Leonhard Euler

(This 875th Buffalo Sunday News column was first published on December 30, 2007.)

Leonhard Euler

painting by Handmann

This year mathematicians celebrated the 300th anniversary of one of the four finest mathematicians of all history, yet you have probably never heard of him. He is a kind of mathematical Rodney Dangerfield -- getting too little respect. The mathematician is Leonhard Euler -- pronounced "oiler".

A delightful poem about him by Charlie Marion and William Dunham appeared in 1997 in Mathematics Magazine. Here is just one of its eight stanzas:

Six dozen volumes, what a feat!

Profound and deep throughout.

Does Leonhard rank with the elite?

Of this there is no doubt.

Indeed, mathematicians rank Euler with Archimedes, Isaac Newton and Karl Friedrich Gauss. He was, in fact, more prolific than any of them and his work has had important applications. John Derbyshire has listed a few: "Euler's discoveries continue to influence such disparate fields as computer networking, harmonics, and statistical analysis, and they did nothing less than transform pure mathematics." You gain some sense of the power of Euler's contributions when you discover that he developed ideas central today to the work of computer scientists two centuries before computers were even invented.

It would take several university mathematics graduate courses to provide the background necessary to understand much of what Euler wrote, but among his discoveries are a few that are easier to illustrate.

One of those is Euler's Formula for Polyhedra. Polyhedra are solid figures like boxes or dice. Euler proved that the number of their corners plus the number of their faces equal two more than the number of their edges. Stated in equation form this is simply: C + F = E + 2. A cube, for example, has 8 corners, 6 faces and 12 edges; thus we have 8 + 6 = 12 + 2. Using this formula, it is easy to prove that the five regular polyhedra displayed are the only ones. They also offer additional opportunities to check Euler's formula.

The five Regular Polyhedra

In solving another problem Euler introduced the mathematical fields of graph theory and topology, now two central subjects of advanced mathematics. The problem is known as The Seven Bridges of Konigsberg.

The Pregel River passes through Konigsberg (now Kaliningrad) creating two islands. In Euler's days there were seven bridges connecting parts of the city as shown in the diagram. The question Euler was asked: Is it possible to walk a route within the city that crosses each bridge exactly once? Euler simplified the problem by letting dots represent the four land areas and connecting lines the bridges. This resulted in a simple diagram involving only dots and lines.

The Konigsberg bridges abstracted

Now, Euler argued, notice that each dot has an odd number of lines running from it. Passing through any dot would use two lines (one incoming, one outgoing) so a dot with an odd number of lines must be a start or end of the walk. Thus you can only have two dots with an odd number of lines to complete a walk. Since there are four here, Euler announced that this walk was impossible. (Note that removing the horizontal line — closing that one bridge — would make the problem possible.)

But Euler is best known for two mathematical constants: e and i. The number i is the so-called imaginary number, the square root of -l. The number e, approximately 2.71828, is applied in such widely diverse applications as banking, atomic decay and the cooling of your cup of coffee. (The letter e was later chosen to honor Euler.)

Euler associated those constants along with the more familiar numbers 0, 1 and pi in a famous formula that relates these five fundamental constants of mathematics:

This equation is as central to mathematics as the equation e = mc² is to physics, yet it can be derived today using high school algebra.

It is appropriate at this time to extend belated birthday wishes to Leonhard Euler and early birthday wishes for the New Year.-- Gerry Rising