Hyperbolic geometryFrom Wikipedia, the free encyclopedia

Lines through a given point P and asymptotic to line R.

A triangle immersed in a saddle-shape plane (ahyperbolic paraboloid), as well as two diverging ultraparallel lines.

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai-Lobachevskian

geometry) is a non-Euclidean geometry, meaning that the parallel postulate ofEuclidean geometry is replaced.

The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for

any given line R and point P not on R, there is exactly one line through P that does not intersect R; i.e., that is

parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, so

the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms

of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid

(assuming that those other postulates are in fact consistent).

Because there is no precise hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and

related terms varies among writers. In this article, the two limiting lines are calledasymptotic and lines sharing a

common perpendicular are called ultraparallel; the simple wordparallel may apply to both.

A characteristic property of hyperbolic geometry is that the angles of a triangle add to less than a straight angle.

In the limit as the vertices go to infinity, there are even ideal hyperbolic triangles in which all three angles are

0°.

Contents

  [hide]

1 Non-intersecting lines

2 Triangles

3 Circles, disks, spheres and balls

4 History

5 Models of the hyperbolic plane

o 5.1 Connection between the models

6 Visualizing hyperbolic geometry

7 Homogeneous structure

8 Universal hyperbolic geometry

9 See also

10 Notes

11 References

12 External links

[edit]Non-intersecting lines

An interesting property of hyperbolic geometry follows from the occurrence of more than one line parallel

to R through a point P, not onR: there are two classes of non-intersecting lines. Let B be the point on R such

that the line PB is perpendicular to R. Consider the linex through P such that x does not intersect R, and the

angle θ between PB and x counterclockwise from PB is as small as possible; i.e., any smaller angle will force

the line to intersect R. This is called an asymptotic line in hyperbolic geometry. Symmetrically, the line ythat

forms the same angle θ between PB and itself but clockwise from PB will also be asymptotic. x and y are the

only two lines asymptotic to R through P. All other lines through P not intersecting R, with angles greater than θ

with PB, are called ultraparallel (or disjointly parallel) to R. Notice that since there are an infinite number of

possible angles between θ and 90°, and each one will determine two lines through P and disjointly parallel

to R, there exist an infinite number of ultraparallel lines.

Thus we have this modified form of the parallel postulate: In hyperbolic geometry, given any line R, and

point P not on R, there are exactly two lines through P which are asymptotic to R, and infinitely many lines

through P ultraparallel to R.

The differences between these types of lines can also be looked at in the following way: the distance between

asymptotic lines shrinks toward zero in one direction and grows without bound in the other; the distance

between ultraparallel lines (eventually) increases in both directions. The ultraparallel theorem states that there

is a unique line in the hyperbolic plane that is perpendicular to each of a given pair of ultraparallel lines.

In Euclidean geometry, the "angle of parallelism" is a constant; that is, any distance   between parallel

lines yields an angle of parallelism equal to 90°. In hyperbolic geometry, the angle of parallelism varies with the

Π(p) function. This function, described byNikolai Ivanovich Lobachevsky, produces a unique angle of

parallelism for each distance p =  . As the distance gets shorter, Π(p) approaches 90°, whereas with

increasing distance Π(p) approaches 0°. Thus, as distances get smaller, the hyperbolic plane behaves more

and more like Euclidean geometry. Indeed, on small scales compared to  , where K is the

(constant) Gaussian curvatureof the plane, an observer would have a hard time determining whether the

environment is Euclidean or hyperbolic.

[edit]Triangles

Distances in the hyperbolic plane can be measured in terms of a unit of length  , analogous to

the radius of the sphere in spherical geometry. Using this unit of length a theorem in hyperbolic geometry can

be stated which is analogous to the Pythagorean theorem. If a, b are the legs and c is the hypotenuse of a right

triangle all measured in this unit then:

The cosh function is a hyperbolic function which is an analog of the standard cosine function. All six of the

standard trigonometric functions have hyperbolic analogs. In trigonometric relations involving the sides and

angles of a hyperbolic triangle the hyperbolic functions are applied to the sides and the standard

trigonometric functions are applied to the angles. For example the law of sines for hyperbolic triangles is:

For more of these trigonometric relationships see hyperbolic triangles.

Unlike Euclidean triangles whose angles always add up to 180° or π radians the sum of the angles of

a hyperbolic triangle is always strictly less than 180°. The difference is sometimes referred to as

the defect. The area of a hyperbolic triangle is given by its defect multiplied by R²

where  . As a consequence all hyperbolic triangles have an area which is less than

R²π. The area of an ideal hyperbolic triangle is equal to this maximum.

As in spherical geometry the only similar triangles are congruent triangles.

[edit]Circles, disks, spheres and balls

In hyperbolic geometry the circumference of a circle of radius r is greater than 2πr. It is in fact equal to

The area of the enclosed disk is

The surface area of a sphere is

The volume of the enclosed ball is

For the measure of an n -1 sphere  in n dimensional space the corresponding

expression is

where the full n dimensional solid angle is

using   for the Gamma function.

