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One might at first tend to think that the growth of plants and animals, because of their elaborate forms, are ruled by highly complex laws. However, this is surprisingly not always true: many aspects of the growth of plants and animals may be described by remarkably simple mathematical laws. An obvious example of this are the seashells and snails, as we show here: with a very simple model it is possible to describe and generate any of the many types of seashells of the classes of Gastropods, Bivalves, Cephalopods and Scaphopods that one may find classified in a good seashell bookguide. Beauty of style and harmony and grace and good rhythm depends on simplicity. — Plato There is much beauty in nature’s clues, and we can all recognize it without any mathematical training. There is beauty, too, in the mathematical stories that start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things. Mathematics is to nature as Sherlock Holmes is to evidence. — I. Stewart [8] 1. How seashells grow The idea that mathematics is deeply implied in the natural forms goes back to the Ancient Greeks. Many aspects of the growth of plants and animals may be described by remarkably simple mathematical laws, in spite of their elaborate forms (cf., for instance, the classical book of D’Arcy Thompson [9] and the recent book of Stephen Wolfram [10]). An obvious example of this are the seashells and snails [6]. Why do so many shells form spirals? As far as the animal that lives in a shell grows he needs the shell to grow with him in order to keep accommodating him. The fact that the animal which lives at the open edge of the shell places new shell material always in that edge, and faster on one side than the other, makes the shell to grow in a spiral. The rates at which shell material is secreted at different points of the open edge are presumably determined by the anatomy of the animal. And, surprisingly, even fairly small changes in such rates can have quite tremendous effects on the overall shape of the shell, which is in the origin of the existence of a great diversity of shells. Date: March 15, 2009. 1 2 JORGE PICADO A two-dimensional version of this fact may be observed in the growth of horns. Like nails and hair, a horn grows by the addition of material as its base. In order to get a perfectly straight structure, the rate at which material is added must be exactly the same on each side of the base: On the other hand, if there is some difference (indicated in percentage, in the figures below), one of the edges of the horn will be longer than the other and, inevitably, the horn will coil to the side where less material is added, following a spiral curve: It is essentially a three-dimensional version of this phenomena that yields the spiral structures of the shells of mollusks. Besides that, the mollusk does not enlarge its shell in a uniform way: it only adds material in one of the edges of the shell (the open or “growth” ending) and makes it in such a way that the new shell is always an exact model, to scale, of the smaller shell. This growth process yields an elegant spiral structure (very visible when the shell is sliced). The widths of the straight lines that link the shell center (the spiral origin) to the points of the shell increase, but the amplitudes of the angles defined by those lines and the corresponding tangents to the shell are constant, that is, shells follow an equiangular spiral: This fact was identified in the 17th century by Christopher Wren. A clear mathematical model of shell growth modes based on equiangular spirals was given by Henry Moseley SEASHELLS: THE PLAINNESS AND BEAUTY OF THEIR MATHEMATICAL DESCRIPTION 3 in 1838, and the model used here is a direct extension of his (M. B. Cortie [1]). Careful studies from the mid-1800s to mid-1900s validated Moseley’s basic model for a wide variety of shells. As indicated in the figure above, given a point O (the origin or the pole of the spiral), an equiangular spiral is a curve such that the angle α (the equiangular angle) between the tangent in each point P of the curve and the radial line OP is constant. Jacob Bernoulli (1654-1705) called it the Spira mirabilis (the marvellous spiral). It was first described mathematically in 1638 by René Descartes (who believed that “only mathematics is certain, so everything must be based on mathematics”). Its parametric equation1 is given, in polar coordinates r and θ, by r(θ) = Ae , θ ≥ 0, (1.1) where A denotes the radius associated to θ = 0. It gives the distance of a curve point to origin O in terms of θ. In cartesian coordinates, the points (x(θ), y(θ)) of the spiral are given by  x(θ) = r(θ) cos θ y(θ) = r(θ) sin θ Note that when α = 90◦ the equiangular spiral degenerates to a circle. Of course the animal would not be very satisfied with a circular shell, because he could not keep growing inside the shell. If α is not a right angle, then a true spiral forms, which corresponds to an enlargement of the shell. This growth process keeps the shape of the shell and is called gnomonic. In geometry, the gnomon (a word of Greek origin, due to Aristotle, meaning “indicator”) of a given picture is a second picture that, added or subtracted to the former, generates a third picture similar to the original one. This growth pattern is so common that it is referred by many as a “law of nature”. The majority of animal horns and nails, corals and snails, among other examples, also follow, basically, equiangular spirals. The next figure shows three of the cases that may occur. The first example is typical of a cone, the second one of the nautilus shell and the third one of the clam of a bivalve. On the right, an horizontal section – in the case of the nautilus and the bivalve – and a vertical section – in the case of the cone – show the corresponding growth spiral. In each case the new shell material is progressively added in the aperture of the shell. 1Equivalently, the equation may be given by log(r(θ)/A) = θ cotα. By that reason, the equiangular spiral is also known as the logarithmic spiral. 4 JORGE PICADO (1) Cone:

Seashells: the Plainness and Beauty of Their Mathematical Description