## Archive for the 'Occasional' Category

### A Zero-Order Front

Sharp gradients known as fronts form in the atmosphere when variations in the wind field bring warm and cold air into close proximity. Much of our interesting weather is associated with the fronts that form in extratropical depressions.

Below, we describe a simple mechanistic model of frontogenesis, the process by which fronts are formed.

### The Flight of the Bumble Bee

Alice and Bob, initially a distance l apart, walk towards each other, each at a speed w. A bumble bee flies from the nose of one to the nose of the other and back again, repeating this zig-zag flight at speed f until Alice and Bob meet. How far does the bumble bee fly?

### Euler’s “Degree of Agreeableness” for Musical Chords

The links between music and mathematics stretch back to Pythagoras and many leading mathematicians have studied the theory of music. Music and mathematics were pillars of the Quadrivium, the four-fold way that formed the basis of higher education for thousands of years. Music was a central theme for Johannes Kepler in his Harmonices Mundi – Harmony of the World, and René Descartes’ first work was a compendium of music.

### Grandi’s Series: A Second Look

In an earlier post, we discussed Grandi’s series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series

$\displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots$

This is a divergent series: the sequence of partial sums is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$, which obviously does not converge, but alternates between ${0}$ and ${1}$.

### Grandi’s Series: Divergent but Summable

Is the Light On or Off?

Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states “on” and “off” by ${1}$ and ${0}$, the sequence of states over the first minute is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$. But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.

### Numbers with Nines

What proportion of all numbers less than a given size N have a 9 in their decimal expansion? A naive argument would be that, since 9 is one of ten distinct digits, the answer must be about 10%. But this is not “remotely close” to the true answer.

### “Dividends and Divisors Ever Diminishing”

Next Saturday is Bloomsday, the anniversary of the date on which the action of Ulysses took place. Mathematical themes occur occasionally throughout Ulysses, most notably in the penultimate episode, Ithaca, where the exchanges between Leopold Bloom and Stephen Dedalus frequently touch on weighty scientific matters. [Last week’s ThatsMaths post]

Joyce in Zurich: did he meet Zermelo?