The measure of the enclosed n ball is:

[edit]History

A number of geometers made attempts to prove the parallel

postulate by assuming its negation and trying to derive a

contradiction, including Proclus, Ibn al-Haytham (Alhacen), Omar

Khayyám,[1] Nasir al-Din al-Tusi, Witelo, Gersonides, Alfonso, and

later Giovanni Gerolamo Saccheri, John Wallis, Johann Heinrich

Lambert, and Legendre.[2] Their attempts failed, but their efforts gave

birth to hyperbolic geometry.

The theorems of Alhacen, Khayyam and al-Tusi on quadrilaterals,

including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–

Saccheri quadrilateral, were the first theorems on hyperbolic

geometry. Their works on hyperbolic geometry had a considerable

influence on its development among later European geometers,

including Witelo, Gersonides, Alfonso, John Wallis and Saccheri.[3]

In the 18th century, Johann Heinrich Lambert introduced

the hyperbolic functions and computed the area of a hyperbolic

triangle.

In the nineteenth century, hyperbolic geometry was extensively

explored by János Bolyai and Nikolai Ivanovich Lobachevsky, after

whom it sometimes is named. Lobachevsky published in 1830, while

Bolyai independently discovered it and published in 1832. Carl

Friedrich Gauss also studied hyperbolic geometry, describing in a

1824 letter to Taurinus that he had constructed it, but did not publish

his work. In 1868, Eugenio Beltrami provided models of it, and used

this to prove that hyperbolic geometry was consistent if Euclidean

geometry was.

The term "hyperbolic geometry" was introduced by Felix Klein in 1871.

[4]

For more history, see article on non-Euclidean geometry, and the

references Coxeter and Milnor.

[edit]Models of the hyperbolic plane

Poincaré disc model of great rhombitriheptagonal tiling

Lines through a given point and asymptotic to a given line, illustrated in the

Poincaré disc model

There are four models commonly used for hyperbolic geometry:

the Klein model, the Poincaré disc model, thePoincaré half-plane

model, and the Lorentz model, or hyperboloid model. These models

define a real hyperbolic space which satisfies the axioms of a

hyperbolic geometry. Despite their names, the first three mentioned

above were introduced as models of hyperbolic space by Beltrami,

not byPoincaré or Klein.

1. The Klein model, also known as the projective disc model

and Beltrami-Klein model, uses the interior of a circle for the

hyperbolic plane, and chords of the circle as lines.

This model has the advantage of simplicity, but the

disadvantage that angles in the hyperbolic plane are

distorted.

The distance in this model is the cross-ratio, which was

introduced by Arthur Cayley in projective geometry.

2. The Poincaré disc model, also known as the conformal disc

model, also employs the interior of a circle, but lines are

represented by arcs of circles that are orthogonal to the

boundary circle, plus diameters of the boundary circle.

3. The Poincaré half-plane model takes one-half of the

Euclidean plane, as determined by a Euclidean line B, to be

the hyperbolic plane (B itself is not included).

Hyperbolic lines are then either half-circles orthogonal

to B or rays perpendicular to B.

Both Poincaré models preserve hyperbolic angles, and

are thereby conformal. All isometries within these

models are therefore Möbius transformations.

The half-plane model is identical (at the limit) to the

Poincaré disc model at the edge of the disc

4. The Lorentz model or hyperboloid model employs a 2-

dimensional hyperboloid of revolution (of two sheets, but

using one) embedded in 3-dimensional Minkowski space.

This model is generally credited to Poincaré, but Reynolds

(see below) says thatWilhelm Killing and Karl

Weierstrass used this model from 1872.

This model has direct application to special relativity, as

Minkowski 3-space is a model for spacetime,

suppressing one spatial dimension. One can take the

hyperboloid to represent the events that various moving

observers, radiating outward in a spatial plane from a

single point, will reach in a fixed proper time. The

hyperbolic distance between two points on the

hyperboloid can then be identified with the

relative rapidity between the two corresponding

observers.

[edit]Connection between the models

Poincare disk, hemispherical and hyperboloid models are related by central projection from −1. Klein disk

model is vertical projection from hemispheric model. Poincare half-plane model here projected from the

hemisphere model by rays from left end of Poincare disk model.

The four models essentially describe the same structure. The

difference between them is that they represent

different coordinate charts laid down on the same metric space,

namely the hyperbolic space. The characteristic feature of the

hyperbolic space itself is that it has a constant negative scalar

curvature, which is indifferent to the coordinate chart used.

The geodesics are similarly invariant: that is, geodesics map to

geodesics under coordinate transformation. Hyperbolic geometry

generally is introduced in terms of the geodesics and their

intersections on the hyperbolic space.[5]

Once we choose a coordinate chart (one of the "models"), we

can always embed it in a Euclidean space of same dimension,

but the embedding is clearly not isometric (since the scalar

curvature of Euclidean space is 0). The hyperbolic space can be

represented by infinitely many different charts; but the

embeddings in Euclidean space due to these four specific charts

show some interesting characteristics.

