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Mathematics in Nature:
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From the Preface:“Mathematics in nature: this book grew out of a course of the same name, with the rather long subtitle “the beauty of nature as revealed by mathematics and the beauty of mathematics as revealed in nature.” That course in turn grew out of an awakened awareness of both such facets of nature, even in a suburban environment, enhanced by occasional trips to National and State Parks armed with binoculars and camera. I decided that it might be fun to develop a course that included some of the mathematics that lies behind some of the phenomena we encounter in the natural world around us… …I wanted to limit the topics covered to those objects that could be seen with the naked eye by anyone who takes their eyes outside; there are many books written on the mathematical principles behind phenomena that take place at the microscopic and submicroscopic levels, and also from planetary to galactic scales. But leaves, trees, spider webs, bubbles, waves, clouds, rainbows…these are elements of the stuff we can easily see. The length scales extend roughly from 0.1 mm (the thickness of a human hair or the size of some ice crystals and diatoms) to almost 1000 km (large storm systems), a factor of about 10^10; the timescales of the phenomena we seek to describe range from a fraction of a second (the period of some ripples on a puddle) through the order of a day (tidal periods) to the time a tree takes to mature (perhaps thirty years in some cases), corresponding to a factor of about 108 or 109. Of course, in all likelihood we would not be around to see a sequoia tree reach maturity (!), so I have drawn the line somewhat arbitrarily at thirty years, but not in order to suggest that we stand around and “watch trees grow. Many patterns are readily identified in nature. A visit to the zoo reminds us, no doubt unnecessarily, that tigers and zebras have striped patterns; leopards and hyenas have spotted patterns; giraffes are very blotchy (as well as very tall), while butterflies and moths may possess them all: spots, blotches, bands and stripes. Everyone notices the wave patterns that move across oceans, lakes, ponds and puddles, but fewer perhaps realize that waves move slowly across deserts in the form of sand dunes. In the sky brightly colored circular arcs -- rainbows -- beautify the sky after rain showers; from an airplane or a high peak there may sometimes be seen “glories” -- small colored concentric rings surrounding the shadow of the airplane or observer -- often called “The Specter of the Brocken” because it is frequently observed by climbers on the Brocken peak in Germany. On occasion circular and non-circular halos around and about the sun can be produced by ice crystals; if still we look up, we may sometimes observe parallel bands of cloud spreading across the sky; these may be billows or lee waves depending on the mechanism producing them -- the latter are sometimes called mountain waves for this very reason. We may also see hexagonal-like clouds hanging suspended from a blue ceiling: this is a manifestation of cellular convection. Patterns of light scattering are exhibited in the pre-dawn and twilight skies, in sunrise and sunset: blue sky and iridescent clouds... If we look down and around in a well-tended garden, we may become aware (upon further investigation) that arrangements of leaves, petals, seeds and florets are intimately associated with spirals patterns, the related sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... and also with an angle of about 137.5 degrees (or its complement, 222.5 degrees). Spirals in three dimensions (helices) with interesting geometric properties plus striped patterns combine to make exquisite sea shells; moving on to bigger plants, we may note that the heights of trees are closely related to their diameters, following sound engineering principles. Branching patterns in trees, leaves, river networks, lungs and blood vessels exhibit similar (fractal-like) features; there is an amazing unity (without uniformity) in nature. There are also to be found some fascinating geometric properties associated with mud cracks and patterns in the bark of trees. Although it developed out of a university course, this is not a textbook per se. It will be very useful as such, I hope, but also it can be dipped into at leisure; simple examples are scattered liberally thoughout the book, and especially so in the early chapters. For whom then is this book written? The answer is that it is for a mixture of communities, academic and otherwise. Certainly I have in mind the college population of undergraduate students in mathematics, science and engineering (and their professors) who may be able to use it as a supplement to their standard texts in various courses, particularly those in mathematical modeling. The material covered here is very broad in its scope, and I hope that it will be of considerable interest to professors and students engaged in both these and interdisciplinary courses. My further hope is that this material will appeal to high school teachers and their students who may have the opportunity for “mathematical enrichment” beyond the normal syllabus, if time permits. Anyone interested in the beauty of nature, regardless of mathematical background also (I trust) will enjoy much in this book. Although the mathematical level ranges across a broad swathe, from “applied arithmetic” to partial differential equations, there is considerable non-mathematical discussion of the basic science behind the equations which I hope will also appeal to many others who might wish to ignore the equations (but not at their peril). Thus those who have no formal mathematical background will find much of value in the descriptive material contained here. The level of mathematics used in this book varies from basic arithmetic, geometry and trigonometry through calculus of a single variable, (and a smidgin of linear algebra) to the occasional senior or first year graduate-level topic in American universities and colleges. A background in geometry, trigonometry and single-variable calculus will suffice for most purposes; familiarity with the theory of linear ordinary and partial differential equations is useful but definitely not necessary in order to appreciate the contents of this book. It should therefore be accessible almost in its entirety to students of mathematics, science and engineering, the occasional advanced topic notwithstanding. One of the other major
reasons for writing this book is to bring together different strands
from the many fascinating books and scientific articles, both technical
and popular that I Continuing this personal theme, I would like to share my philosophy for both writing about and teaching applied mathematics. It is a simple one: (i) try to understand the material to be presented at as many levels of description as is reasonable, and (ii) attempt to communicate that understanding with enthusiasm, gentleness and humor. Like most others in my profession, I continue to be fascinated by the beauty, power and applicability of mathematics, and try to induce that fascination in others (often with mixed success in the classroom). Mathematics is a subject that is misperceived, sadly, by many both inside and outside the academic world. It is either thought to involve “doing long sums” or to be a cold, austere subject of little interest in its own right and no practical application whatsoever. These extremes could not be further from the truth, and one of my goals in teaching mathematics is to try and open students' minds to the above-mentioned triad of beauty, power and applicability (even one out of three would be valuable!). My goals are the same in writing. Nowadays a great deal of what is taught in universities and colleges by applied mathematicians comes under the general description mathematical modeling. Mathematical modeling is as much “art” as “science”: it requires the practitioner to (i) identify a so-called “real world” problem (whatever the context may be), (ii) formulate it in mathematical terms (the “word problem” so beloved of undergraduates), (iii) solve the problem thus formulated (if possible; perhaps approximate solutions will suffice, especially if the complete problem is intractable), and (iv) interpret the solution in the context of the original problem (i.e. what does this answer tell me? What does it really mean? Is it consistent with what I know already about the problem? What predictions can be made?). The formulation stage is often the most difficult: it involves making judicious simplifications to ``get a handle'' on the salient features of the problem. This in turn provides the basis in principle for a more sophisticated model, and so on. Whether the class is at an elementary, intermediate or advanced level, it is important to convey aspects of the modeling process that are relevant to a particular mathematical result or technique that may be discussed in that class. Often this is most easily accomplished by illustrating the result in the context of a particular application. Consequently there are plenty of applications in this book. Some will wonder why
I have not included a chapter on “Fractals and Chaos in Nature” since
fractals, chaos and “complexity” are of such interest within the
scientific community. There are two basic reasons: (i) many others
have done an excellent job already on this topic, and (ii) it is
a huge subject that could occupy several volumes if carried out
properly. Fractal geometry has been characterized by some as the
only realistic way to mimic nature and describe it in mathematical
terms, and without wishing to question the foundations upon which
this book is based (non-fractal mathematics), there is a measure
of truth to that statement. But it has always been a strongly-held
opinion of mine that in order to gain the most understanding of
a physical phenomenon, it is necessary to view it with as many
complementary levels of description and explanation as possible; however,
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