Math made visual: creating images for understanding mathematics

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. Challenges

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

3.4. Some inequalities

3.5. Sums of squares

3.6. Sums of cubes

3.7. Challenges

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. Challenges

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. Challenges

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. 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. Challenges

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. Challenges

12. Sequential frames

12.1. The parallelogram law

12.2. An unknown angle

12.3. Determinants

12.4. Challenges

13. Geometric dissections

13.1. Cutting with ingenuity

13.2. The "smart Alec" puzzle

13.3. The area of a regular dodecagon

13.4. Challenges

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. Challenges

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. Challenges

18. Ingenuity in 3 D

18.1. From 3D with love

18.2. Folding and cutting paper

18.3. Unfolding polyhedra

8.4. Challenges

19. Using 3D models

19.1. Platonic secrets

19.2. The rhombic dodecahedron

19.3. The Fermat point again

19.4. Challenges

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

Visualization in the classroom

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.