Since the four models describe the same metric space, each can

be transformed into the other. See, for example, the Beltrami–

Klein model's relation to the hyperboloid model, the Beltrami–

Klein model's relation to the Poincaré disk model, and the

Poincaré disk model's relation to the hyperboloid model.

[edit]Visualizing hyperbolic geometry

M.C. Escher's Circle Limit III, 1959

A collection of crocheted hyperbolic planes, in imitation of a coral

reef, by the Institute For Figuring

A coral with similar geometry on the Great Barrier Reef

M. C. Escher's famous prints Circle Limit III and Circle Limit

IV illustrate the conformal disc model quite well. The white lines

in III are not quite geodesics (they are hypercycles), but are quite

close to them. It is also possible to see quite plainly the

negative curvature of the hyperbolic plane, through its effect on

the sum of angles in triangles and squares.

For example, in Circle Limit III every vertex belongs to three

triangles and three squares. In the Euclidean plane, their angles

would sum to 450°; i.e., a circle and a quarter. From this we see

that the sum of angles of a triangle in the hyperbolic plane must

be smaller than 180°. Another visible property is exponential

growth. In Circle Limit III, for example, one can see that the

number of fishes within a distance of n from the center rises

exponentially. The fishes have equal hyperbolic area, so the area

of a ball of radius n must rise exponentially in n.

There are several ways to physically realize a hyperbolic plane

(or approximation thereof). A particularly well-known paper model

based on the pseudosphere is due toWilliam Thurston. The art

of crochet has been used to demonstrate hyperbolic planes with

the first being made by Daina Taimina,[6] whose book Crocheting

Adventures with Hyperbolic Planes won the

2009 Bookseller/Diagram Prize for Oddest Title of the Year.[7] In

2000, Keith Henderson demonstrated a quick-to-make paper

model dubbed the "hyperbolic soccerball". Instructions on how to

make a hyperbolic quilt, designed byHelaman Ferguson,[8] has

been made available by Jeff Weeks.[9]

[edit]Homogeneous structure

Hyperbolic space of dimension n is a special case of a

Riemannian symmetric space of noncompact type, as it is

isomorphic to the quotient

The orthogonal group O(1,n) acts by norm-preserving

transformations on Minkowski space R1,n, and it acts

transitively on the two-sheet hyperboloid of norm 1 vectors.

Timelike lines (i.e., those with positive-norm tangents)

through the origin pass through antipodal points in the

hyperboloid, so the space of such lines yields a model of

hyperbolic n-space. The stabilizer of any particular line is

isomorphic to the product of the orthogonal groups O(n) and

O(1), where O(n) acts on the tangent space of a point in the

hyperboloid, and O(1) reflects the line through the origin.

Many of the elementary concepts in hyperbolic geometry can

be described in linear algebraic terms: geodesic paths are

described by intersections with planes through the origin,

dihedral angles between hyperplanes can be described by

inner products of normal vectors, and hyperbolic reflection

groups can be given explicit matrix realizations.

In small dimensions, there are exceptional isomorphisms of

Lie groups that yield additional ways to consider symmetries

of hyperbolic spaces. For example, in dimension 2, the

isomorphisms SO+(1,2) ≅ PSL(2,R) ≅ PSU(1,1) allow one to

interpret the upper half plane model as the quotient

SL(2,R)/SO(2) and the Poincaré disc model as the quotient

SU(1,1)/U(1). In both cases, the symmetry groups act by

fractional linear transformations, since both groups are the

orientation-preserving stabilizers in PGL(2,C) of the

respective subspaces of the Riemann sphere. The Cayley

transformation not only takes one model of the hyperbolic

plane to the other, but realizes the isomorphism of symmetry

groups as conjugation in a larger group. In dimension 3, the

fractional linear action of PGL(2,C) on the Riemann sphere is

identified with the action on the conformal boundary of

hyperbolic 3-space induced by the isomorphism O+(1,3) ≅

PGL(2,C). This allows one to study isometries of hyperbolic

3-space by considering spectral properties of representative

complex matrices. For example, parabolic transformations

are conjugate to rigid translations in the upper half-space

model, and they are exactly those transformations that can

be represented by unipotent upper triangular matrices.

[edit]Universal hyperbolic geometry

A recent approach (2009) has been outlined by N. J.

Wildberger, termed universal hyperbolic geometry,[10] based

on rational trigonometry, his reformulation of the main

metrical properties from Euclidean

geometry (quadrance and spread). These two measures,

which appear quite distinct in their Euclidean setting,

become completely dual when transferred to

a projective (hyperbolic) one. All 'points' and 'lines'

(projectively, lines and planes) become related simply as

'pole and polar', their correspondences being determined by

a 'null circle'. The duality in the metrical properties derived

('hyperbolic quadrance' and 'hyperbolic spread') reflect this

complete point-line correspondence from the non-metrical

point of view. Though naturally synthetic however, the

approach is developed purely algebraicallyand 'without

pictures'. As expected, hyperbolic analogues exist for the

five main laws of rational trigonometry.

