Blogging the Highlights: Alex Through the Looking Glass

Another in the occasional series of blog posts about the highlights I make in books I read. This time it’s Alex Bellos’ Alex Through The Looking Glass (called The Grapes of Math in the US), a look at the hidden patterns in mathematics, and mathematical patterns in life. There’s some quite complex maths in it and I found it quite tough going at times, but there are also many fascinating facts and stories.

Another clever menu strategy is to show the prices immediately after the description of each dish, rather than listing them in a column, since listing prices facilitates price comparison.

I find these little psychological manipulations endlessly interesting. It’s about the concept of bounded rationality, how we think we make rational decisions but are in fact constantly manipulated.

Benford argued that the phenomenon must be evidence of a universal law, which  he called the Law of Anomalous Numbers. The coinage didn’t catch on. His name, however, did. The phenomenon is known as Benford’s law.

Benford’s Law is a law of frequency of digits in many data sets, and has been used reliably to detect falsified data in accounting, science, economics and more. It’s quite fascinating.

In other words, well-connected nodes become even better connected. The rich get richer. The famous get more famous. The node with the most links has the highest chance of getting new links, and the more links it gets the more attractive it becomes.

Networks – whether that’s hyperlinked pages on the Web or followers on Twitter – tend to follow power laws and grow in very predictable ways.

More recently it has been argued that 360 was chosen because six equilateral triangles fit snugly within a circle, as shown below, and that each of these angles was divided into 60  as demanded by sexagesimal fractions.

Why do circles have 360 degrees? It could be because the Babylonians counted in sixties rather than tens. Ancient origins of everyday concepts fascinate me.

In the second century BCE the Greeks appropriated Babylonian fractions, which have been in use ever since. The degree was traditionally divided into sixty smaller units,  each a pars minuta prima, or first minute part, which were then divided into sixty smaller units, each a pars minuta secunda, or second minute part. From the translation of these  Latin phrases we get the words minute and second, our units of time, which are the most prominent modern relics of the ancient practice of counting in groups of sixty.

Again.

[Tycho] Brahe was a flamboyant aristocrat. He wore a prosthetic gold and silver nose, after a cousin sliced the original one off in a duel about a mathematical formula.

Actually they exhumed the astronomer in 2010 and found out that his nose was probably brass.

… Christopher Wren, a young English astronomy professor…

I had no idea Wren was an astronomer before an architect. Seems to have been a very common profession.

To keep his position as professor of mathematics at the Collège de France, the  country’s most prestigious seat of learning, [Gilles Personne de Roberval] had to provide the best answer to a problem announced publicly every three years.

He set the problems himself, but still this is a great test of suitability for a role.

[John] Whitney could adjust the speed and size of the sinusoids electronically, giving him much more control and eliminating the effects of damping. The patterns he produced were dazzling and  became some of the most iconic images in the history of mathematical art. They were famously used in the title sequence and posters for Alfred Hitchcock’s 1958 movie Vertigo.

The first computer-generated art used in a Hollywood feature film, in 1958.

It was not long, however, before engineers were using catenaries. Before the computer age the quickest way to make one was to hang a chain, trace out the curve, build a model  using a rigid material and stand it upside down.

Catenaries are a curve where the tension is so perfectly distributed that it makes an arch which needs no brace or buttresses. Gaudí used them extensively in designing the Colònia Güell.

Underlying the whimsy, however, is a whole field of incredibly useful theory, called ‘optimal stopping’, or the maths of when is the best time  to stop.

There really is scientific theory around the best time to take a particular action.

Since [1976] about 200 looping roller-coasters have been built around the world, all of them using Stengel’s principle.

Werner Stengel invented the first looping roller-coaster, when he used a clothoid instead of a circle for the loops.

The term for a word that only appears once in a text is hapax legomenon.

There’s a word for everything (and a Wikipedia page).