The Universe of Discourse

Happy birthday Perl!
In case you hadn't yet heard, today is the 20th anniversary of the first release of Perl.

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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!!.

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The brief explanantion is something like this: the exponential function e^ct 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:

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

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: a^φ(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	=	0·	9.
26 - 1	=	7·	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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MadHatterDay 2006

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Happy Birthday Benjamin Franklin!

Today is Benjamin Franklin's 300th birthday.

Franklin was born on 6 January, 1706. When they switched to the Gregorian calendar in 1752, everyone had their birthday moved forward eleven days, so Franklin's moved up to 17 January. (You need to do this so that, for example, someone who is entitled to receive a trust fund when he is thirty years old does not get access to it eleven days before he should. This adjustment is also why George Washington's birthday is on 22 February even though he was born 11 February 1732.)

(You sometimes hear claims that there were riots when the calendar was changed, from tenants who were angry at paying a month's rent for only 19 days of tenancy. It's not true. The English weren't stupid. The law that adjusted the calendar specified that monthly rents and such like would be pro-rated for the actual number of days.)

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Since I live in Philadelphia, Franklin is often in my thoughts. In the 18th century, Franklin was Philadelphia's most important citizen. (When I first moved here, my girlfriend of the time sourly observed that he was still Philadelphia's most important citizen. Philadelphia's importance has faded since the 18th century, leaving it with a forlorn nostalgia for Colonial days.) When you read Franklin's Autobiography, you hear him discussing places in the city that are still there:

So not considering or knowing the difference of money, and the greater cheapness nor the names of his bread, I made him give me three-penny worth of any sort. He gave me, accordingly, three great puffy rolls. I was surpriz'd at the quantity, but took it, and, having no room in my pockets, walk'd off with a roll under each arm, and eating the other.

Thus I went up Market-street as far as Fourth-street, passing by the door of Mr. Read, my future wife's father; when she, standing at the door, saw me, and thought I made, as I certainly did, a most awkward, ridiculous appearance.

Heck, I was down at Fourth and Market just last month.

Franklin's personality comes across so clearly in his Autobiography and other writings that it's easy to imagine what he might have been like to talk to. I sometimes like to pretend that Franklin and I are walking around Philadelphia together. Wouldn't he be surprised at what Philadelphia looks like, 250 years on! What questions does Franklin have? I spend a lot of time explaining to Franklin how the technology works. (People who pass me in the street probably think I'm insane, or else that I'm on the phone.) Some of the explaining is easy, some less so. Explaining how cars work is easy. Explaining how cell phones work is much harder.

Here's my favorite quotation from Franklin:

I believe I have omitted mentioning that, in my first voyage from Boston, being becalm'd off Block Island, our people set about catching cod, and hauled up a great many. Hitherto I had stuck to my resolution of not eating animal food, and on this occasion consider'd, with my master Tryon, the taking every fish as a kind of unprovoked murder, since none of them had, or ever could do us any injury that might justify the slaughter. All this seemed very reasonable. But I had formerly been a great lover of fish, and, when this came hot out of the frying-pan, it smelt admirably well. I balanc'd some time between principle and inclination, till I recollected that, when the fish were opened, I saw smaller fish taken out of their stomachs; then thought I, "If you eat one another, I don't see why we mayn't eat you." So I din'd upon cod very heartily, and continued to eat with other people, returning only now and then occasionally to a vegetable diet. So convenient a thing it is to be a reasonable creature, since it enables one to find or make a reason for everything one has a mind to do.

Happy birthday, Dr. Franklin.

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Happy Birthday Universe of Discourse
Today is the 12th anniversary of my web site. Early attractions included a CGI version of the venerable "Guess-the-Animal" game and what I believe was the first "guestbook" application on the web.

It's been a wild ride! I hope the next twelve years are as much fun.

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