In contrast to the usual models, universal hyperbolic

geometry does not require real numbers, calculus,

or transcendental functions. In turn an algebraic approach

allows any valid result obtained over one field not of

characteristic two (and not merely the usual infinite field of

the rational numbers) to in principle apply directly over any

other field.

(This 'field independence' of theorems is what qualifies use

of the term universal)

[edit]See also

Hyperbolic space

Elliptic geometry

Gyrovector space

Hjelmslev transformation

Horocycle

Hyperbolic 3-manifold

Hyperbolic manifold

Hyperbolic set

Hyperbolic tree

Kleinian group

Open universe

Poincaré metric

Pseudosphere

Saccheri quadrilateral

Spherical geometry

Systolic geometry

[edit]Notes

1. ̂  See for instance, "Omar Khayyam 1048-1131".

Retrieved 2008-01-05.

2. ̂  http://www.math.columbia.edu/~pinkham/teaching/

seminars/NonEuclidean.html

3. ̂  Boris A. Rosenfeld and Adolf P. Youschkevitch

(1996), "Geometry", in Roshdi Rashed,

ed., Encyclopedia of the History of Arabic Science,

Vol. 2, p. 447-494 [470], Routledge, London and New

York:

"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 this

postulate and the sum of the angles of a triangle and a

quadrangle. By their works on the theory of parallel

lines Arab mathematicians directly influenced the

relevant investigations of their European counterparts.

The first European attempt to prove the postulate on

parallel lines - made by Witelo, the Polish scientists of

the 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

both J. Wallis's and G. Saccheri's studies of the theory

of parallel lines."

4. ̂  F. Klein, Über die sogenannte Nicht-Euklidische,

Geometrie, Math. Ann. 4, 573-625 (cf. Ges. Math.

Abh. 1, 244-350).

5. ̂  Arlan Ramsay, Robert D. Richtmyer, Introduction to

Hyperbolic Geometry, Springer; 1 edition (December

16, 1995)

6. ̂  "Hyperbolic Space". The Institute for Figuring.

December 21, 2006. Retrieved January 15, 2007.

7. ̂  Bloxham, Andy (March 26, 2010). "Crocheting

Adventures with Hyperbolic Planes wins oddest book

title award". The Telegraph

8. ̂  "Helaman Ferguson, Hyperbolic Quilt".

9. ̂  "How to sew a Hyperbolic Blanket".

10. ̂  "Universal Hyperbolic Geometry I: Trigonometry".

[edit]References

Wikimedia Commons has

media related to: Hyperbolic

geometry

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,

SBN ISBN 978-3-03719-105-7, DOI 10.4171/105.

COXETER, H. S. M. , (1942) Non-Euclidean geometry,

University of Toronto Press, Toronto

Fenchel , Werner (1989). Elementary geometry in

hyperbolic space. De Gruyter Studies in

mathematics. 11. Berlin-New York: Walter de Gruyter &

Co..

Fenchel , Werner; Nielsen, Jakob; edited by Asmus L.

Schmidt (2003). Discontinuous groups of isometries in

the hyperbolic plane. De Gruyter Studies in

mathematics. 29. Berlin: Walter de Gruyter & Co..

LOBACHEVSKY, NIKOLAI I., (2010) Pangeometry, Edited

and translated by Athanase Papadopoulos, Heritage of

European Mathematics, Vol. 4. Zürich: European

Mathematical Society (EMS). xii, 310~p, ISBN 978-3-

03719-087-6/hbk

MILNOR, JOHN W. , (1982) Hyperbolic geometry: The first

150 years, Bull. Amer. Math. Soc. (N.S.) Volume 6,

Number 1, pp. 9–24.

REYNOLDS, WILLIAM F., (1993) Hyperbolic Geometry on

a Hyperboloid, American Mathematical

Monthly 100:442-455.

Stillwell, John  (1996). Sources of hyperbolic geometry.

History of Mathematics. 10. Providence, R.I.: American

Mathematical Society. ISBN 978-0-8218-0529-

9. MR 1402697

SAMUELS, DAVID., (March 2006) Knit Theory Discover

Magazine, volume 27, Number 3.

JAMES W. ANDERSON, Hyperbolic Geometry, Springer

2005, ISBN 1-85233-934-9

JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON,

and WALTER R. PARRY (1997) Hyperbolic Geometry,

MSRI Publications, volume 31.

[edit]External links

Java freeware for creating sketches in both the Poincaré

Disk and the Upper Half-Plane Models of Hyperbolic

Geometry University of New Mexico

"The Hyperbolic Geometry Song"  A short music video

about the basics of Hyperbolic Geometry available at

YouTube.

