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).