Non-Euclidean geometry

# Non-Euclidean geometry

Overview
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

, namely hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

and elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

. This is one term which, for historical reasons, has a meaning in mathematics which is much narrower than it appears to have in the general English language. There are a great many geometries which are not Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, but only these two are referred to as the non-Euclidean geometries.
Discussion
 Ask a question about 'Non-Euclidean geometry' Start a new discussion about 'Non-Euclidean geometry' Answer questions from other users Full Discussion Forum

Encyclopedia
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

, namely hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

and elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

. This is one term which, for historical reasons, has a meaning in mathematics which is much narrower than it appears to have in the general English language. There are a great many geometries which are not Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, but only these two are referred to as the non-Euclidean geometries.

The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...

lines. Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

's fifth postulate, the parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point A, which is not on , there is exactly one line through A that does not intersect . In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting , while in elliptic geometry, any line through A intersects (see the entries on hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

, elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

, and absolute geometry
Absolute geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832...

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
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to a third line:
• In Euclidean geometry the lines remain at a constant distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical 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.

### Early history

While Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, named after the Greek mathematician
Greek mathematics
Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the Eastern shores of the Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to...

Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

, 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
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

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 (proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...

s) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate," or simply the "parallel postulate
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

", 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
Parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry...

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
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...

mathematicians Omar Khayyám
Omar Khayyám
Omar Khayyám was aPersian polymath: philosopher, mathematician, astronomer and poet. He also wrote treatises on mechanics, geography, mineralogy, music, climatology and theology....

(12th century) and Nasīr al-Dīn al-Tūsī
Nasir al-Din al-Tusi
Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī , better known as Naṣīr al-Dīn al-Ṭūsī , was a Persian polymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed...

(13th century), and the Italian
Italy
Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

mathematician Giovanni Girolamo Saccheri
Giovanni Girolamo Saccheri
Giovanni Girolamo Saccheri was an Italian Jesuit priest, scholastic philosopher, and mathematician....

(18th century).

The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...

s, including the Lambert quadrilateral
In geometry, a Lambert quadrilateral,named after Johann Heinrich Lambert,is a quadrilateral three of whose angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel...

A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book Euclid vindicatus , an attempt to prove the parallel postulate using the method Reductio ad absurdum...

, were "the first few theorems of the hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

and the elliptic geometries." These theorems along with their alternative postulates, such as Playfair's axiom
Playfair's axiom
Playfair's axiom is a geometrical axiom, intended to replace the fifth postulate of Euclides :Given a line and a point not on it, at most one parallel to the given line can be drawn through the point....

, 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
Witelo
Witelo was a friar, theologian and scientist: a physicist, natural philosopher, mathematician. He is an important figure in the history of philosophy in Poland...

, Levi ben Gerson, Alfonso
Alfonso
Alfonso , Alfons , Afonso , Affonso , Alphonse, Alfonse , Αλφόνσος , Alphonsus , Alphons , Alfonsu in ,...

, 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, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...

): "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 which he didn't realize was equivalent to his own postulate. Another example is 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 another 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
Rome
Rome is the capital of Italy and the country's largest and most populated city and comune, with over 2.7 million residents in . The city is located in the central-western portion of the Italian Peninsula, on the Tiber River within the Lazio region of Italy.Rome's history spans two and a half...

in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis.

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, but did not realize it.

In 1766 Johann Lambert
Johann Heinrich Lambert
Johann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.-Biography:...

wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further.

At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.

### Creation of non-Euclidean geometry

The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Around 1830, the Hungarian
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

mathematician János Bolyai
János Bolyai
János Bolyai was a Hungarian mathematician, known for his work in non-Euclidean geometry.Bolyai was born in the Transylvanian town of Kolozsvár , then part of the Habsburg Empire , the son of Zsuzsanna Benkő and the well-known mathematician Farkas Bolyai.-Life:By the age of 13, he had mastered...

and the Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...

n mathematician Nikolai Ivanovich Lobachevsky
Nikolai Ivanovich Lobachevsky
Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, renowned primarily for his pioneering works on hyperbolic geometry, otherwise known as Lobachevskian geometry...

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.

Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

, in a famous lecture in 1854, founded the field of Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, discussing in particular the ideas now called manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s, Riemannian metric, and curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

.
He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. The simplest of these is called elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

and it is considered to be a non-Euclidean geometry due to its lack of parallel lines.

### Terminology

It was Gauss who coined the term "non-euclidean geometry". He was referring to his own work which today we call hyperbolic geometry. Several modern authors still consider "non-euclidean geometry" and "hyperbolic geometry" to be synonyms. In 1871, Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

, by adapting a metric discussed by Arthur Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

in 1852, was able to bring metric properties into a projective setting and was therefore able to unify the treatments of hyperbolic, euclidean and elliptic geometry under the umbrella of projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

. Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry "parabolic", a term which has not survived the test of time). His influence has led to the current usage of the term "non-euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.

