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A non-Euclidean geometry is the study of shapes and constructions that do not map directly to
any n-dimensional Euclidean system, characterized by a non-vanishing Riemann curvature
tensor. Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry,
which are contrasted with a Euclidean geometry.
The essential difference between Euclidean and non-Euclidean geometry 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 (see the entries on hyperbolic geometry, elliptic geometry, and absolute
geometry for more information).
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 Euclidean geometry the lines remain at a constant distance from each other even if
extended to infinity, 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 eventually intersect.
For the layman, non-euclidean geometry can be understood by picturing the drawing of
geometric figures on curved surfaces, for example, the surface of a sphere or the inside surface of
 Concepts of non-Euclidean geometry
Non-Euclidean geometry systems differ from Euclidean geometry in that they modify Euclid's
fifth postulate, which is also known as the parallel postulate.
In general, there are two forms of (homogeneous) non-Euclidean geometry, hyperbolic geometry
and elliptic geometry. In hyperbolic geometry there are many distinct lines through a particular
point that will not intersect with another given line. In elliptic geometry there are no lines that
will not intersect, as all that start separate will converge. In addition, elliptic geometry modifies
Euclid's first postulate so that two points determine at least one line. Riemannian geometry deals
with geometries which are not homogeneous, which means that in some sense not all the points
are the same. For example, consider the surface formed by gluing one end of a cylinder to a half
sphere. Then points on the sphere locally obey elliptic geometry, but points on the cylinder
locally obey Euclidean geometry. Bernhard Riemann, building on the work of Gauss, determined
a method of describing such spaces.
Basing new systems on these assumptions, each is constructed with its own rules and postulates.
Non-Euclidean geometries and in particular elliptic geometry play an important role in relativity
theory and the geometry of spacetime.
The concepts applied to certain non-Euclidean planes can only be shown in three or even four
dimensions. The Möbius strip and Klein bottle are both complete one-sided objects, impossible
in a Euclidean plane. The Möbius strip can be shown in three dimensions, but the Klein bottle
 Non-Euclidean geometry of spacetime
The scope of non-Euclidean geometry includes the spacetime theory of Herman Minkowski.
This geometry substitutes a bilinear form for the usual metric distance. Concepts in this
geometry refer to a hyperbolic angle rather than the usual Euclidean angle; for example the
squeeze mapping moves these angles as does Euclidean rotation move ordinary angles. Instead
of perpendicular lines, the spacetime geometry uses hyperbolic-orthogonal lines which determine
hyperplanes of simultaneity. Foundations of a planar version of spacetime were explored, using
synthetic geometry, in 1912 by Gilbert N. Lewis and Edwin B. Wilson in the Proceedings of the
American Academy of Arts and Sciences 48:387–507. See the references for an excerpt,
"Synthetic Spacetime", including definitions, 16 axioms, 21 theorems, and various corollaries by
Lewis and Wilson.
Furthermore, hyperbolic geometry arises in special relativity as follows: an inertial frame of
reference is determined by a velocity, and given a unit of time, each velocity corresponds to a
future event from the origin that is the position of an observer with that velocity after the
temporal unit. These future events form a hyperboloid, the basis of the hyperboloid model of
hyperbolic geometry. Herman Minkowski made this connection on his famous paper of 1908.
While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the
oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate
until the 19th century.
The debate that eventually led to the discovery of non-Euclidean geometries began almost as
soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited
number of assumptions (23 definitions, five common notions, and five postulates) and sought to
prove all the other results (propositions) in the work. The most notorious of the postulates is
often referred to as "Euclid's Fifth Postulate," or simply the "parallel postulate", which in
Euclid's original formulation is:
If a straight line falls on two straight lines in such a manner that the interior angles on the same
side are together less than two right angles, then the straight lines, if produced indefinitely, meet
on that side on which are the angles less than the two right angles.
Other mathematicians have devised simpler forms of this property (see parallel postulate for
equivalent statements). Regardless of the form of the postulate, however, it consistently appears
to be more complicated than Euclid's other postulates (which include, for example, "Between any
two points a straight line may be drawn").
For at least a thousand years, geometers were troubled by the disparate complexity of the fifth
postulate, and believed it could be proved as a theorem from the other four. Many attempted to
find a proof by contradiction, including the Arabic mathematician Ibn al-Haytham (Alhazen,
11th century), the Persian mathematicians Omar Khayyám (12th century) and Nasīr al-Dīn al-
Tūsī (13th century), and the Italian mathematician Giovanni Girolamo Saccheri (18th century).
The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert
quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the
elliptic geometries." These theorems along with their alternative postulates, such as Playfair's
axiom, played an important role in the later development of non-Euclidean geometry. These
early attempts at challenging the fifth postulate had a considerable influence on its development
among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis and
Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry
however provided flawed proofs of the parallel postulate, containing assumptions that were
essentially equivalent to the parallel postulate. These early attempts did, however, provide some
early properties of the hyperbolic and elliptic geometries.
Khayyam, however, may be somewhat of an exception. Unlike many commentators on Euclid
before and after him (including Saccheri), Khayyam was not trying to prove the parallel
postulate as such but to derive it from an equivalent postulate he formulated from "the principles
of the Philosopher" (Aristotle): "Two convergent straight lines intersect and it is impossible for
two convergent straight lines to diverge in the direction in which they converge." Khayyam
then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri
quadrilateral can take and after proving a number of theorems about them, he correctly refuted
the obtuse and acute cases based on his postulate and hence derived the classic postulate of
Euclid. Another exception may be al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-
Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which
presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel
postulate. "He essentially revised both the Euclidean system of axioms and postulates and the
proofs of many propositions from the Elements." His work was published in Rome in 1594
and was studied by European geometers, including Saccheri.
