SOME WHIMSICAL GEOMETRY

THIS article deals with a method of approximating, by folding a paper strip, any regular polygon whose number of sides is of the form 2n + 1, where is any natural number; or 2W+1 ? 1, where is any natural number greater than 1. Since there are exact Euclidean con structions, and exact folding processes (Johnson 1970; Row 1966) for some of the polygons mentioned above, one might ar gue that there is very little point in using the methods outlined here to obtain their approximations. On the other hand this method requires no special tools other than ordinary add ing machine tape; and, above all, it is sim ple, which makes it fun. Indeed, it can be taught even to very young students who have no knowledge of geometry. Any grade school student who is capable of folding a paper airplane can certainly be taught to fold a very respectable approxi mation of a regular seven-sided polygon. It is not necessary that the student also be taught that a ruler and compass con struction of this particular polygon is im possible (Eves 1969, pp. 106, 107). That fact can be saved as a surprise, when he studies geometry?meanwhile, he may de velop enough interest in geometry to want to study it. This method may also be used as a rec reational device for motivating more ad vanced students. In geometry classes, for instance, there are often some students for whom the ruler and compass construc tions, which are perfect in the mind, often fall somewhat short of perfection when executed on real paper with their soft, not too sharp pencils. Oddly enough those students are frequently just the ones who are best at folding paper (possibly due to their experience with the paper airplanes). It is hoped that this method, with the accent on the actual folding, will be used by teachers as a recreational tool to stimu late more student interest in the study of exact geometric constructions?and, for the better students who want to pursue the proofs?more interest in geometric se quences and limits. In order to appreciate this article the reader is advised to obtain some adding machine tape and to fold it in the manner described in each of the examples. Then, having witnessed that the method does work, pursue the proofs if you feel so in clined. Don't fret if you enjoy folding the paper, but you don't wish to know the proofs, or if the proofs are beyond the level of your students' comprehension. The former is surely a personal matter (the author will not be offended) and, the latter will be taken care of when your students become more knowledgeable. In either event, the proofs will still be accessible if you wish to refer to them later. If this attitude alarms you consider the fact that students are routinely taught how to use the formula for the area of a circle with far less evidence of its validity. Besides, in this case, you are just supposed to be having fun ! For the most accurate results in all of the examples, fold the tape so that when ever transversals are formed the tape re mains folded on those creases. The next fold always occurs on the portion of the paper strip that comes from the top of the existing configuration. Thus, the triangles that are formed will either stack up or form a zig-zag type pattern in the folding plane, but the configuration will never need to be turned over during the folding process. One quickly discovers, however, that certain rotations of the configuration