclidean and Non-Euclidean Geometries: Development and History, 4th ed., New York: W. H. Freeman, 2007. ISBN 0-7167-9948-0
• Morris Kline (1972) Mathematical Thought from Ancient to Modern Times, Chapter 36 Non-Euclidean Geometry, pp 861–81, Oxford University Press.
• Bernard H. Lavenda, (2012) " A New Perspective on Relativity : An Odyssey In Non-Euclidean Geometries", World Scientific, pp. 696, ISBN 9789814340489.
• Nikolai Lobachevsky (2010) Pangeometry, Translator and Editor: A. Papadopoulos, Heritage of European Mathematics Series, vol. 4, European Mathematical Society.
• Manning, Henry Parker (1963), Introductory Non-Euclidean Geometry, New York: Dover
• Meschkowski, Herbert (1964), Noneuclidean Geometry, New York: Academic Press
• Milnor, John W. (1982) Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24.
• Richards, Joan L. (1988), Mathematical Visions: The Pursuit of Geometry in Victorian England, Boston: Academic Press, ISBN 0-12-587445-6
• Smart, James R. (1997), Modern Geometries (5th Ed.), Pacific Grove: Brooks/Cole, ISBN 0-534-35188-3
• Stewart, Ian (2001) Flatterland, New York: Perseus Publishing ISBN 0-7382-0675-X (softcover)
• John Stillwell (1996) Sources of Hyperbolic Geometry, American Mathematical Society ISBN 0-8218-0529-0 .
• Trudeau, Richard J. (1987), The Non-Euclidean Revolution, Boston: Birkhauser, ISBN 0-8176-3311-1
• A. Papadopoulos et Guillaume Théret (2014) La théorie des parallèles de Johann Heinrich Lambert, Critical edition of Lambert's memoir with a French translation, with historical and mathematical notes and commentaries éd. Blanchard, coll. Sciences dans l'Histoire, Paris ISBN 978-2-85367-266-5

• Media related to Non-Euclidean geometry at Wikimedia Commons

Wikiquote has quotations related to Non-Euclidean geometry.

• Roberto Bonola (1912) Non-Euclidean Geometry, Open Court, Chicago.
• MacTutor Archive article on non-Euclidean geometry
• Non-euclidean geometry at PlanetMath.
• Non-Euclidean geometries from Encyclopedia of Math of European Mathematical Society and Springer Science+Business Media
• Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite.