Some Isonemal Fabrics on Polyhedral Surfaces

The motivation for the mathematics presented here should really be viewed as originating with the practitioners of the weaver’s craft. The catalyst that resulted in this particular effort, however, was some recent work of Branko Grünbaum and G. C. Shephard [4,5]. They have carefully analyzed certain geometric objects which represent an idealization of woven fabrics in the plane and their investigations lead, among other things, to remarkable theorems concerning the number and nature of the different kinds of what they call “isonemal”1 fabrics in the plane. They have posed many open problems. The models described and pictured here (see Plates A-E, following page 120) were the result of my investigating one of their problems. The resulting models were a joy to discover and are truly beautiful to behold, but as so frequently happens in mathematics, as the existence of the answer to the original question was unveiled other similar questions seemed to spring forth. And herein lies the major difficulty involved with presenting such embryonic material. It is tempting (and, of course, desirable in the long run) to attack the problem with a great deal of mathematical rigor and preciseness (a) because it will certainly yield to that kind of discussion and (b) because there are beautiful and psychologically satisfying results. I will choose not to do that here because I believe that it is beneficial for the reader to observe first some of the natural beauty and surprise that is felt when viewing these models for the first time (unencumbered by technical detail). My second reason is that I wish, right now, to write an article—not a book.