Google's amazing mathematical doodle

by Burkard Polster and Marty Ross

The Age, 22 April 2013

Last Monday was the birthday of the great mathematician Leonhard Euler … and your Maths Masters forgot. Unforgivable! To make it even more embarrassing, Google remembered, honouring Euler with a stunning doodle.

Still, we have a bit of an excuse. It was Leonhard’s 306th birthday, which is hardly an obvious one to mark in the calendar. (It was also commonly reported to be Euler’s 360th birthday, but positional notation has always been a little tricky for some people.)

We’ve tended to celebrate round number birthdays (in spite of our deciphobia). Unfortunately Euler’s 300th birthday predated the birth of our column by a few months, but we’ve made firm plans to celebrate Euler’s 350th birthday. In our 2000th column. However, just in case things don't go according to plan, we’ll wish Euler a belated happy birthday here, and we’ll piggy back upon Google’s wonderful doodle.

No column can begin to describe Euler's brilliant and massive output. Newton and Leibniz may have invented the calculus, but it was Euler who then applied it with an effortless brilliance. He has aptly been described as the Mozart of mathematics. Wikipedia and the Mactutor archive are excellent sites to beginning reading about Euler's life and work. There is also an excellent archived series How Euler Did It, and there's the incredible Euler archive.

We won't attempt to compete with those and the many other excellent sources. Instead, we'll concentrate upon explaining the various components of Google's doodle. We'll begin with probably the least familar ingredient, at the very centre of the doodle:

Euler angles

The spherical diagram in the central O is intended to picture Euler angles at work. (In the doodle itself, the sphere is animated.) This was Euler's approach to recording and analysing the rotation of an object about any axis through a central point; he reduced any such rotation to a sequence of three rotations (giving the three Euler angles) around coordinate axes. It was in some sense just an issue of mathematical bookkeeping. Nonetheless, Euler's approach was elegant and simple, so much so that it is still in use today.

The Seven Bridges of Königsberg

This may be Euler's most famous contribution to mathematics. The left diagram represents the city of Königsberg, divided into four regions by the river Pregel and with those regions connected by seven bridges. The Königsberg Bridge Problem was to determine whether it was possible to take a tour of the city, crossing each bridge exactly once.

In 1736 Euler proved that such a tour of Königsberg was impossible, and then generalised and solved the problem for any conceivable network of regions and bridges. Euler's paper is a delight to read, and his beautiful analysis gave birth to graph theory, now a hugely important field of mathematics.

Euler begins by stripping away the nonessentials. This amounts to picturing Königsberg as in the diagram on the right: the vertices A, B, C and D respresent the four regions and the connecting edges represent the seven bridges. (This corresponds to Euler's description, though Euler did not include such a schematic diagram in his paper.)

Euler described two related methods of solving the Königsberg problem. His second and more famous method was to consider regions/vertices that can be reached by an odd number of bridges/edges. Euler noted that any such region must either appear at the start of the tour or at the end, implying that there can be at most two such regions. So, since all four regions of Königsberg are connected by an odd number of bridges, the tour of Königsberg must be impossible.

That is a beautifully simple solution to the problem, but Euler's first method is perhaps even more stunning. Euler imagines a tour of Königsberg over the seven bridges, noting the eight regions visited along the way. The tour can be represented by a string of eight letters, with no concern for recording the actual bridges employed. Euler then notes that, since region A is connected by five bridges, the letter A must appear in the string 3 times. (Drawing a quick picture makes this clear.) Similarly, each of the letters B, C and D must appear 2 times. But that means our string must have 3 + 2 + 2 + 2 = 9 letters. Since the string is only 8 letters in length, the tour must be impossible. Beautiful!

Euler’s polyhedral formula

Consider a tetrahedron (the pyramidy thing in the O). It has V = 4 vertices, E = 6 edges and F = 4 faces. So, V – E + F = 2, just like the doodle says. Tada!

But just a minute. V – E + F will obviously be some number, so what's the big deal that it's 2? Well, consider an icosahedron (the triangle-sphere thing in the G); in this case V = 12, E = 30 and F = 20, and again we find that V – E + F = 2. Now it's getting interesting.

The quantity V - E + F is now known as the Euler characteristic. In 1752, Euler (employing somewhat different concepts) proved that any convex polyhedron has an Euler characteristic of 2. (By "convex" we mean the polyhedron does not have any indentations. Employing the Euler characteristic to analyse non-convex polyhedra turns out to be a tricky but very fruitful enterprise.)

The beautifully simple formula V – E + F = 2 also leads to beautiful applications. For example, it permits a very elegant proof that there can be at most five Platonic solids.

It is another instance of Euler's work giving birth to a hugely important mathematical field, in this case the field of topology. However, it is only fair to disclose that in this case Euler was beaten to the punch: René Descartes had effectively discovered the same formula over a hundred years earlier. (Party pooper!)

Euler’s (other) formula

The graph in the google doodle is difficult to make out, but it is intended to be a diagram of the following, amazing formula:

Here, e is the fundamental mathematical constant (annoyingly and erroneously referred to as "Euler's number" by people who should know better), and i = √-1 is that weird "imaginary" number. (We'll have much more to say about i in a future column.)

Euler's formula associates imaginary numbers, the trig functions sin and cos and exponentials in an unexpected and incredibly powerful manner. (Actually, as it is commonly taught, the formula is boringly expected, being simply true by definition. A true appreciation of the formula requires a proper treatment of e and exponentiation, which is seldom provided.)

Euler’s identity

Substituting the angle φ = π in Euler's formula, we obtain the equation in the bottom right of the doodle. Or, rearranging slightly,

This may be the most amazing of all of Euler's amazing work. It unites the five fundamental constants of mathematics – 0, 1, π, e and i – in one stunning equation. It is widely regarded as most beautiful equation in mathematics, and your Maths Masters agree. It is a wonderful way to close Google's superb tribute to the great Leonhard Euler.

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.