Bücher der Reihe Design Science Collection

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 "e;handout"e; 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.

In a broad sense design science is the grammar of a language of images rather than of words. Modem communication techniques enable us to transmit and reconstitute images without needing to know a specific verbal sequence language such as the Morse code or Hungarian. Inter- national traffic signs use international image symbols which are not An image language differs specific to any particular verbal language. from a verbal one in that the latter uses a linear string of symbols, whereas the former is multidimensional. Architectural renderings commonly show projections onto three mutually perpendicular planes, or consist of cross sections at different altitudes capable of being stacked and representing different floor plans. Such renderings make it difficult to imagine buildings compris- ing ramps and other features which disguise the separation between and consequently limit the creative process of the architect. floors, Analogously, we tend to analyze natural structures as if nature had used similar stacked renderings, rather than, for instance, a system of packed spheres, with the result that we fail to perceive the system of organization determining the form of such structures.

na broad sense Design Science is the grammar of a language of images Irather than of words. Modern communication techniques enable us to transmit and reconstitute images without needing to know a specific verbal sequence language such as the Morse code or Hungarian. International traffic signs use international image symbols which are not specific to any particular verbal language. An image language differs from a verbal one in that the latter uses a linear string of symbols, whereas the former is multi- dimensional. Architectural renderings commonly show projections onto three mutual- ly perpendicular planes, or consist of cross sections at different altitudes capa- ble of being stacked and representing different floor plans. Such renderings make it difficult to imagine buildings comprising ramps and other features which disguise the separation between floors, and consequently limit the cre- ative process of the architect. Analogously, we tend to analyze natural struc- tures as if nature had used similar stacked renderings, rather than, for instance, a system of packed spheres, with the result that we fail to perceive the system of organization determining the form of such structures. Perception is a complex process. Our senses record; they are analogous to audio or video devices. We cannot, however, claim that such devices perceive.