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# A gentle introduction to hyperbolic geometry

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A gentle introduction to hyperbolic geometry
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Knudson, Kevin
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Presented April 12, 2011 at the Gainesville Coral Reef Exhibit Opening Celebration at Marston Science Library.

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Full Text

A Gentle Introduction to Hyperbolic Geometry

Kevin P. Knudson

Director of the Honors Program and Professor of Mathematics
University of Florida
kknudson@honors.ufl.edu
http://www.math.ufl.edu/~kknudson/

April 12, 2011

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Let's recall basic high school geometry.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Let's recall basic high school geometry.

Euclid's postulates form the basis for classical plane geometry. There is
one that stands out though:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Let's recall basic high school geometry.

Euclid's postulates form the basis for classical plane geometry. There is
one that stands out though:

The Parallel Postulate For any given line f and a point P not on f, there
is exactly one line through P that does not intersect .

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Here's how we usually look at it:

line 1 line 2

a b

If: a + b = 1800
Then: line 1 and line 2 are parallel

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

For centuries, mathematicians attempted to prove that the Parallel
Postulate followed from the other four postulates, but were unable to do
so.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

For centuries, mathematicians attempted to prove that the Parallel
Postulate followed from the other four postulates, but were unable to do
so.

Mathematicians being mathematicians, they began to wonder what would
happen if they tried to drop the postulate, or replace it with a different
version.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

For centuries, mathematicians attempted to prove that the Parallel
Postulate followed from the other four postulates, but were unable to do
so.

Mathematicians being mathematicians, they began to wonder what would
happen if they tried to drop the postulate, or replace it with a different
version.

Hence, non-Euclidean geometries were born.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

So how would you change the parallel postulate? Two possibilities:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

So how would you change the parallel postulate? Two possibilities:

1. Given a line f and a point P not on f, there are no lines through P
that do not intersect .

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

So how would you change the parallel postulate? Two possibilities:

1. Given a line f and a point P not on f, there are no lines through P
that do not intersect .
2. Given f and P, there exist multiple lines through P that do not
intersect .

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

So how would you change the parallel postulate? Two possibilities:

1. Given a line f and a point P not on f, there are no lines through P
that do not intersect .
2. Given f and P, there exist multiple lines through P that do not
intersect .

Of course, this means that we have to decide what a "line" is.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

It gets pounded into us at an early age that the shortest path between two
points in the plane is a straight line, and that any two points determine a
unique line. Well, that's how mathematicians define it:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

It gets pounded into us at an early age that the shortest path between two
points in the plane is a straight line, and that any two points determine a
unique line. Well, that's how mathematicians define it:

Given two points P and Q in some space, a line joining them is the
shortest path in the space from P to Q.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

It gets pounded into us at an early age that the shortest path between two
points in the plane is a straight line, and that any two points determine a
unique line. Well, that's how mathematicians define it:

Given two points P and Q in some space, a line joining them is the
shortest path in the space from P to Q.

In the usual two-dimensional plane, this is exactly what we think of, but in
other contexts it might be something else.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Example:

On the surface of a sphere (like our planet), the shortest path between two
points isn't a straight line, but rather an arc of a longitude through the
two points.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

Example:

On the surface of a sphere (like our planet), the shortest path between two
points isn't a straight line, but rather an arc of a longitude through the
two points.

This is an example of elliptic or spherical geometry. In this case, every
line through a point not on a given line intersects the line.

Notice also that the sum of the angles of a triangle add up to more than
1800 in this case. Since we are so small relative to the size of the earth, we
don't really notice this, and we generally observe that the shortest distance
between points is a straight line in the usual sense.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

What about the other case-where more than one line can exist?

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

What about the other case-where more than one line can exist?

This leads to hyperbolic geometry, and examples exist in nature.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Coral reefs:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

Lettuce:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

Pringles:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

The Pringle is a realization of a hyperbolic paraboloid:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

The Pringle is a realization of a hyperbolic paraboloid:

Note that if we draw a triangle on this surface, the angles add up to less
than 180.

