Phyllotaxis and the Fibonacci SeriesG. J. MitchisonScience, New Series, Vol. 196, No. 4287. (Apr. 15, 1977), pp. 270-275.Stable URL:http://links.jstor.org/sici?sici=0036-8075%2819770415%293%3A196%3A4287%3C270%3APATFS%3E2.0.CO%3B2-HScience is currently published by American Association for the Advancement of Science.Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtainedprior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content inthe JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. 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If the marketplace phenomenon of few job op- portunities for scientists and engineers is allowed to reign unchecked (that is, ap- preciably reduce the input of graduate students), the nation will lose that enormous research effort now contrib-uted by graduate and postdoctoral stu- dents. One solution to this problem, as I mentioned earlier, would be an increase in federal formula grants, from which we could support these young people. A direct effect of the lead institution concept that may cause consternation in research universities is that departments would feel pressure to bend their efforts toward interdisciplinary research, per- haps at the expense of "small science," the kind of basic unarticulated research that has been the lifeblood of research universities. While I clearly do not advo- cate diminishing such research, the pres- sure, on balance, may be a healthy one. With some very notable exceptions, the traditional departmental structures of universities often remain as barriers to in- terdisciplinary research. To help us over- come these barriers-and still preserve the departments as basic academic and administrative units-we need to rethink ways of subsidizing our research efforts in the interdisciplinary, problem-focused mode. If the research universities do not adjust to society's needs, society's dol- lars for high-priority research may simply go elsewhere. Ideally, the university should be the focus for both basic and interdiscipli- nary research on long-range issues. Fed- eral laboratories, corporations, and special university institutes and centers would be responsible for the shorter-term Phyllotaxis and the Fibonacci Series An explanation is offered for the characteristic spiral leaf arrangement found in many plants. The spiral patterns of leaves, bracts, or florets of plants are a familiar mathemati- cal curiosity of nature. Anyone who has counted the spirals which catch the eye on the head of a sunflower, or on a pine cone, will have discovered that their num- ber is generally a term of the series 1, 1,2, 3, 5, 8, 13, 21. . . . This is the famous Fibonacci series, each of whose terms is the sum of the preceding two. Although the study of phyllotaxis (leaf arrange- ment) goes back to classical antiquity, the attempt to find aplausible mechanism or a mathematical explanation for this Fibo- nacci phyllotaxis began more recently. It is perhaps to Richards that we owe the most lucid treatment of the subject. In particular, his paper on the "Geometry of phyllotaxis" (I)seems to offer a key to the problem. However, it falls short of an The author is a member of the scientific staffat the MRC Laboratory of Molecular Biology, Hills Road, Cambridge, CB2 ZQH, England. G.J. Milchison explanation: a suggestive diagram and several pregnant sentences culminate in the assertion that Fibonacci phyllotaxis must inevitably occur, given certain plau- sible assumptions. It is clear that even Richards' authority has not convinced later investigators, for the problem has continued to be regarded as ilnsolved. In response to this, Adler (2) recently pro- posed a somewhat elaborate mathemati- cal theory. I show here that such com- plexities are unnecessary, and that a simple geometric argument, in the spirit of Richards' paper, suffices to explain the phenomenon. A proviso is necessary, however, since this argument only applies when the mechanism which positions new leaves meets a certain condition: loosely speak- ing, the influence of existing leaves in de- termining the position of a new leaf must be short-range. Experimental evidence suggests that this is so (in some plants at approaches closer to practical applica- tion. It will not speak well of the scientific community if we must be dragged into the global age kicking and screaming, with a debilitating case of future shock. If we can protect and strengthen basic re-search-and you will recall I made partic- ular note of the health of our universities and the support of young investigators- if we can encourage more problem-on- cnted research and better articulation be- tween research sectors, then, I believe, we can do better than muddle through my so-called global age. In my optimism I think we are bending in this direction now, but my great hope is that these ten- dencies will accelerate-for our sake and for the sake of future generations. References 1. Philip H. Abelson and Allen L. Hammond, "The electronics revolution," Science 195, 1087 (1977). least), but it is clearly important to know what happens otherwise. Here, simple geometrical reasoning does not