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Symmetry, Shape, and Space: An Introduction to Mathematics Through Geometry



Symmetry, Shape, and Space: An Introduction to Mathematics Through Geometry

L. Christine Kinsey, Teresa E. Moore

ISBN: 978-0-470-41238-1 June 2008 494 Pages


Symmetry, Shape, and Space uses the visual nature of geometry to involve students in discovering mathematics.  The text allows students to study and analyze patterns for themselves, which in turn teaches creativity, as well as analytical and visualization skills.  Varied content, activities, and examples lead students into an investigative process and provide the experience of doing and discovering mathematics as mathematicians do. Exercises requiring students to express their ideas in writing and to create drawings or physical models make math a hands-on experience.  Assuming no mathematics beyond the high school level, Symmetry, Shape, and Space is the perfect introduction to mathematics in the liberal arts course of study, and it is designed so that each chapter is independent of the others, allowing great flexibility.


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To the Reader.

1. The Basics.

1.1 Measurement.

1.2 Polygons.

2. Grids.

2.1. Billiards.

2.2. Celtic Knots.

3. Constructions.

3.1. Ruler and Compass Constructions.

3.2. The Pentagon and the Golden Ratio.

3.3. Theoretical Origami.

3.4. Knots and Stars.

3.5. Linkages.

4. Tesselations.

4.1. Regular and Semiregular Tilings.

4.2. Irregular Tilings.

4.3. Penrose Tilings.

5. Two-Dimensional Symmetry.

5.1. Kaleidoscopes.

5.2. Rosette Groups: Point Symmetry.

5.3. Frieze Patterns: Line Symmetry.

5.4. Wallpaper Patterns: Plane Symmetry.

5.5. Islamic Lattice Patterns.

6. Other Dimensions, Other Worlds.

6.1. Flatlands.

6.2. The Fourth Dimension.

7. Polyhedra.

7.1. Pyramids, Prisms, and Antiprisms.

7.2. The Platonic Solids.

7.3. The Archimedean Solids.

7.4. Polyhedral Transformations.

7.5. Models of Polyhedra.

7.6. Infinite Polyhedra.

8. Three-Dimensional Symmetry.

8.1. Symmetries of Polyhedra.

8.2. Three-Dimensional Kaleidoscopes.

9. Spiral Growth.

9.1. Spirals and Helices.

9.2. Fibonacci Numbers and Phyllotaxis.

10. Drawing Three Dimensions in Two.

10.1. Perspective.

10.2. Optical Illusions.

11. Shape.

11.1. Noneuclidean Geometry.

11.2. Map Projections.

11.3. Curvature of Curves.

11.4. Curvature of Surfaces.

11.5. Soap Bubbles.

12. Graph Theory.

12.1. Graphs.

12.2. Trees.

12.3. Mazes.

13. Topology.

13.1. Dimension.

13.2. Surfaces.

13.3. More About Surfaces.

13.4. Map Coloring Problems.

Hints and Solutions to Selected Problems.




Exercises and activities link students' intuitive and analytical capabilities