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Math made visual : creating images for understanding mathematics
- Author / Creator
- Available as
- Online
- Physical
- Summary
-
The object of this book is to show how visualization techniques may be employed to produce pictures that have interest for the creation, communication, and teaching of mathematics.
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- Creator
- Claudi Alsina and Roger B. Nelsen
- Format
- Books
- Language
- English
- Contributors
- Publication
-
- [Washington, DC] : Mathematical Association of America, ©2006
- Series
- Physical Details
-
- xv, 173 pages : illustrations ; 27 cm
- ISBNs
- 0883857464, 9781614441007, 9780883857465, 1614441006, 0883857820
- OCLC
- ocm69137417
- Includes bibliographical references (pages 161-168) and index.
- Introduction -- pt. 1. Visualizing mathematics by creating pictures -- 1. Representing numbers by graphical elements -- 1.1. Sums of odd integers -- 1.2. Sums of integers -- 1.3. Alternating sums of squares -- 1.4. Challenges -- 2. Representing numbers by lengths of segments -- 2.1. Inequalities among means -- 2.2. The mediant property -- 2.3. A Pythagorean inequality -- 2.4. Trigonometric functions -- 2.5. Numbers as function values -- 2.6. 3. Representing numbers by areas of plane figures -- 3.1. Sums of integers revisited -- 3.2. The sum of terms in arithmetic progression -- 3.3. Fibonacci numbers -- 3.4. Some inequalities -- 3.5. Sums of squares -- 3.6. Sums of cubes -- 3.7. 4. Representing numbers by volumes of objects -- 4.1. From two dimensions to three -- 4.2. Sums of squares of integers revisited -- 4.3. Sums of triangular numbers -- 4.4. A double sum -- 4.5. Challenges
- 5. Identifying key elements -- 5.1. On the angle bisectors of a convex quadrilateral -- 5.2. Cyclic quadrilaterals with perpendicular diagonals -- 5.3. A property of the rectangular hyperbola -- 5.4. Challenges -- 6. Employing isometry -- 6.1. The Chou Pei Suan Ching proof of the Pythagorean theorem -- 6.2. A theorem of Thales -- 6.3. Leonardo da Vinci's proof of the Pythagorean theorem -- 6.4. The Fermat point of a triangle -- 6.5. Viviani's theorem -- 6.6. 7. Employing similarity -- 7.1. Ptolemy's theorem -- 7.2. The golden ratio in the regular pentagon -- 7.3. The Pythagorean theorem -- again -- 7.4. Area between sides and cevians of a triangle -- 7.5. 8. Area-preserving transformations -- 8.1. Pappus and Pythagoras -- 8.2. Squaring polygons -- 8.3. Equal areas in a partition of a parallelogram -- 8.4. The Cauchy-Schwarz inequality -- 8.5. A theorem of Gaspard Monge -- 8.6. Challenges
- 9. Escaping from the plane -- 9.1. Three circles and six tangents -- 9.2. FAir division of a cake -- 9.3. Inscribing the regular heptagon in a circle -- 9.4. The spider and the fly -- 9.5. Challenges -- 10. Overlaying tiles -- 10.1. Pythagorean tilings -- 10.2. Cartesian tilings -- 10.3. Quadrilateral tilings -- 10.4. Triangular tilings -- 10.5. Tiling with squares and parallelograms -- 10.6. 11. Playing with several copies -- 11.1. From Pythagoras to trigonometry -- 11.2. Sums of odd integers revisited -- 11.3 Sums of squares again -- 11.4. The volume of a square pyramid -- 11.5. 12. Sequential frames -- 12.1. The parallelogram law -- 12.2. An unknown angle -- 12.3. Determinants -- 12.4. 13. Geometric dissections -- 13.1. Cutting with ingenuity -- 13.2. The "smart Alec" puzzle -- 13.3. The area of a regular dodecagon -- 13.4. 14. Moving frames -- 14.1. Functional composition -- 14.2. The Lipschitz condition -- 14.3. Uniform continuity -- 14.4. Challenges
- 15. Iterative procedures -- 15.1. Geometric series -- 15.2. Growing a figure iteratively -- 15.3. A curve without tangents -- 15.4. Challenges -- 16. Introducing colors -- 16.1. Domino tilings -- 16.2. L-Tetromino tilings -- 16.3. Alternating sums of triangular numbers -- 16.4. In space, four colors are not enough -- 16.5. 17. Visualization by inclusion -- 17.1. The genuine triangle inequality -- 17.2. The mean of the squares exceeds the square of the mean -- 17.3. The arithmetic mean-geometric mean inequality for three numbers -- 17.4. 18. Ingenuity in 3 D -- 18.1. From 3D with love -- 18.2. Folding and cutting paper -- 18.3. Unfolding polyhedra -- 18.4. 19. Using 3D models -- 19.1. Platonic secrets -- 19.2. The rhombic dodecahedron -- 19.3. The Fermat point again -- 19.4. 20. Combining techniques -- 20.1. Heron's formula -- 20.2. The quadrilateral law -- 20.3. Ptolemy's inequality -- 20.4. Another minimal path -- 20.5. Slicing cubes -- 20.6. Vertices, faces, and polyhedra -- 20.7. challenges
- pt. 2. Visualization in the classroom -- Mathematical drawings : a short historical perspective -- On visual thinking -- On the role of hands-on materials -- Everyday life objects as resources -- Making models of polyhedra -- Using soap bubbles -- Lighting results -- Mirror images -- Towards creativity -- pt. 3. Hints and solutions to the challenges -- References -- Index -- About the authors