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).

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.