Concepts & Images : Visual Mathematics

1. Introduction. 1 2. Areas and Angles.. 6 3. Tessellations and Symmetry 14 4. The Postulate of Closest Approach 28 5. The Coexistence of Rotocenters 36 6. A Diophantine Equation and its Solutions 46 7. Enantiomorphy........ 57 8. Symmetry Elements in the Plane 77 9. Pentagonal Tessellations. 89 10. Hexagonal Tessellations 101 11. Dirichlet Domain 106 12. Points and Regions 116 13. A Look at Infinity. 122 14. An Irrational Number 128 15. The Notation of Calculus 137 16. Integrals and Logarithms 142 17. Growth Functions... 149 18. Sigmoids and the Seventh-year Trifurcation, a Metaphor 159 19. Dynamic Symmetry and Fibonacci Numbers 167 20. The Golden Triangle 179 21. Quasi Symmetry 193 Appendix I: Exercise in Glide Symmetry. 205 Appendix II: Construction of Logarithmic Spiral. 207 Bibliography. 210 Index.................... 225 Concepts and Images is the result of twenty years of teaching at Harvard's Department of Visual and Environmental Studies in the Carpenter Center for the Visual Arts, a department devoted to turning out students articulate in images much as a language department teaches reading and expressing one­ self in words. It is a response to our students'requests for a'handout'and to l our colleagues'inquiries about the courses : Visual and Environmental Studies 175 (Introduction to Design Science), YES 176 (Synergetics, the Structure of Ordered Space), Studio Arts 125a (Design Science Workshop, Two-Dimension­ al), Studio Arts 125b (Design Science Workshop, Three-Dimensional),2 as well as my freshman seminars on Structure in Science and Art.