Hazewinkel, Michiel, ed. (2001), "Lobachevskii

geometry", Encyclopedia of

Mathematics, Springer, ISBN 978-1-55608-010-4

Weisstein, Eric W. , "Gauss-Bolyai-Lobachevsky Space"

from MathWorld.

Weisstein, Eric W. , "Hyperbolic Geometry"

from MathWorld.

More on hyperbolic geometry, including movies and

equations for conversion between the different

models University of Illinois at Urbana-Champaign

Hyperbolic Voronoi diagrams made easy, Frank Nielsen

Stothers, Wilson (2000). Hyperbolic

geometry. University of Glasgow, interactive

instructional website.

Universal Hyperbolic Geometry II: A pictorial overview

Universal Hyperbolic Geometry III: First Steps in

Projective Triangle Geometry

3: What is Non-Euclidean Geometry

1.1 Euclidean Geometry: The geometry with which we are most familiar is called Euclidean geometry. Euclidean geometry was named after Euclid, a Greek mathematician who lived in 300 BC. His book, called "The Elements", is a collection of axioms, theorems and proofs about squares, circles acute angles, isosceles triangles, and other such things. Most of the theorems which are taught in high schools today can be found in Euclid's 2000 year old book.

Euclidean geometry is of great practical value. It has been used by the ancient Greeks through modern society to design buildings, predict the location of moving objects and survey land. 

1.2 Non-Euclidean Geometry: non-Euclidean geometry is any geometry that is different from Euclidean geometry. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. The essential difference between

Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it. In spherical geometry there are no such lines. In hyperbolic geometry there are at least two distinct lines that pass through the point and are parallel to (in the same plane as and do not intersect) the given line. 

1.3 Spherical Geometry: Spherical geometry is a plane geometry on the surface of a sphere. In a plane geometry, the basic concepts are points and lines. In spherical geometry, points are defined in the usual way, but lines are defined such that the shortest distance between two points lies along them. Therefore, lines in spherical geometry are great circles. A great circle is the largest circle that can be drawn on a sphere. The longitude lines and the equator are great circles of the Earth. Latitude lines, except for the equator, are not great circles. Great circles are lines that divide a sphere into two equal hemispheres.

Spherical geometry is used by pilots and ship captains as they navigate around the globe. Working in spherical geometry has some non-intuitive results. For example, did you know that the shortest flying distance from Florida to the Philippine Islands is a path across Alaska? The Philippines are south of Florida - why is flyingnorth to Alaska a short-cut? The answer is that Florida, Alaska, and the Philippines are collinear locations in spherical geometry (they lie on a great circle). Another odd property of spherical geometry is that the sum of the angles of a triangle is always greater then 180°. Small triangles, like those drawn on a football field, have very, very close to 180°. Big triangles, however, (like the triangle with veracities: New York, L.A. and Tampa) have significantly more than 180°.

1.4 Hyperbolic Geometry: hyperbolic geometry is the geometry of which the NonEuclid software is a model. Hyperbolic geometry is a "curved" space, and plays an important role in Einstein's General theory of Relativity. hyperbolic geometry is also has many applications within the field of Topology.

Hyperbolic geometry shares many proofs and theorems with Euclidean geometry, and provides a novel and beautiful prospective from which to view those theorems. Hyperbolic geometry also has many differences from Euclidean geometry. The following sections discuss and explore hyperbolic geometry in some detail.

Hyperbolic Geometry

Figure 1

In the Fun Fact on Spherical Geometry, we saw an example of a space which is curved in such a way that the sum of angles in a triangle is greater than 180 degrees, where the sides of the triangle are "intrinsically" straight lines, or geodesics.

Is it also possible to have a space that "curves" in such a way that the sum of angles in a triangle is less than 180 degrees?

Yes! For instance, consider a saddle-shaped surface. A triangle that extends over the saddle of this surface (whose edges are geodesics) will have this property.

Another space with this property is something called the hyperbolic plane. This can be modeled by disc in which is "curved" in such a strange way that a bug on this disc would think that the "straight" lines are the pieces of circles or straight lines (viewed in planar geometry) that intersect the disc boundary at right angles. Any 3-sided figure using such lines will have angles in the corners that sum to less than 180 degrees!

Presentation Suggestions:Convince students of the triangle assertion by drawing a saddle-shaped surface and a triangle on it. Alternatively, you could show that the angles of a square do not add to 360 degrees. Follow by showing drawing the hyperbolic disc and explaining what the "straight lines" are. You can also construct and bring to class an approximate physical model of a hyperbolic plane; the references discuss ways to construct them.

The Math Behind the Fact:These spaces are examples of spaces with a kind of non-Euclidean geometry called hyperbolic geometry. Unlike planar geometry, the parallel postulate does not hold in hyperbolic geometry. Two lines are said to be parallel if they do not intersect. In Euclidean geometry, given a line L there is exactly one line through any given point P that is parallel to L (the parallel postulate). However in hyperbolic geometry,

there are infinitely many lines parallel to L passing through P.

