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Geometry and Symmetry

L. Christine Kinsey, Teresa E. Moore, Efstratios Prassidis

ISBN: 978-0-470-49949-8 April 2010 480 Pages


This new book for mathematics and mathematics education majors helps students 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 a rigorous introduction of Euclidean geometry, the second covers various noneuclidean geometries, and the last part delves into symmetry and polyhedra. Historical contexts accompany each topic. Exercises and activities are interwoven with the text to enable the students to explore geometry. Some of the activities take advantage of geometric software so students - in particular, future teachers - gain a better understanding of its capabilities. Others explore the construction of simple models or use manipulatives allowing students to experience the hands-on, creative side of mathematics. While this text contains a rigorous mathematical presentation, key design features and activities allow it to be used successfully in mathematics for teachers courses as well.

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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.