There are some mathematicians who would extend the list of geometries that should be called "non-euclidean" in various ways. In other disciplines, most notably mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, the term "non-euclidean" is often taken to mean not Euclidean.

## Axiomatic basis of non-Euclidean geometry

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 which should also have been taken as axioms. Hilbert's system
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...

consisting of 20 axioms 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
Primitive notion
In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or...

obtain the same geometry by different paths. In all approaches, however, there is an axiom which is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

uses the Playfair axiom form, while Birkhoff
Garrett Birkhoff
Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....

, for instance, uses the axiom which says that "there exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom which is equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry
Absolute geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry that does not assume the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832...

. 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.

To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...

. Negating the Playfair's axiom
Playfair's axiom
Playfair's axiom is a geometrical axiom, intended to replace the fifth postulate of Euclides :Given a line and a point not on it, at most one parallel to the given line can be drawn through the point....

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
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

. 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, 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". In order to obtain a consistent set of axioms which includes this axiom about having no parallel lines, some of the other axioms must be tweaked. The adjustments to be made depend upon the axiom system being used. Among others these tweaks will 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
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

emerges as the most natural geometry satisfying this axiom.

## Models of non-Euclidean geometry

Two dimensional Euclidean geometry is modelled by our notion of a "flat plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

."

### Elliptic geometry

The simplest model for elliptic geometry
Elliptic geometry
Elliptic geometry is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which can be interpreted as asserting that there is exactly one...

is a sphere, where lines are "great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s" (such as the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

or the meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

s on a globe
Globe
A globe is a three-dimensional scale model of Earth or other spheroid celestial body such as a planet, star, or moon...

), and points opposite each other (called antipodal points) are identified (considered to be the same). This is also one of the standard models of the real projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...

. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric.

In the elliptic model, for any given line and a point A, which is not on , all lines through A will intersect .

### Hyperbolic geometry

Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: does such a model exist for hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

? The model for hyperbolic geometry
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...

was answered by Eugenio Beltrami
Eugenio Beltrami
Eugenio Beltrami was an Italian mathematician notable for his work concerning differential geometry and mathematical physics...

, in 1868, who first showed that a surface called the pseudosphere
Pseudosphere
In geometry, the term pseudosphere is used to describe various surfaces with constant negative gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid....

has the appropriate curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

to model a portion of hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

, and in a second paper in the same year, defined the Klein model
Klein model
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points in the interior of the n-dimensional unit ball and lines are represented by the...

, the Poincaré disk model
Poincaré disk model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of n-dimensional hyperbolic geometry in which the points of the geometry are in an n-dimensional disk, or unit ball, and the straight lines of the hyperbolic geometry are segments of circles contained in the disk...

, and the Poincaré half-plane model
Poincaré half-plane model
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane , together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry....

which model the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent
Equiconsistency
In mathematical logic, two theories are equiconsistent if, roughly speaking, they are "as consistent as each other".It is not in general possible to prove the absolute consistency of a theory T...

, 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 and a point A, which is not on , there are infinitely many lines through A that do not intersect .

In these models the concepts of non-Euclidean geometries are being represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are being represented by Euclidean curves which visually bend. This "bending" is not a property of the non-Euclidean lines, only an artifice of the way they are being represented.

## Uncommon properties

There are many similar properties to be found in Euclidean and non-Euclidean geometries, naturally those which do not depend upon the nature of parallelism. However, the properties which distinguish one geometry from the others are the ones which 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
In geometry, a Lambert quadrilateral,named after Johann Heinrich Lambert,is a quadrilateral three of whose angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel...

is a quadrilateral which has three right angles. The fourth angle of a Lambert quadrilateral is acute if the geometry is hyperbolic, a right angle
Right angle
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...

if the geometry is Euclidean or obtuse if the geometry is elliptic. Consequently, rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

s only exist in Euclidean geometry.

• A Saccheri quadrilateral
A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his book Euclid vindicatus , an attempt to prove the parallel postulate using the method Reductio ad absurdum...

is a quadrilateral which has 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.

## Importance

Non-Euclidean geometry is an example of a paradigm shift
A Paradigm shift is, according to Thomas Kuhn in his influential book The Structure of Scientific Revolutions , a change in the basic assumptions, or paradigms, within the ruling theory of science...

in the history of science
History of science
The history of science is the study of the historical development of human understandings of the natural world and the domains of the social sciences....

. Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

of space
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...