Mathematicians sometimes work with strange geometries by defining them in terms of a Riemannian metric, which gives a local notion of how to measure "distance" and "angles" on an arbitrary set. You can learn more about such metrics by taking a first course on real analysis, then following with an advanced course in differential geometry.

How to Cite this Page: Su, Francis E., et al. "Hyperbolic Geometry." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Hyperbolic Geometry

The McDonnell Planetarium at the St. Louis Science Center.

Contents

 [hide]

1 Explorations

2 Hyperbolic Space

3 Models of Hyperbolic Space

o 3.1 Poincare Disk Model

o 3.2 Upper Half Space Model

o 3.3 Other Models

4 Polygons and Defect

o 4.1 Ideal Polygons

5 Hyperbolic Tessellations

o 5.1 Examples

o 5.2 Escher's Circle Limit III

o 5.3 Ideal Tessellations

6 Exercises

7 Relevant examples from Escher's work

8 Related Sites

9 Notes

Explorations

You may begin exploring hyperbolic geometry with the following explorations. We recommend

doing some or all of the basic explorations before reading the section. Gaining some intuition about

the nature of hyperbolic space before reading this section will be more effective in the long run.

Basic Explorations

Hyperbolic Paper Exploration

Escher's Circle Limit Exploration  This exploration is designed to help the student gain an

intuitive understanding of what hyperbolic geometry may look like. Escher's prints are nice

examples that illustrate what we would see when looking down on a hyperbolic universe.

Hyperbolic Geometry Exploration  NonEuclid allows us to draw acurate pictures of objects in

hyperbolic space.

Further explorations that help give a better understanding of hyperbolic geometry.

Hyperbolic Geometry II with NonEuclid Exploration

Hyperbolic Tessellations Exploration

Ideal Hyperbolic Tessellations Exploration

Hyperbolic Escher Exploration

Hyperbolic Space

We have seen two different geometries so far: Euclidean and spherical geometry. Geometry is

meant to describe the world around us, and the geometry then depends on some fundamental

properties of the world we are describing. Objects that live in a flat world are described by

Euclidean (or flat) geometry, while objects that live on a spherical world will need to be described

by spherical geometry.

M.C. Escher, Circle Limit IV (Heaven and Hell), 1960.

In two dimensions there is a third geometry. This geometry is called hyperbolic geometry. If

Euclidean geometry describes objects in a flat world or a plane, and spherical geometry describes

objects on the sphere, what world does hyperbolic geometry describe? Like spherical geometry,

which takes place on a sphere, hyperbolic geometry takes place on a curved two dimensional

surface calledhyperbolic space.

We will describe hyperbolic space in several different ways. In Escher's work, hyperbolic space is a

distorted disk. All of the angels in Circle Limit IV (Heaven and Hell) live in hyperbolic space, where

they are actually the same size, as do the devil figures. The image that Escher presents is a

distorted map of the hyperbolic world.

You can explore Escher's hyperbolic Circle Limit prints and get an introduction to hyperbolic

geometry in the Escher's Circle Limit Exploration

Models of Hyperbolic Space

On a sphere, a small neighborhood of a point looks like a cap. In hyperbolic space, every point

looks like a saddle:

A piece of a sphere A piece of hyperbolic space

Unfortunately, while you can piece caps together to make a sphere, piecing saddles together

quickly runs out of space. Try this yourself with the Hyperbolic Paper Exploration.

Hyperbolic paper

Hyperbolic paper is a floppy, saddle like object. Eventually, it contains too many triangles in too

small a space to continue any further, although most people and run out of patience before running

out of room. It's also possible to crochet models of hyperbolic space.

Because globes are unwieldy, navigators use flat maps of the spherical earth. Maps of the Earth

are necessarily distorted, for example Greenland appears extremely large on the

common Mercator map of the Earth, shown with red dots to indicate the distortion of area.

Mercator projection map with equal area dots.

Poincare Disk Model

Because models of hyperbolic space are unwieldy (not to mention infinite), we will do all of our

work with a map of hyperbolic space called the Poincaré disk. The Poincaré disk is the inside of a

circle (although the circle is not included) and is badly distorted near its edge.

Objects near the edge of the Poincaré disk are larger than they appear.

Hyperbolic man takes a walk

The picture shows a stick man as he walks towards the edge of the disk. He appears to shrink, as

does the distance he moves with each step. But this disk is a distorted map, and in the actual

hyperbolic space his steps are all the same length and he stays the same size. The man will never

reach the edge of the disk, because it is infinitely far away. The edge is drawn dashed because it is

not actually part of hyperbolic space.

The geodesics in hyperbolic space play the role of straight lines. Geodesics appear straight to an

inhabitant of hyperbolic space, and they are the shortest paths between points. In the Poincaré

disk model, geodesics appear curved. They are arcs of circles. Specifically:

Geodesics are arcs of circles which meet the edge of the disk at 90°.

