The Universe of Discourse

Sun, 15 Apr 2007

Happy birthday Leonhard Euler
Leonhard Euler, one of the greatest and most prolific mathematicians ever to walk the earth, was born 300 years ago today in Basel, Switzerland.

Euler named the constant e (not for himself; he used vowels for constants and had already used a for something else), and discovered the astonishing formula !!e^{ix} = \cos x + i \sin x!!, which is known as Euler's formula. A special case of this formula is the Euler identity: !!e^{i\pi} + 1 = 0!!.

I never really understood what was going on there until last year, when I read the utterly brilliant book Visual Complex Analysis, by Tristan Needham. This was certainly the best math book I read in all of 2006, and probably the best one I've read in the past five years. (Many thanks to Dan Schmidt for rcommending it.)

The brief explanantion is something like this: the exponential function ect is exactly the function that satisfies the differential equation df/dt = cf(t). That is, it is the function that describes the motion of a particle whose velocity is proportional to its position at all times.

Imagine a particle moving on the real line. If its velocity is proportional to its position, it will speed away from the origin at an exponentially increasing rate. Or, if the proportionality constant is negative, it will rapidly approach the origin, getting closer (but never quite reaching it) at an exponentially increasing rate.

Now, suppose we consider a particle moving on the complex plane instead of on the real line, again with velocity proportional to position. If the proportionality constant is real, the particle will speed away from the origin (or towards it, if the constant is negative), as before. But what if the proportionality constant is imaginary?

A proportionality constant of i means that the velocity of the particle is at right angles to the position, because multiplication by i in the complex plane corresponds to a counterclockwise rotation by 90°, as always. In this case, the path of the particle is a circle, and so its position as a function of t is described by something like cos t + i sin t. But this function must satisfy the differential equation also, with c = i, and we have Euler's formula.

Another famous and important formula named after Euler is also called Euler's formula, and states that for any simply-connected polyhedron with F faces, E edges, and V vertices, F - E + V = 2. For example, the cube has 6 faces, 12 edges, and 8 vertices, and indeed 6 - 12 + 8 = 2. The formula also holds for all planar graphs and is the fundamental result of planar graph theory.

Spheres in this case behave like planes, and graphs that cover spheres also satisfy F - E + V = 2. One then wonders whether the theorem holds for more complex surfaces, such as tori; this is equivalent to asking about polyhedra that have a single hole. In this case, the theorem is a little different, and the identity becomes F - E + V = 0.

It turns out that every surface S has a value χ(S), called the Euler characteristic, such that graphs on the surface all satisfy F - E + V = χ(S).

Euler also discovered that the sum of the first n terms of the harmonic series, 1 + 1/2 + 1/3 + ... + 1/n, is approximately log n. We might like to say that it becomes arbitrarily close to log n, as so many things do, but it does not. It is always a bit larger than log n, and you cannot make it as close as you want. The more terms you take, the closer the sum gets to log n + γ, where γ is approximately 0.577216. This γ is Euler's constant:

$$\gamma = \lim_{n\rightarrow\infty}\left({\sum_{i=1}^n {1\over i} - \ln n}\right)$$

This is one of those numbers that shows up all over the place, and is easy to calculate, but is a big fat mystery. Is it rational? Everyone would be shocked if it were, but nobody knows for sure.

The Euler totient function φ(x) counts the number of integers less than x that have no divisors in common with x. It is of tremendous importance in combinmatorics and number theory. One of the most fundamental and astonishing facts about the totient function is Euler's theorem: aφ(n) - 1 is a multiple of n whenever a and n have no divisors in common. For example, since &phi(9) = 6, a6 - 1 is a multiple of 9, except when a is divisible by 3:

16 - 1= 9.
26 - 1= 9.
46 - 1= 455·9.
56 - 1= 1736·9.
76 - 1= 13072·9.

Euler's solution in 1736 of the "bridges of Königsberg" problem is often said to have begun the study of topology. It is also the source of the term "Eulerian path".

Wikipedia lists forty more items that are merely named for Euler. The list of topics that he discovered, invented, or contributed to would be far too large to actually construct.

Happy birthday, Leonhard Euler.

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