The most popular model of the hyperbolic plane is the Poincar6 disc
model. To build it we begin with the interior of the unit circle and declare
that the following paths are straight lines:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

The most popular model of the hyperbolic plane is the Poincar6 disc
model. To build it we begin with the interior of the unit circle and declare
that the following paths are straight lines:

1. diameters of the circle

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

The most popular model of the hyperbolic plane is the Poincar6 disc
model. To build it we begin with the interior of the unit circle and declare
that the following paths are straight lines:

1. diameters of the circle
2. circles perpendicular to the boundary circle

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

In this geometry, it is possible for there to be infinitely many lines passing
through a given point "parallel" to a given line:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

In this geometry, it is possible for there to be infinitely many lines passing
through a given point "parallel" to a given line:

,, I

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Once you define the notion of "line" you then have a notion of distance
(the length of the line between two points). And once you have distance
you have area. Let's look at this tiling of the hyperbolic plane:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

Once you define the notion of "line" you then have a notion of distance
(the length of the line between two points). And once you have distance
you have area. Let's look at this tiling of the hyperbolic plane:

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

Every red region has the same area in hyperbolic space. Note that they
look "smaller" as you go out towards the boundary circle. What this
means is that the boundary circle is infinitely far away in this space.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Every red region has the same area in hyperbolic space. Note that they
look "smaller" as you go out towards the boundary circle. What this
means is that the boundary circle is infinitely far away in this space.

Note also that the number of red (or green or blue) regions increases
exponentially as you head toward the boundary circle.

This model of hyperbolic space is most famous for inspiring the Dutch
artist M. C. Escher. Here are two examples of wood cuts he produced
from this theme.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

This model of hyperbolic space is most famous for inspiring the Dutch
artist M. C. Escher. Here are two examples of wood cuts he produced
from this theme.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

It's this notion that the regions really all have the same area that creates
the unusual structures we observe in lettuce and coral. If we try to embed
this geometry into ordinary Euclidean space, we run into trouble. So the
space is forced to curve to preserve the areas and parallel lines.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

It's this notion that the regions really all have the same area that creates
the unusual structures we observe in lettuce and coral. If we try to embed
this geometry into ordinary Euclidean space, we run into trouble. So the
space is forced to curve to preserve the areas and parallel lines.

But I'll leave it to the biologists to explain why this is advantageous for
the organisms.

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geometry

Fin

Kevin P. Knudson University of Florida
A Gentle Introd-tion to Hyperbolic Geornetry

Full Text

PAGE 1

AGentleIntroductiontoHyperbolicGeometry KevinP.Knudson DirectoroftheHonorsProgramandProfessorofMathematics UniversityofFlorida kknudson@honors.ufl.edu http://www.math.ufl.edu/ kknudson/ April12,2011 KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 2

Let'srecallbasichighschoolgeometry. Euclid'spostulatesformthebasisforclassicalplanegeometry.Thereis onethatstandsoutthough: TheParallelPostulate Foranygivenline ` andapoint P noton ` ,there isexactlyonelinethrough P thatdoesnotintersect ` KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 3

Let'srecallbasichighschoolgeometry. Euclid'spostulatesformthebasisforclassicalplanegeometry.Thereis onethatstandsoutthough: TheParallelPostulate Foranygivenline ` andapoint P noton ` ,there isexactlyonelinethrough P thatdoesnotintersect ` KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 4

Let'srecallbasichighschoolgeometry. Euclid'spostulatesformthebasisforclassicalplanegeometry.Thereis onethatstandsoutthough: TheParallelPostulate Foranygivenline ` andapoint P noton ` ,there isexactlyonelinethrough P thatdoesnotintersect ` KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 5

Here'showweusuallylookatit: KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 6

Forcenturies,mathematiciansattemptedtoprovethattheParallel Postulatefollowedfromtheotherfourpostulates,butwereunabletodo so. Mathematiciansbeingmathematicians,theybegantowonderwhatwould happeniftheytriedtodropthepostulate,orreplaceitwithadierent version. Hence, non-Euclidean geometrieswereborn. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 7