Geodesics which pass through the center of the disk appear straight.

Some geodesics in the Poincaré disk

Practice drawing geodesics in the Poincaré disk with Hyperbolic Geometry Exploration.

Upper Half Space Model

Ideal hyperbolic triangle in the upper half space model. From a Roman mosaic, Real Alcazar, Cordoba, Spain

Another commonly used model for hyperbolic space in the upper half space model. In this model,

hyperbolic space is mapped to the upper half of the plane. The model includes all points (x,y)

where y>0. In other words, everything above the x-axis.

The geodesics in the upper half space model are lines perpendicular to the x-axis and semi-circles

perpendicular to the x-axis. The image of the mosaic to the right shows three geodesics. The

lighter semi-circles at the bottom create 2 geodesics, and the dark semi-circle in the background

creates the third geodesic. Geodesics measure shortest distance and play the role of straight lines

in Euclidean geometry, hence these three geodesics form an (ideal) triangle. Note how the

geometric figures have rather small angle measures just as in the Poincaré disk model discussed

above.

Non-Euclid also allows the user to experiment with this model of hyperbolic space.

Other Models

There are several other models that can be used to represent hyperbolic space.

The Pseudosphere is a model that accurately shows how hyperbolic space curves, but only

models a portion of the whole space.

The Beltrami-Klein model represents hyperbolic space as the interior of a disk, just as the Poincare

disk model, but it chooses to distort angles rather than geodesics. So the geodesics actually

appear as straight lines, but the angles between them are no longer correctly shown.

Plenty of other models exist, just like there are many ways to make maps of the spherical geometry

of the Earth, but the Poincare disk model is the one that Escher uses exclusively.

Polygons and Defect

Polygon

A polygon in hyperbolic geometry is a sequence of points and geodesic segments joining

those points. The geodesic segments are called the sides of the polygon.

A triangle in hyperbolic geometry is a polygon with three sides, a quadrilateral is a polygon

with four sides, and so on, as in Euclidean geometry. Here are some triangles in hyperbolic

space:

From these pictures, you can see that:

The sum of the angles in any hyperbolic triangle is less than 180°.

Defect

The defect of a hyperbolic triangle is 180° – (angle sum of the triangle).

By cutting other polygons into triangles, we see that a hyperbolic polygon has angle

sum less than that of the corresponding Euclidean polygon.

Define the defect for a hyperbolic polygon with n sides to

be  .

Putting this together with the defect in spherical geometry:

The defect of a polygon is the difference between its angle sum and the angle sum for a

Euclidean polygon with the same number of sides.

This statement works in spherical and hyperbolic geometry, for polygons with any

number of sides. It even works for biangles, because a biangle in Euclidean geometry

must have two 0° angles. The area of a hyperbolic polygon is still proportional to its

defect:

Area of a hyperbolic polygon =  .

This equality is a special case of the Gauss-Bonnet theorem.

In spherical geometry, we had a formula relating the defect of a polygon to the fraction

of the sphere's area covered by the polygon. For a sphere of radius 1, the total surface

area of the sphere is 4π, and so the area of a polygon is   which (after a

little simplification) is exactly the same formula as in hyperbolic space!

Ideal Polygons

Ideal Triangle

An ideal triangle consists of three geodesics that touch at the boundary of the Poincaré

disk.

The three geodesics are called the sides of the ideal triangle. Since the boundary

of the disk isn’t part of hyperbolic space, the sides of an ideal triangle are infintely

long and never actually meet. However, the do get closer together as they head

towards the edge. The three points on the boundary are called the ideal

vertices of the ideal triangle, and play a similar role as the vertices of an ordinary

triangle. Since the sides are all perpendicular to the Poincaré disk boundary, they

make an angle of 0° with each other.

You can create other ideal polygons in a similar manner:

Ideal triangles and an ideal hexagon

The area formula implies that any ideal triangle has area π, because the angle

sum is zero and its defect is 180°. Both ideal triangles shown above have the

same area even though the distortion of the Poincaré disk makes one look much

smaller than the other. The ideal hexagon shown has angle sum zero, so it’s

defect is 720° and its area is 4π.

Hyperbolic Tessellations

Hyperbolic Tessellations Exploration

A hyperbolic tessellation is a covering of hyperbolic space by tiles, with no

overlapping tiles and no gaps. Escher’s Circle Limit prints are examples of

hyperbolic tessellations. Like his other tessellations, Escher began with a

geometric tessellation by polygons and worked from there.

Consider Escher’s Circle Limit I, shown with geometric scaffolding consisting of

geodesics drawn in red:

The spines of the fish, emphasized with red lines, form a tessellation of hyperbolic

space by quadrilaterals. Since these quadrilaterals meet four or six at a vertex,

they have corner angles 90°-60°-90°-60°. They are not regular polygons because

regular polygons have all sides and all angles equal.

