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Preface to the Second Edition of 

by Jay Kappraff

Since Connections was published in 1990, I have been gratified to observe the rising interest in the discipline of Design Science. Numerous conferences on the interface between mathematics, science, art, architecture and design have fostered a sense of community among the participants. This has led to new research and collaborations, the creation of works of art, the creation of new journals, and the establishment of new courses in mathematics and design. 

Perhaps the most fundamental changes in the field since 1990 are the ease of computer visualization; the communication made possible through the Internet; and the access to building kits and other constructive materials. Much software is now available with which anyone who is interested can create and explore fractals, tessellations, polyhedra, minimal surfaces, etc. The Zometool kit created by Mark Pelletier has also revolutionized the study of polyhedra, and George Hart has just published a Zometool Geometry book (2001) to facilitate its use. The Zome system upon which this kit is built was initially invented by Steve Baer around 1970. These resources make courses in design science more accessible and easier to teach. There is also a greater sense of community because internet researchers are more aware of what others are doing and can easily disseminate their results to each other and to the world. Two excellent websites, ISAMA (The International Society for the Arts, Mathematics, and Architecture, and the George Hart's website (, provide links to the web pages of many people making connections between art, mathematics, and science. 

Arthur Loeb has been a pioneer in the field of Design Science and many of his contributions were documented in the first edition of Connections. Soon after the publishing of Connections, Professor Loeb published two excellent books: Space Structures, Their Harmony and Counterpoint (1991) and Concepts and Images (1993). Eric Weisstein has also accumulated a wealth of knowledge in his Concise Encyclopedia of Mathematics (1998). I reported on the exquisite art of origami in the first edition of Connectionsbut neglected to mention the application of origami to polyhedron construction. Although there is a substantial literature of such books, I offer two references: Unit Origami Multidimensional Transformations by T.Fuse (1990) and Modular Origami Polyhedra by L.Simon, B.Arnstein and R.Gurkewitz (1999). I have also included three additional references of interest to polyhedra specialists: Polyhedra by P.Cromwell (1997), Build Your Owm Polyhedra by P.Hilton and J.Pedersen(1988), and Spherical Models by M.Wenninger (1999). 

Hardly a summer goes by without four or five conferences convening. At the time of this writing, ISIS-Symmetry (International Society for Interdisciplinary Study of Symmetry) under the leadership of Denes Nagy is holding its 5th Congress subtitled Intersections of Art and Science, in Sydney, Australia organized by Liz Ashburn; the 3rd International Conference on Mathematics and Design will be held in Melbourne, Australia organized by Vera De Spinadel, Javier Barrallo, Mark Burry and others; the 4th Bridges Conference subtitled Mathematical Connections between Art, Music, and Science will be held at Southwestern College under the direction of Reza Sarhangi; Symmetry 2000 was held last September in Stockholm organized by Istvan Hargittai; ISAMA 2000 was held last August in Albany under the direction of Nat Friedman, his tenth consecutive conference in art and mathematics; the MOSAIC 2000 Conference was held in Seattle; and the 3rd biannual Nexus Conference was held in Ferrara under the direction of Kim Williams. 

There are two new electronic journals devoted to the intersection of art, architecture, mathematics, science and design. The Nexus Network Journal (, edited by Kim Williams was created in 1997, and the on-line journal Visual Mathematics (, edited by Slavik Jablan and Denes Nagy, was created in 1998 as a continuation of the ISIS-Symmetry printed journal Symmetry: Art and Science (Symmetry: Culture and Science). 

When I wrote Connections, few courses in Mathematics and Design were being taught. Now many faculty have discovered the satisfaction that can be derived from engaging students in the constructive activity of creating their own designs based on mathematical principles. Several textbooks are now available to help teach these courses (see in Geometry by Discovery by D.Gay, 1998). However, there is still a need for additional texts to help guide prospective teachers at both the college and pre-college levels. With the help of a grant from the National Endowment for the Arts, I wrote a Workbook on Mathematics of Design (1997) and also, with the help the Media Center of The New Jersey Institute of Technology and a grant from the Graham Foundation, created an eleven part series of videotapes entitled Mathematics of Design (1994) to aid faculty who wish to use Connections as a primary text. 

As I mentioned in the introduction to Connections, the discipline of Design Science has advanced through the energy and creativity of many individuals, each focusing on a single idea. Several researchers not mentioned in the original edition of Connections have made important contributions to the field over the past ten years. Carlo Sequin has created an amazing computer program: Sculpture Generator 1 and 2 in which he is able to generate three-dimensional models for sculpture using his program and 3-D fabrication techniques. Bathesheba Grossman has used that technology in order to make jewelry and small bronze sculptures. Brent Collins has created extraordinary sculptures by hand from wood reminiscent of mathematical surfaces and knotted structures. He has also collaborated with Sequin to fabricate his sculptures with the aid of the computer. Vladimir Bulatov has created many polyhedral studies which can be found on his website. Charles Perry’s polyhedral sculptures are now found throughout the world. His most recent work is based on knots and minimal surfaces leading to new shell sculptures in limestone. Nat Friedman, a mathematician and sculptor, has played a major role through his conferences and his assistance to others in the field to further the objectives of design science. George Hart, another polyhedral sculptor and computer scientist, has enriched the field with his creative work that can be seen on his website He has also developed new fabrication techniques and he is also working on a history of polyhedra in art, in his book Euclid's Kiss

My only regret upon publishing the original edition of Connections was that when referring to the various crystalline states of carbon in Section 10.10, I mentioned only diamond and graphite and not the crystalline states known as the Buckminsterfullerenes. I was aware of the existence of this remarkable family of molecules as far back as 1986. However, they burst onto the mainstream of science only in 1990 just as Connectionswas in its final editing. This oversight is remedied in the current edition where I have placed several additions to the first edition in a Supplement section at the end of the book. I have also included in this Supplement additional material on the snub figures, a section on uniform polyhedra, and additional discussion of the Dorman Luke method of constructing the faces of dual polyhedra. I have also added some new material within the text on dihedral angles and orthoschemes. References not included in the first edition are found at the end of the reference section and are referred to in the text by an "S" after the date. Other changes are minor. I wish to acknowledge, once again, the generous help that I received from Branko Grünbaum in writing the first edition of Connections , and to thank Peter W. Messer for his invaluable help editing this new edition.  

Since writing Connectionsmy own professional life has been enriched by the many contacts that I have made as a result of the visibility that the book has offered to me and the wide approval with which its publication has been met. I was pleased that in 1991 the National Association of Publishers selected Connections as the best book in Mathematics and Science in the division of Professional and Reference. As a result of Connections, I made the acquaintance and began collaborations with researchers such as the Kim Williams, Ben Nicholson, Anne Macaulay, Lawrence Edwards, Ernest McClain, Tons Brunes, Louis Kauffman, Stan Tenen, and Gary Adamson. Some of their work will be featured in my new book, Beyond Measure: A Guided Tour through Nature, Myth and Numberto be published by World Scientific. It is my hope that the second edition of Connections will continue to play a role in breaking down the barriers between the arts and the sciences, and encourage others to explore the interfaces between these two human endeavors. 

January 2001