Forcenturies,mathematiciansattemptedtoprovethattheParallel Postulatefollowedfromtheotherfourpostulates,butwereunabletodo so. Mathematiciansbeingmathematicians,theybegantowonderwhatwould happeniftheytriedtodropthepostulate,orreplaceitwithadierent version. Hence, non-Euclidean geometrieswereborn. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 8

Forcenturies,mathematiciansattemptedtoprovethattheParallel Postulatefollowedfromtheotherfourpostulates,butwereunabletodo so. Mathematiciansbeingmathematicians,theybegantowonderwhatwould happeniftheytriedtodropthepostulate,orreplaceitwithadierent version. Hence, non-Euclidean geometrieswereborn. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 9

Sohowwouldyouchangetheparallelpostulate?Twopossibilities: 1.Givenaline ` andapoint P noton ` ,therearenolinesthrough P thatdonotintersect ` 2.Given ` and P ,thereexistmultiplelinesthrough P thatdonot intersect ` Ofcourse,thismeansthatwehavetodecidewhataline"is. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 10

Sohowwouldyouchangetheparallelpostulate?Twopossibilities: 1.Givenaline ` andapoint P noton ` ,therearenolinesthrough P thatdonotintersect ` 2.Given ` and P ,thereexistmultiplelinesthrough P thatdonot intersect ` Ofcourse,thismeansthatwehavetodecidewhataline"is. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 11

Sohowwouldyouchangetheparallelpostulate?Twopossibilities: 1.Givenaline ` andapoint P noton ` ,therearenolinesthrough P thatdonotintersect ` 2.Given ` and P ,thereexistmultiplelinesthrough P thatdonot intersect ` Ofcourse,thismeansthatwehavetodecidewhataline"is. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 12

Sohowwouldyouchangetheparallelpostulate?Twopossibilities: 1.Givenaline ` andapoint P noton ` ,therearenolinesthrough P thatdonotintersect ` 2.Given ` and P ,thereexistmultiplelinesthrough P thatdonot intersect ` Ofcourse,thismeansthatwehavetodecidewhataline"is. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 13

Itgetspoundedintousatanearlyagethattheshortestpathbetweentwo pointsintheplaneisastraightline,andthatanytwopointsdeterminea uniqueline.Well,that'showmathematiciansdeneit: Giventwopoints P and Q insomespace,a line joiningthemisthe shortestpathinthespacefrom P to Q Intheusualtwo-dimensionalplane,thisisexactlywhatwethinkof,butin othercontextsitmightbesomethingelse. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 14

Itgetspoundedintousatanearlyagethattheshortestpathbetweentwo pointsintheplaneisastraightline,andthatanytwopointsdeterminea uniqueline.Well,that'showmathematiciansdeneit: Giventwopoints P and Q insomespace,a line joiningthemisthe shortestpathinthespacefrom P to Q Intheusualtwo-dimensionalplane,thisisexactlywhatwethinkof,butin othercontextsitmightbesomethingelse. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 15

Itgetspoundedintousatanearlyagethattheshortestpathbetweentwo pointsintheplaneisastraightline,andthatanytwopointsdeterminea uniqueline.Well,that'showmathematiciansdeneit: Giventwopoints P and Q insomespace,a line joiningthemisthe shortestpathinthespacefrom P to Q Intheusualtwo-dimensionalplane,thisisexactlywhatwethinkof,butin othercontextsitmightbesomethingelse. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 16

Example: Onthesurfaceofaspherelikeourplanet,theshortestpathbetweentwo pointsisn'tastraightline,butratheranarcofalongitudethroughthe twopoints. Thisisanexampleof elliptic or spherical geometry.Inthiscase,every linethroughapointnotonagivenlineintersectstheline. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 17

Example: Onthesurfaceofaspherelikeourplanet,theshortestpathbetweentwo pointsisn'tastraightline,butratheranarcofalongitudethroughthe twopoints. Thisisanexampleof elliptic or spherical geometry.Inthiscase,every linethroughapointnotonagivenlineintersectstheline. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 18