Giordano Vitale, in his book Euclide restituo (1680, 1686), used the Saccheri quadrilateral to
prove that if three points are equidistant on the base AB and the summit CD, then AB and CD
are everywhere equidistant.
In a work titled Euclides ab Omni Naevo Vindicatus (Euclid Freed from All Flaws), published in
1733, Saccheri quickly discarded elliptic geometry as a possibility (some others of Euclid's
axioms must be modified for elliptic geometry to work) and set to work proving a great number
of results in hyperbolic geometry. He finally reached a point where he believed that his results
demonstrated the impossibility of hyperbolic geometry. His claim seems to have been based on
Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove
Euclidean geometry he instead unintentionally discovered a new viable geometry. At this time it
was widely believed that the universe worked according to the principles of Euclidean geometry.
The beginning of the 19th century would finally witness decisive steps in the creation of non-
Euclidean geometry. Around 1830, the Hungarian mathematician János Bolyai and the Russian
mathematician Nikolai Ivanovich Lobachevsky separately published treatises on hyperbolic
geometry. Consequently, hyperbolic geometry is called Bolyai-Lobachevskian geometry, as both
mathematicians, independent of each other, are the basic authors of non-Euclidean geometry.
Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had
developed such a geometry about 20 years before, though he did not publish. While
Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai
worked out a geometry where both the Euclidean and the hyperbolic geometry are possible
depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide
through mathematical reasoning alone if the geometry of the physical universe is Euclidean or
non-Euclidean; this is a task for the physical sciences. Between 1820 and 1823 Bolyai prepared a
treatise on a complete system of non-Euclidean geometry.
In the 1840s, Hermann Grassmann wrote a Ph.D. thesis on abstract algebra and exterior algebra,
wherein he argued that the dimensionality of the physical Universe was not necessarily three, but
may be unbounded. In 1846 he derived a coordinate and metric-free geometric calculus, suitable
for a class of spaces including affine and projective spaces. Unfortunately although Grassmann's
work was fundamental to several 20th century branches of mathematics, it was so far ahead of its
time that his peers couldn't understand it.
Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry,
discussing in particular the ideas now called manifolds, Riemannian metric, and curvature. He
constructed an infinite family of non-Euclidean geometries by giving a formula for a family of
Riemannian metrics on the unit ball in Euclidean space. Sometimes he is unjustly credited with
only discovering elliptic geometry; but in fact, this construction shows that his work was far-
reaching, with his theorems holding for all geometries.
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
For more details on this topic, see Models of non-Euclidean geometry.
Euclidean geometry is modelled by our notion of a "flat plane."
 Elliptic geometry
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).
In the elliptic model, for any given line ℓ and a point A, which is not on ℓ, all lines through A will
 Hyperbolic geometry
Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: does such a
model exist for hyperbolic geometry? The model for hyperbolic geometry was answered by
Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the
appropriate curvature to model a portion of hyperbolic space, and in a second paper in the same
year, defined the Klein model, the Poincaré disk model, and the Poincaré half-plane model which
model the entirety of hyperbolic space, and used this to show that Euclidean geometry and
hyperbolic geometry were equiconsistent, so that hyperbolic geometry was logically consistent if
and only if Euclidean geometry was. (The reverse implication follows from the horosphere
model of Euclidean geometry.)
In the hyperbolic model, within a two-dimensional plane, for any given line ℓ and a point A,
which is not on ℓ, there are infinitely many lines through A that do not intersect ℓ.
 Other models
There are other mathematical models of the plane in which the parallel postulate fails, for
example the Dehn plane consisting of all points (x,y), where x and y are finite surreal numbers.
The development of non-Euclidean geometries proved important to physics in the 20th century.
Albert Einstein's general theory of relativity describes space as generally not flat (i.e.,
Euclidean), but as elliptically curved (i.e., non-Euclidean) near regions where energy is present.
This kind of geometry, where the curvature changes from point to point, is called Riemannian
Non-Euclidean geometry often makes appearances in works of science fiction and fantasy. Its
usage is most clearly tied 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. This
is said to be a profoundly unsettling sight, often to the point of driving those who look upon it
insane. 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 main
1. ^ Scott Walter (1999) Non-Euclidean Style of Special Relativity
2. ^ Eder, Michelle (2000), Views of Euclid'sParallel Postulate in Ancient Greece and in Medieval
Islam, Rutgers University, http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html,
3. ^ Boris A. Rosenfeld & Adolf P. Youschkevitch, "Geometry", p. 470, in Roshdi Rashed & Régis
Morelon (1996), Encyclopedia of the History of Arabic Science,Vol. 2, pp. 447–494, Routledge,
London and New York:
"Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable
contribution to this branch of geometry whose importance came to be completely recognized only
in the nineteenth 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 tthis
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 investiagtions of their
European couterparts. The first European attempt to prove the postulate on parallel lines – made
by Witelo, the Polish scientists of the thirteenth 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 fourteenth century by the Jewish scholar 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 borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines."
"But in a manuscript probably written by his son Sadr al-Din in 1298, based on Nasir al-Din's
later thoughts on the subject, there is a new argument based on another hypothesis, also
equivalent to Euclid's, [...] The importance of this latter work is that it was published in Rome in
1594 and was studied by European geometers. In particular, it became the starting point for the
work of Saccheriand ultimately for the discovery of non-Euclidean geometry."