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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 Introdtion to Hyperbolic Geometry Let's recall basic high school geometry. Kevin P. Knudson University of Florida A Gentle Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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, nonEuclidean geometries were born. Kevin P. Knudson University of Florida A Gentle Introdtion to Hyperbolic Geometry So how would you change the parallel postulate? Two possibilities: Kevin P. Knudson University of Florida A Gentle Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 twodimensional 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 Introdtion 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 Introdtion 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 Introdtion to Hyperbolic Geometry What about the other casewhere more than one line can exist? Kevin P. Knudson University of Florida A Gentle Introdtion to Hyperbolic Geometry What about the other casewhere more than one line can exist? This leads to hyperbolic geometry, and examples exist in nature. Kevin P. Knudson University of Florida A Gentle Introdtion to Hyperbolic Geometry Coral reefs: Kevin P. Knudson University of Florida A Gentle Introdtion to Hyperbolic Geornetry Lettuce: Kevin P. Knudson University of Florida A Gentle Introdtion to Hyperbolic Geornetry Pringles: Kevin P. Knudson University of Florida A Gentle Introdtion to Hyperbolic Geornetry The Pringle is a realization of a hyperbolic paraboloid: Kevin P. Knudson University of Florida A Gentle Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion 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 Introdtion to Hyperbolic Geometry Kevin P. Knudson University of Florida A Gentle Introdtion 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 Introdtion 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 Introdtion to Hyperbolic Geometry Fin Kevin P. Knudson University of Florida A Gentle Introdtion to Hyperbolic Geornetry 
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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, nonEuclidean geometrieswereborn. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry PAGE 7 Forcenturies,mathematiciansattemptedtoprovethattheParallel Postulatefollowedfromtheotherfourpostulates,butwereunabletodo so. Mathematiciansbeingmathematicians,theybegantowonderwhatwould happeniftheytriedtodropthepostulate,orreplaceitwithadierent version. Hence, nonEuclidean geometrieswereborn. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry PAGE 8 Forcenturies,mathematiciansattemptedtoprovethattheParallel Postulatefollowedfromtheotherfourpostulates,butwereunabletodo so. Mathematiciansbeingmathematicians,theybegantowonderwhatwould happeniftheytriedtodropthepostulate,orreplaceitwithadierent version. Hence, nonEuclidean 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 Intheusualtwodimensionalplane,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 Intheusualtwodimensionalplane,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 Intheusualtwodimensionalplane,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 Whatabouttheothercase{wheremorethanonelinecanexist? Thisleadsto hyperbolicgeometry ,andexamplesexistinnature. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry PAGE 20 Whatabouttheothercase{wheremorethanonelinecanexist? Thisleadsto hyperbolicgeometry ,andexamplesexistinnature. KevinP.Knudson UniversityofFlorida AGentleIntroductiontoHyperbolicGeometry 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 