. Furthermore, since the substance of the subject in synthetic geometry
Synthetic geometry
Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...

was a chief exhibit of rationality, the Euclidean point of view represented absolute authority. Non-Euclidean geometry, though assimilated by learned investigators, continues to be suspect for those not having exposure to hyperbolic and elliptical concepts.

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
Immanuel Kant
Immanuel Kant was a German philosopher from Königsberg , researching, lecturing and writing on philosophy and anthropology at the end of the 18th Century Enlightenment....

'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 mathematics that was a result of this paradigm shift.

The existence of non-Euclidean geometries impacted the "intellectual life" of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

. This curriculum issue was hotly debated at the time and was even the subject of a play, Euclid and his Modern Rivals, written by the author of Alice in Wonderland
Alice's Adventures in Wonderland
Alice's Adventures in Wonderland is an 1865 novel written by English author Charles Lutwidge Dodgson under the pseudonym Lewis Carroll. It tells of a girl named Alice who falls down a rabbit hole into a fantasy world populated by peculiar, anthropomorphic creatures...

.

## Planar algebras

In analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

a plane is described with Cartesian coordinates : C = {(x,y) : x, y in R}. The point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

s are sometimes identified with hypercomplex numbers z = x + y ε where the square of ε is in {−1, 0, +1}.
The Euclidean plane corresponds to the case ε2 = −1 since the modulus of z is given by
and this quantity is the square of the Euclidean distance
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

between z and the origin.
For instance, {z : z z* = 1} is the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

.

For planar algebra, non-Euclidean geometry arises in the other cases.
When , then z is a split-complex number
Split-complex number
In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...

and conventionally j replaces epsilon. Then
and {z : z z* = 1} is the unit hyperbola
Unit hyperbola
In geometry, the unit hyperbola is the set of points in the Cartesian plane that satisfies x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial lengthWhereas the unit circle surrounds its center, the unit hyperbola requires the...

.

When , then z is a dual number
Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...

.

This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

in the dual number plane and hyperbolic angle
Hyperbolic angle
In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle...

in the split-complex plane correspond to angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

in Euclidean geometry. Indeed, they each arise in polar decomposition of a complex number z.

## Kinematic geometries

Hyperbolic geometry found an application in kinematics
Kinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....

with the cosmology
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

introduced by Herman Minkowski in 1908. Minkowski introduced terms like worldline and proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

into mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

. He realized that the submanifold
Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required...

, of events one moment of proper time into the future, could be considered a hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

of three dimensions.
Already in the 1890s Alexander Macfarlane
Alexander Macfarlane
Alexander Macfarlane was a Nova Scotia lawyer and political figure. He was a member of the Canadian Senate from 1870 to 1898. His surname also appears as McFarlane in some sources....

was charting this submanifold through his Algebra of Physics and hyperbolic quaternion
Hyperbolic quaternion
In the abstract algebra of algebras over a field, the hyperbolic quaternionq = a + bi + cj + dk, \quad a,b,c,d \in R \!is a mutated quaternion wherei^2 = j^2 = k^2 = +1 \! instead of the usual −1....

s, though Macfarlane didn’t use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model
Hyperboloid model
In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model , is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in -dimensional Minkowski space and m-planes are...

of hyperbolic geometry.

The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the split-complex number
Split-complex number
In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...

z = eaj can represent a spacetime event one moment into the future of a frame of reference
Frame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...

of rapidity
Rapidity
In relativity, rapidity is an alternative to speed as a framework for measuring motion. On parallel velocities rapidities are simply additive, unlike speeds at relativistic velocities. For low speeds, rapidity and speed are proportional, but for high speeds, rapidity takes a larger value. The...

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 number
Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...

s to represent the classical description of motion in absolute time and space
Absolute time and space
Originally introduced by Sir Isaac Newton in the Philosophiæ Naturalis Principia Mathematica, the concepts of absolute time and space provided a theoretical foundation that facilitated Newtonian mechanics...

:
The equations are equivalent to a shear mapping in linear algebra:
With dual numbers the mapping is

Another view of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

as a non-Euclidean geometry was advanced by E. B. Wilson
E. B. Wilson
E. B. Wilson may refer to:* Edmund Beecher Wilson, American zoologist and geneticist* Edwin Bidwell Wilson, American mathematician and pioneer in vector analysis.* E. B. Wilson and Company, British locomotive manufacturer...

and Gilbert Lewis
Gilbert Lewis
Gilbert Lewis is an American actor who is best known for playing The King of Cartoons in the first season of the 1986 children's show, Pee-wee's Playhouse. Lewis played the King of Cartoons in thirteen episodes before being replaced by actor, William Marshall...

in Proceedings of the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...

in 1912. They revamped the analytic geometry implicit in the split-complex number algebra into synthetic geometry
Synthetic geometry
Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...

of premises and deductions.

