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Informationen zum Autor Dr. L. Christine Kinsey is in the Mathematics and Statistics department at Canisius University. Teresa E. Moore is the author of Geometry and Symmetry, published by Wiley. Efstratios Prassidis is the author of Geometry and Symmetry, published by Wiley. Klappentext This new book for mathematics teachers helps them gain an appreciation of geometry and its importance in the history and development of mathematics. The material is presented in three parts. The first is devoted to Euclidean geometry. The second covers non-Euclidean geometry. The last part explores symmetry. Exercises and activities are interwoven with the text to enable them to explore geometry. The activities take advantage of geometric software so they'll gain a better understanding of its capabilities. Mathematics teachers will be able to use this material to create exciting and engaging projects in the classroom. Zusammenfassung * This new book helps learners gain an appreciation of geometry and its importance in the history and development of mathematics. * The material is presented in three parts: The first is devoted to Euclidean geometry. The second covers non-Euclidean geometry. Inhaltsverzeichnis Preface xi I Euclidean geometry 1 1 A brief history of early geometry 3 1.1 Prehellenistic mathematics 3 1.2 Greek mathematics before Euclid 5 1.3 Euclid 9 1.4 The Elements 11 1.5 Projects 14 2 Book I of Euclid's The Elements 15 2.1 Preliminaries 15 2.2 Propositions I.5-I.26: Triangles 39 2.3 Propositions I.27-I.32: Parallel lines 54 2.4 Propositions I.33-I.46: Area 59 2.5 The Pythagorean Theorem 64 2.6 Hilbert's axioms for euclidean geometry 71 2.7 Distance and geometry 75 2.8 Projects 78 3 More euclidean geometry 80 3.1 Circletheorems 80 3.2 Similarity 88 3.3 More triangle theorems 92 3.4 Inversioninacircle 101 3.5 Projects 107 4 Constructions 109 4.1 Straightedge and compass constructions 109 4.2 Trisections 121 4.3 Constructions with compass alone 124 4.4 Theoretical origami 129 4.5 Knots and star polygons 139 4.6 Linkages 144 4.7 Projects 153 II Noneuclidean Geometries 155 5 Neutral geometry 157 5.1 Viewsongeometry 157 5.2 Neutralgeometry 159 5.3 Alternate parallel postulates 169 5.4 Projects 176 6 Hyperbolic geometry 178 6.1 The history of hyperbolic geometry 178 6.2 Strangenewuniverse 181 6.3 Models of the hyperbolic plane 186 6.4 Consistency of geometries 197 6.5 Asymptoticparallels 199 6.6 Biangles 203 6.7 Divergentparallels 208 6.8 Triangles in hyperbolic space 210 6.9 Projects 218 7 Other geometries 220 7.1 Exploring the geometry of a sphere 220 7.2 Ellipticgeometry 226 7.3 Comparative geometry 239 7.4 Areaanddefect 242 7.5 Taxicab geometry 254 7.6 Finite geometries 258 7.7 Projects 264 III Symmetry 265 8 Isometries 267 8.1 Transformationgeometry 267 8.2 Rosette groups 289 8.3 Frieze patterns 294 8.4 Wallpaper patterns 301 8.5 Isometries in hyperbolic geometry 314 8.6 Projects 319 9 Tilings 320 9.1 Tilings on the plane 320 9.2 Tilings by irregular tiles 327 9.3 Tilings of noneuclidean spaces 341 9.4 Penrose tilings 345 9.5 Projects 355 10 Geometry in three dimensions 357 10.1 Euclidean geometry in three dimensions 357 10.2 Polyhedra 369 10.3 Volume 387 10.4 Infinite polyhedra 393 10.5 Isometries in three dimensions 397 10.6 Symmetri...
List of contents
Preface.
I Euclidean geometry.
1 A brief history of early geometry.
1.1 Prehellenistic mathematics.
1.2 Greek mathematics before.
1.3 Euclid.
1.4 The Elements
1.5 Projects
2 Book I of Euclid's The Elements.
2.1 Preliminaries.
2.2 Propositions I.5-26: Triangles.
2.3 Propositions I.27-32: Parallel lines.
2.4 Propositions I.33-46: Area.
2.5 The Pythagorean Theorem.
2.6 Hilbert's axioms for euclidean geometry.
2.7 Distance and .
2.8 Projects.
3 More euclidean geometry.
3.1 Circle theorems.
3.2 Similarity.
3.3 More triangle theorems.
3.4 Inversion in a circle.
3.5 Projects.
4 Constructions.
4.1 Straightedge and compass constructions.
4.2 Trisections.
4.3 Constructions with compass alone.
4.4 Theoretical origami.
4.5 Knots and star polygons.
4.6 Linkages.
4.7 Projects.
II Noneuclidean geometries.
5 Neutral geometry.
5.1 Views on geometry.
5.2 Neutral geometry.
5.3 Alternate parallel postulates.
5.4 Projects.
6 Hyperbolic geometry.
6.1 The history of hyperbolic geometry.
6.2 Strange new universe.
6.3 Models of the hyperbolic plane.
6.4 Consistency of geometries.
6.5 Asymptotic parallels.
6.6 Biangles.
6.7 Divergent parallels.
6.8 Triangles in hyperbolic space.
6.9 Projects.
7 Other Geometries.
7.1 Exploring the geometry of a sphere.
7.2 Elliptic geometry.
7.3 Comparative geometry.
7.4 Area and defect.
7.5 Taxicab geometry.
7.6 Finite geometries.
7.7 Projects.
III Symmetry.
8 Isometries.
8.1 Transformation Geometry.
8.2 Rosette groups.
8.3 Frieze patterns.
8.4 Wallpaper patterns.
8.5 Isometries in hyperbolic geometry.
8.6 Projects.
9 Tilings.
9.1 Tilings on the plane.
9.2 Tilings by irregular tiles.
9.3 Tilings of noneuclidean spaces.
9.4 Penrose tilings.
9.5 Projects.
10 Geometry in three dimensions.
10.1 Euclidean geometry in three dimensions.
10.2 Polyhedra.
10.3 Volume.
10.4 Infinite polyhedra.
10.5 Isometries in three dimensions.
10.6 Symmetries of polyhedra.
10.7 Four-dimensional figures.
10.8 Projects.
A Logic and proofs.
A.1 Mathematical .
A.2 Logic.
A.3 Structuring proofs.
A.4 Inventing proofs.
A.5 Writing proofs.
A.6 Geometric diagrams.
A.7 Using geometric software.
A.8 Van Hiele levels of geometric thought.
B Postulates and theorems.
B.1 Postulates.
B.2 Book I of Euclid's The Elements.
B.3 More euclidean geometry.
B.4 Constructions.
B.5 Neutral geometry.
B.6 Hyperbolic geometry.
B.7 Other geometries.
B.8 Isometries.
B.9 Tilings.
B.10 Geometry in three dimensions.
Bibliography.
