Concepts & Images

1. Front Matter

   Pages i-xi

   PDF

2. Introduction

   • Arthur L. Loeb

   Pages 1-5

3. Areas and Angles

   • Arthur L. Loeb

   Pages 6-13

4. Tessellations and Symmetry

   • Arthur L. Loeb

   Pages 14-27

5. The Postulate of Closest Approach

   • Arthur L. Loeb

   Pages 28-35

6. The Coexistence of Rotocenters

   • Arthur L. Loeb

   Pages 36-45

7. A Diophantine Equation and its Solutions

   • Arthur L. Loeb

   Pages 46-56

8. Enantiomorphy

   • Arthur L. Loeb

   Pages 57-76

9. Symmetry Elements in the Plane

   • Arthur L. Loeb

   Pages 77-88

10. Pentagonal Tessellations

    • Arthur L. Loeb

    Pages 89-100

11. Hexagonal Tessellations

    • Arthur L. Loeb

    Pages 101-105

12. Dirichlet Domains

    • Arthur L. Loeb

    Pages 106-115

13. Points and Regions

    • Arthur L. Loeb

    Pages 116-121

14. A Look at Infinity

    • Arthur L. Loeb

    Pages 122-127

15. An Irrational Number

    • Arthur L. Loeb

    Pages 128-136

16. The Notation of Calculus

    • Arthur L. Loeb

    Pages 137-141

17. Integrals and Logarithms

    • Arthur L. Loeb

    Pages 142-148

18. Growth Functions

    • Arthur L. Loeb

    Pages 149-158

19. Sigmoids and the Seventh-year Trifurcation, a Metaphor

    • Arthur L. Loeb

    Pages 159-166

20. Dynamic Symmetry and Fibonacci Numbers

    • Arthur L. Loeb

    Pages 167-178

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.

• design
• mathematics
• synergetics

• Department of Visual and Environmental Studies, Carpenter Center for the Visual Arts, Harvard University, Cambridge, USA

  Arthur L. Loeb

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