There are plenty of other tessellations of hyperbolic space, including regular

tessellations. In fact, there are infinitely many regular tessellations of hyperbolic

space. This stands in sharp contrast to Euclidean space, which has only three, and

to spherical geometry, where there are only five non-degenerate possibilities

(corresponding to the Platonic solids). Hyperbolic space is easy to tessellate

because the corner angles of polygons want to be small, and small angles fit nicely

around a vertex.

As in the other two geometries, we describe regular tessellations by the number of

sides in each polygon and the number of polygons that meet at a vertex.

Schläfli Symbol

The Schläfli symbol {n,k} describes the regular tessellation by n-gons where k meet at

each vertex.

Examples

This tessellation has seven triangles at each vertex, and so its Schläfli

symbol is {3,7}.

The triangles have

360°/7 ≈51.4°

angles and angle sums of approximately

3×51.4°=154.3°

This gives a defect of approximately 180°–154.3° = 25.7° and an area of

approximately 0.45.

This tessellation has four pentagons at each vertex, and so its Schläfli

symbol is {5,4}.

Each pentagon has five 90° angles.

The angle sum for each pentagon is 450°.

Since Euclidean pentagons have angle sum 540°, these pentagons have a

defect of 540°-450° = 90° and an area of  .

This tessellation has six quadrilaterals at each vertex, and so its Schläfli

symbol is {4,6}.

Each quadrilateral has four 60° angles.

The angle sum for each quadrilateral is 240°.

Since Euclidean quadrilaterals have angle sum 360°, these quadrilaterals

have a defect of 360°-240° = 120° and an area of  .

Escher's Circle Limit III

M.C. Escher, Circle Limit III, 1959.

Escher's Circle Limit III is his most accomplished print based on hyperbolic

space. It is also his most subtle.

Looking at the white spines of the fish, it appears to be a tessellation of

hyperbolic space by triangles and squares, with three triangles and three

squares coming together at each vertex. This would mean that the corner

angles of both polygons are 60° each, but this gives the triangle an angle

sum of 180°, which cannot happen in hyperbolic space!

However, Circle Limit III clearly has the flavor of a hyperbolic

tessellation. H.S.M Coxeter, a mathematician and friend of Escher’s

analyzed the print[1], and discovered that the white circular arcs along the

spines meet the boundary of the disk at 80° each, rather than the 90°

required to be geodesic. Since the white lines are not geodesics, neither the

“triangle” or the “square” is really a polygon at all! Even Escher does not

seem to have realized this, although most probably he would not have cared,

as he was very satisfied with the print and the suggestion of infinity it

presents.

A natural underlying tessellation for Circle Limit III is the {8,3} tessellation by

octagons:

A re-creation of Circle Limit III [2]

A geodesic and two (non-geodesic) curves at equal distance from the geodesic.

The image on the right shows a hyperbolic geodesic that runs through the

midpoints of the sides of the octagons, surrounded by two curves (not

geodesics) at a fixed distance from the geodesic. The rightmost of these

curves forms the spines of one row of fish in Circle Limit III.

Ideal Tessellations

A hyperbolic tessellation by ideal triangles.

Drawing hyperbolic tessellations by hand is quite difficult. However,

tessellations by ideal polygons are somewhat easier to work with. The

advantage of an ideal tessellation is that all the vertices lie on the boundary

(or, more precisely, approach the boundary) of the Poincaré disk. Since the

tiles in ideal tessellations are not polygons, but ideal polygons, they do not

have Schläfli symbols. However, they do fit into the scheme of regular

tessellations, with the odd feature that infinitely many tiles now meet at an

ideal vertex.

Learn more about these in the Ideal Hyperbolic Tessellations Exploration.

Exercises

Hyperbolic Geometry Exercises

Relevant examples from Escher's work

Circle Limit I

Circle Limit II

Circle Limit III

Circle Limit IV (Heaven and Hell)

Related Sites

NonEuclid : A java applet for working with the Poincaré disk, by Joel

Castellanos.

Hyperbolic Geometry Applet : by Paul Garrett

Circull : An iPad/iPhone app to play a game with hyperbolic tessellations,

by Anthony Thibault.

Online Exhibit: Hyperbolic Space  by The Institute For Figuring.

CurvedSpaces  A non-Euclidean space flight simulator by Jeff Weeks.

Doug Dunham  has a collection of articles on hyperbolic tessellations and

Escher's Circle Limit series.

Hyperbolic Tessellations , by David Joyce.

Hyperbolic Tessellations Applet  by Don Hatch.

Capturing Infinity , the Circle Limit Series of M.C. Escher, by T. Wieting,

Reed Magazine 2010.

Notes

1. ↑  Coxeter, H.S.M. "The Non-Euclidean Symmetry of Escher's

Picture 'Circle Limit III'". Leonardo Vol. 12 No. 1, 1979, pg 19-25.

2. ↑  Dunham, D. "More 'Circle Limit III' patterns". Proceedings of

Bridges 2006 pg 451.