Noticealsothatthesumoftheanglesofatriangleadduptomorethan 180 inthiscase.Sincewearesosmallrelativetothesizeoftheearth,we don'treallynoticethis,andwegenerallyobservethattheshortestdistance betweenpointsisastraightlineintheusualsense. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 19

PAGE 20

PAGE 21

Coralreefs: KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 22

Lettuce: KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 23

Pringles: KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 24

ThePringleisarealizationofa hyperbolicparaboloid : Notethatifwedrawatriangleonthissurface,theanglesadduptoless than180 KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 25

ThePringleisarealizationofa hyperbolicparaboloid : Notethatifwedrawatriangleonthissurface,theanglesadduptoless than180 KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 26

Themostpopularmodelofthehyperbolicplaneisthe Poincaredisc model .Tobuilditwebeginwiththeinterioroftheunitcircleanddeclare thatthefollowingpathsarestraightlines: 1.diametersofthecircle 2.circlesperpendiculartotheboundarycircle KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 27

Themostpopularmodelofthehyperbolicplaneisthe Poincaredisc model .Tobuilditwebeginwiththeinterioroftheunitcircleanddeclare thatthefollowingpathsarestraightlines: 1.diametersofthecircle 2.circlesperpendiculartotheboundarycircle KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 28

Themostpopularmodelofthehyperbolicplaneisthe Poincaredisc model .Tobuilditwebeginwiththeinterioroftheunitcircleanddeclare thatthefollowingpathsarestraightlines: 1.diametersofthecircle 2.circlesperpendiculartotheboundarycircle KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 29

Inthisgeometry,itispossiblefortheretobeinnitelymanylinespassing throughagivenpointparallel"toagivenline: KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 30

Inthisgeometry,itispossiblefortheretobeinnitelymanylinespassing throughagivenpointparallel"toagivenline: KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 31

Onceyoudenethenotionofline"youthenhaveanotionofdistance thelengthofthelinebetweentwopoints.Andonceyouhavedistance youhavearea.Let'slookatthistilingofthehyperbolicplane: KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 32

Onceyoudenethenotionofline"youthenhaveanotionofdistance thelengthofthelinebetweentwopoints.Andonceyouhavedistance youhavearea.Let'slookatthistilingofthehyperbolicplane: KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 33

Everyredregionhasthesameareainhyperbolicspace.Notethatthey looksmaller"asyougoouttowardstheboundarycircle.Whatthis meansisthattheboundarycircleisinnitelyfarawayinthisspace. Notealsothatthenumberofredorgreeenorblueregionsincreases exponentiallyasyouheadtowardtheboundarycircle. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 34

Everyredregionhasthesameareainhyperbolicspace.Notethatthey looksmaller"asyougoouttowardstheboundarycircle.Whatthis meansisthattheboundarycircleisinnitelyfarawayinthisspace. Notealsothatthenumberofredorgreeenorblueregionsincreases exponentiallyasyouheadtowardtheboundarycircle. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 35

ThismodelofhyperbolicspaceismostfamousforinspiringtheDutch artistM.C.Escher.Herearetwoexamplesofwoodcutsheproduced fromthistheme. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 36

ThismodelofhyperbolicspaceismostfamousforinspiringtheDutch artistM.C.Escher.Herearetwoexamplesofwoodcutsheproduced fromthistheme. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 37

KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 38

It'sthisnotionthattheregionsreallyallhavethesameareathatcreates theunusualstructuresweobserveinlettuceandcoral.Ifwetrytoembed thisgeometryintoordinaryEuclideanspace,werunintotrouble.Sothe spaceisforcedtocurvetopreservetheareasandparallellines. ButI'llleaveittothebiologiststoexplainwhythisisadvantageousfor theorganisms. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 39

It'sthisnotionthattheregionsreallyallhavethesameareathatcreates theunusualstructuresweobserveinlettuceandcoral.Ifwetrytoembed thisgeometryintoordinaryEuclideanspace,werunintotrouble.Sothe spaceisforcedtocurvetopreservetheareasandparallellines. ButI'llleaveittothebiologiststoexplainwhythisisadvantageousfor theorganisms. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry

PAGE 40

Fin KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry