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Mathematical Evolutions (Spectrum Series)

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Advances in mathematics are no different from those in other fields. A new result is generally the consequence of years, sometimes centuries of effort, by many people. Revolutionary results are rare, with evolutionary being a far more accurate descriptor of the movement of mathematical progress. Although mathematical progress happens faster than evolutionary changes in organisms, there are many parallels.
When a new environment becomes available, many different creatures move to occupy it and within a very short time, new species have branched off and are flourishing. Upon the development of a new area of mathematics, many branches rapidly rise and shoot off in many unexpected directions. Different creatures in isolated environments evolve to fill a common ecological niche and possess almost identical characteristics. Different mathematicians often work on the same problem independently and arrive at similar breakthroughs at roughly the same time.
With all of these similarities, it is only natural that mathematical topics be examined from an evolutionary perspective. In 1993, John Ewing, then editor of the "American Mathematical Monthly" approached one of the editors of this book (Abe Shentizer) with the idea to start a regular column "The Evolution Of . . . ", which was to chronicle the development of a mathematical idea. The columns were to be accessible to a general, albeit knowledgeable mathematical audience. This book is a collection of those columns.
The historical development of some of the basic concepts such as function, integration, optimization, rings and elliptical curves are described in great detail. While formulas are used, they appear only when necessary, which is fairly rare. This is a tribute to the quality of the exposition, which generally renders any use of equations redundant. The authors also are to be commended for their spending the time to explain the historical context of the discovery as well as some of the consequences.
Properly done, essays in the history of mathematics explain where the idea was first described, the background that led the discoverer to the discovery, the personal and professional interactions of the discoverers and where the discovery fits in that fascinating, interrelated jigsaw puzzle called mathematics. The essays in this book are properly done.

Published in the recreational mathematics newsletter, reprinted with permission.

I shall indicate some features of my favourite articles. Stillwell, "Modular Miracles". Historically, modular functions grew out of the theory of elliptic functions by considering the value of the elliptic integral of 1/(1-x^2)(1-k^2x^2) as a function of the "modulus" k^2. Now, elliptic integrals are analogous to arc trigonometric integrals, and indeed modular functions satisfy equations analogous to trigonometric identities. But trig identities can be used to solve the cubic equation (reduce general cubic to 4x^3-3x=c and use 4cos^3(x)-3cos(x)=cos(3x)) and indeed Hermite discovered that modular equations can be used to solve the general quintic. Geometrically, modular functions are functions periodic with respect to the modular tessellation, which is a tessellation of the upper half-plane by congruent triangles in the sense of hyperbolic geometry; in other words they are invariant under integer linear fractional transformations with determinant 1, which is precisely the equivalence relation for lattice shapes thought of as the ratio of its two generating complex numbers, which explains the connection with elliptic functions. But lattices are also fundamental in the theory of quadratic integers as follows. The set of integers O in Q(sqrt(-D)) is a lattice because it can be formed as the closure under + and *, and so is any of its ideals. Algebraically, one has unique factorisation if every ideal is principal, i.e. of the form aO, i.e., geometrically, Q(sqrt(-D)) has unique prime factorisation if and only if all its ideal have the same shape. Perhaps, then, one could hope that modular functions will be of use in algebraic number theory. Knonecker discovered that they are in a most remarkable way: plug an integer of Q(sqrt(-D)) into the prototypical modular function and you will get back and algebraic integer whose degree is the class number of Q(sqrt(-D)) (which of course we recall is 1 if and only if there is unique factorisation). If we feel rusty on algebraic number theory there is a separate Stillwell article on this topic. Shenitzer, "How Hyperbolic Geometry Became Respectable". Beltrami discovered that the only surfaces that can be mapped to the plane in such a way that geodesics are preserved (i.e., go to lines) are surfaces of constant curvature. This is easy for surfaces with zero curvature or positive curvature, but interesting for surfaces of negative curvature. For a sphere, for instance, we can map it onto the plane by projection from its center. By this map, the metric on the sphere induces a metric on the plane whose length element we can calculate. So spherical geometry is modelled by the plane with this new metric, which is pointless, to be sure, since we already had the sphere itself. But Beltrami discovered that any geodesic-preserving map onto the plane will have the same form of its length element. Putting negative curvature in the length element expression, we see that this only exists inside the unit circle. So with this metric the unit circle is a model of hyperbolic geometry. This is wonderful, because it circumvents the fact that there actually doesn't exist any surface of constant negative curvature mapping onto the complete unit disc. From here one easily derives the other standard models of hyperbolic geometry, as Beltrami did. First, put a hemisphere over the unit disc and suck up the geodesics (lines) by vertical projection, so that geodesics are now vertical sections of the hemisphere. Stereographic projection of this northern hemisphere from the south pole onto the tangent plane at the north pole gives the conformal disc model. And projection from a point on the equator onto the tangent plane at the diametrically opposite point gives the half plane model.