Thinking Geometrically: A Survey of Geometries

584 Pages·2015·5.64 MB·English

by  Thomas Q. Sibley

Thinking Geometrically
A Survey of Geometries
c 2015by
⃝
TheMathematicalAssociationofAmerica(Incorporated)
LibraryofCongressControlNumber:2015936100
PrintISBN:978-1-93951-208-6
ElectronicISBN:978-1-61444-619-4
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10 9 8 7 6 5 4 3 2 1
Thinking Geometrically
A Survey of Geometries
Thomas Q. Sibley
St. John’s University
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Contents
Preface xv
1 EuclideanGeometry 1
1.1 OverviewandHistory........................................................................... 1
1.1.1 ThePythagoreansandZeno.......................................................... 2
1.1.2 PlatoandAristotle...................................................................... 4
1.1.3 ExercisesforSection1.1.............................................................. 5
1.2 Euclid’sApproachtoGeometryI:CongruenceandConstructions................... 10
1.2.1 Congruence............................................................................... 11
1.2.2 Constructions............................................................................ 12
1.2.3 EqualityofMeasure.................................................................... 15
1.2.4 TheGreekLegacy....................................................................... 16
1.2.5 ExercisesforSection1.2.............................................................. 17
1.2.6 Archimedes............................................................................... 25
1.3 Euclid’sApproachII:ParallelLines......................................................... 26
1.3.1 ExercisesforSection1.3.............................................................. 30
1.4 SimilarFigures.................................................................................... 33
1.4.1 ExercisesforSection1.4.............................................................. 37
1.5 Three-DimensionalGeometry................................................................. 42
1.5.1 Polyhedra.................................................................................. 42
1.5.2 GeodesicDomes........................................................................ 46
1.5.3 TheGeometryoftheSphere.......................................................... 48
1.5.4 ExercisesforSection1.5.............................................................. 51
1.5.5 BuckminsterFuller..................................................................... 59
1.5.6 ProjectsforChapter1.................................................................. 59
1.5.7 SuggestedReadings.................................................................... 65
2 AxiomaticSystems 67
2.1 FromEuclidtoModernAxiomatics......................................................... 67
2.1.1 OverviewandHistory.................................................................. 67
2.1.2 AxiomaticSystems..................................................................... 68
2.1.3 ASimplifiedAxiomaticSystem..................................................... 71
2.1.4 ExercisesforSection2.1.............................................................. 73
vii
viii Contents
2.2 AxiomaticSystemsforEuclideanGeometry.............................................. 75
2.2.1 SMSGPostulates........................................................................ 76
2.2.2 Hilbert’sAxioms........................................................................ 77
2.2.3 ExercisesforSection2.2.............................................................. 78
2.2.4 DavidHilbert............................................................................. 81
2.3 ModelsandMetamathematics................................................................. 82
2.3.1 ExercisesforSection2.3.............................................................. 88
2.3.2 KurtGo¨del................................................................................ 94
2.3.3 ProjectsforChapter2.................................................................. 94
2.3.4 SuggestedReadings.................................................................... 96
3 AnalyticGeometry 97
3.1 OverviewandHistory........................................................................... 98
3.1.1 TheAnalyticModel.................................................................... 98
3.1.2 ExercisesforSection3.1.............................................................. 100
3.1.3 Rene´ Descartes.......................................................................... 104
3.2 ConicsandLocusProblems................................................................... 104
3.2.1 ExercisesforSection3.2.............................................................. 110
3.2.2 PierredeFermat......................................................................... 114
3.3 FurtherTopicsinAnalyticGeometry........................................................ 114
3.3.1 ParametricEquations................................................................... 114
3.3.2 PolarCoordinates....................................................................... 116
3.3.3 BarycentricCoordinates............................................................... 118
3.3.4 OtherAnalyticGeometries............................................................ 120
3.3.5 ExercisesforSection3.3.............................................................. 121
3.4 CurvesinComputer-AidedDesign.......................................................... 126
3.4.1 ExercisesforSection3.4.............................................................. 131
3.5 HigherDimensionalAnalyticGeometry.................................................... 133
3.5.1 AnalyticGeometryinRn.............................................................. 133
3.5.2 RegularPolytopes....................................................................... 137
3.5.3 ExercisesforSection3.5.............................................................. 140
3.5.4 GaspardMonge.......................................................................... 145
3.5.5 ProjectsforChapter3.................................................................. 145
3.5.6 SuggestedReadings.................................................................... 149
4 Non-EuclideanGeometries 151
4.1 OverviewandHistory........................................................................... 152
4.1.1 TheAdventofHyperbolicGeometry............................................... 153
4.1.2 ModelsofHyperbolicGeometry.................................................... 155
4.1.3 ExercisesforSection4.1.............................................................. 158
4.1.4 CarlFriedrichGauss.................................................................... 160
4.2 PropertiesofLinesandOmegaTriangles.................................................. 161
4.2.1 OmegaTriangles........................................................................ 164
4.2.2 ExercisesforSection4.2.............................................................. 167
4.2.3 NikolaiLobachevskyandJa´nosBolyai............................................ 169
Contents ix
4.3 SaccheriQuadrilateralsandTriangles....................................................... 169
4.3.1 ExercisesforSection4.3.............................................................. 173
4.3.2 OmarKhayyam.......................................................................... 174
4.3.3 GiovanniGirolamoSaccheri.......................................................... 175
4.4 Area,Distance,andDesigns................................................................... 176
4.4.1 ExercisesforSection4.4.............................................................. 183
4.5 SphericalandSingleEllipticGeometries................................................... 185
4.5.1 ExercisesforSection4.5.............................................................. 189
4.5.2 GeorgFriedrichBernhardRiemann................................................. 189
4.5.3 ProjectsforChapter4.................................................................. 190
4.5.4 SuggestedReadings.................................................................... 192
5 TransformationalGeometry 195
5.1 OverviewandHistory........................................................................... 195
5.1.1 ExercisesforSection5.1.............................................................. 200
5.2 Isometries.......................................................................................... 201
5.2.1 ClassifyingIsometries................................................................. 203
5.2.2 CongruenceandIsometries........................................................... 208
5.2.3 Klein’sDefinitionofGeometry...................................................... 209
5.2.4 ExercisesforSection5.2.............................................................. 209
5.2.5 FelixKlein................................................................................ 212
5.3 AlgebraicRepresentationofTransformations............................................. 213
5.3.1 Isometries................................................................................. 217
5.3.2 ExercisesforSection5.3.............................................................. 220
5.4 SimilaritiesandAffineTransformations.................................................... 223
5.4.1 Similarities................................................................................ 223
5.4.2 AffineTransformations................................................................ 226
5.4.3 IteratedFunctionSystems............................................................. 228
5.4.4 ExercisesforSection5.4.............................................................. 231
5.4.5 SophusLie................................................................................ 234
5.5 TransformationsinHigherDimensions;Computer-AidedDesign................... 235
5.5.1 IsometriesoftheSphere............................................................... 235
5.5.2 TransformationsinThreeandMoreDimensions................................ 239
5.5.3 Computer-AidedDesignandTransformations................................... 241
5.5.4 ExercisesforSection5.5.............................................................. 243
5.6 InversionsandtheComplexPlane........................................................... 245
5.6.1 ExercisesforSection5.6.............................................................. 252
5.6.2 AugustusMo¨bius........................................................................ 254
5.6.3 ProjectsforChapter5.................................................................. 254
5.6.4 SuggestedReadings.................................................................... 259
6 Symmetry 261
6.1 OverviewandHistory........................................................................... 262
6.1.1 ExercisesforSection6.1.............................................................. 264