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.


[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).


Blogging the Highlights: Smarter Than You Think

I make no secret of the fact that I love Russell Davies’ blog, and recently he’s been running a series of posts in which he blogs the portions he highlights in books on his Kindle. I think this is a great idea, so I’m stealing it wholesale, except I have a Kobo.

The first book is Clive Thompson’s Smarter Than You Think, which looks at common complaints against modern technology (It makes us stupid! It makes us antisocial!) and gently attempts to debunk them. It’s not cyber-utopian, but it is pro-technology. I really enjoyed the book, and agree with its conclusions.

Here are the bits I highlighted:

In 1915, a Spanish inventor unveiled a genuine, honest-to-goodness robot that could easily play Chess – a simple endgame involving only three pieces, anyway. A writer for Scientific American fretted that the inventor “Would Substitute Machinery for the Human Mind.”

I have a hobby of collecting dire predictions about the perils of technology. This is an example.

The mathematician Gottfried Wilhelm Leibniz bemoaned “that horrible mass of books which keeps on growing,” which would doom the quality writers to “the danger of general oblivion” and produce “a return to barbarism.”

That’s another example.

Each time we’re faced with bewildering new thinking tools, we panic – then quickly set about deducing how they can be used to help us work, meditate, and create.

This is kind of a distillation of the book. Each new technology seems overwhelming, there is a small outcry against it, then we adapt ourselves to it (and it to us).

“Blogging forces you to write down your arguments and assumptions. This is the single biggest reason to do it, and I think it alone makes it worth it.”

Gabriel Weinberg of DuckDuckGo said this, and I endorse this message. That’s what this very blog is for.

U.S. neurologist George Miller Beard diagnosed America’s white-collar population as suffering from neurasthenia. The disorder was, he argued, a depletion of the nervous system by its encounters with the unnatural forces of modern civilization, most particularly “steam power”, “the telegraph”, “the periodical press”, and “the sciences.”

Today we blame modern technology for memory and attention disorders instead.

Sociologists have a name for this problem: pluralistic ignorance. It occurs whenever a group of people underestimate how much others around them share their attitudes and beliefs.

“I’m not racist myself, but I couldn’t employ a black person as my colleagues wouldn’t accept it.”

Complaining is easy – much easier than getting out of your chair. Many critics have worried about the rise of so-called slacktivism, a generation of people who think clicking “like” on a Facebook page is enough to foment change. Dissent becomes a social pose.

The book’s position is that online activism helps act as an instigator of, rather than a replacement for, real-life protest. Really, I just liked the phrasing of the last sentence.

“It strikes me that social media embodies the connection between action and expression.”

Charlie Beckett said this, about the theory in the previous quote.

… this reflexively dystopian view is just as misleading as the giddy boosterism of Silicon Valley. Its nostalgia is false; it pretends these cultural prophecies of doom are somehow new and haven’t occurred with metronomic regularity, and in nearly identical form, for centuries.

(Standing ovation) I share this opinion, and I was delighted to read this in the epilogue. We’ve always had scares about new technologies, and we always will; just read some history and you’ll find it’s an inescapable solution. There never was a more innocent time, we’re not all doomed because we read on our smartphones instead of newspapers, no-one is becoming more stupid because we have better tools to outsource some of our processing to. Everything old is new again.