## Fiction

Non-Euclidean geometry often makes appearances in works of science fiction
Science fiction
Science fiction is a genre of fiction dealing with imaginary but more or less plausible content such as future settings, futuristic science and technology, space travel, aliens, and paranormal abilities...

and fantasy
Fantasy
Fantasy is a genre of fiction that commonly uses magic and other supernatural phenomena as a primary element of plot, theme, or setting. Many works within the genre take place in imaginary worlds where magic is common...

.

In 1895 H. G. Wells
H. G. Wells
Herbert George Wells was an English author, now best known for his work in the science fiction genre. He was also a prolific writer in many other genres, including contemporary novels, history, politics and social commentary, even writing text books and rules for war games...

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
Horror fiction
Horror fiction also Horror fantasy is a philosophy of literature, which is intended to, or has the capacity to frighten its readers, inducing feelings of horror and terror. It creates an eerie atmosphere. Horror can be either supernatural or non-supernatural...

writer H. P. Lovecraft
H. P. Lovecraft
Howard Phillips Lovecraft --often credited as H.P. Lovecraft — was an American author of horror, fantasy and science fiction, especially the subgenre known as weird fiction....

. In his works, many unnatural things follow their own unique laws of geometry: In Lovecraft's Cthulhu Mythos
Cthulhu Mythos
The Cthulhu Mythos is a shared fictional universe, based on the work of American horror writer H. P. Lovecraft.The term was first coined by August Derleth, a contemporary correspondent of Lovecraft, who used the name of the creature Cthulhu - a central figure in Lovecraft literature and the focus...

, the sunken city of R'lyeh
R'lyeh
R'lyeh is a fictional lost city that first appeared in the H. P. Lovecraft short story "The Call of Cthulhu", first published in Weird Tales in 1928. According to Lovecraft's short story, R'lyeh is a sunken city in the South Pacific and the prison of the malevolent entity called Cthulhu.R'lyeh is...

is characterized by its non-Euclidean 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
Zen and the Art of Motorcycle Maintenance
Zen and the Art of Motorcycle Maintenance: An Inquiry into Values is a 1974 philosophical novel, the first of Robert M. Pirsig's texts in which he explores his Metaphysics of Quality.The book sold 5 million copies worldwide...

mentioned Riemannian Geometry on multiple occasions.

In The Brothers Karamazov
The Brothers Karamazov
The Brothers Karamazov is the final novel by the Russian author Fyodor Dostoyevsky. Dostoyevsky spent nearly two years writing The Brothers Karamazov, which was published as a serial in The Russian Messenger and completed in November 1880...

, Dostoevsky discusses non-Euclidean geometry through his main character Ivan.

Christopher Priest's The Inverted World
The Inverted World (novel)
The Inverted World is a 1974 science fiction novel by Christopher Priest, expanded from a short story by the same name included in New Writings in SF 22. In 2010 it was included in the SF Masterworks collection.-Plot summary:...

describes the struggle of living on a planet with the form of a rotating pseudosphere
Pseudosphere
In geometry, the term pseudosphere is used to describe various surfaces with constant negative gaussian curvature. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid....

.

Robert Heinlein's The Number of the Beast
The Number of the Beast (novel)
The Number of the Beast is a science fiction novel by Robert A. Heinlein published in 1980. The first edition featured a cover and interior illustrations by Richard M. Powers...

utilizes non-Euclidean geometry to explain instantaneous transport through space and time and between parallel and fictional universes.

Alexander Bruce's Antichamber uses non-Euclidean geometry to create a brilliant, minimal, Escher-like world, where geometry and space follow unfamiliar rules.

In the Renegade Legion
Renegade Legion is a series of science fiction games that were designed by Sam Lewis, produced by FASA, and published from 1989 to 1993. The line was then licensed to Nightshift games, a spin-off of the garage company Crunchy Frog Enterprises by Paul Arden Lidberg, which published one scenario...

science fiction
Science fiction
Science fiction is a genre of fiction dealing with imaginary but more or less plausible content such as future settings, futuristic science and technology, space travel, aliens, and paranormal abilities...

setting for FASA
FASA
FASA Corporation was an American publisher of role-playing games, wargames and board games between 1980 and 2001. Originally the name FASA was an acronym for "Freedonian Aeronautics and Space Administration", a joking allusion to the Marx Brothers film Duck Soup. This tongue-in-cheek attitude was...

'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 twenty-second century.

• Hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

• Projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

• Schopenhauer's criticism of the proofs of the Parallel Postulate
Schopenhauer's criticism of the proofs of the parallel postulate
Arthur Schopenhauer criticized mathematicians' attempts to prove Euclid's Parallel Postulate because they try to prove from indirect concepts that which is directly evident from perception....