Celebratio Mathematica

Irving Kaplansky

A Song about Pi

by Ivars Peterson

Irving “Kap” Ka­plansky was a prom­in­ent math­em­atician — a lead­ing al­geb­ra­ist — who died in 2006.

I met Ka­plansky in 1999, when he was 82 and still act­ively en­gaged in math­em­at­ic­al re­search. At that time, he was Dir­ect­or Emer­it­us of the Math­em­at­ic­al Sci­ences Re­search In­sti­tute (MSRI) in Berke­ley, Cali­for­nia, where I was spend­ing the sum­mer as Journ­al­ist in Res­id­ence.

Ka­plansky spent much of his time then in the MSRI lib­rary, pok­ing in­to vari­ous nooks and cran­nies of math­em­at­ic­al his­tory. Tidy­ing up loose ends and filling in gaps in the math­em­at­ic­al lit­er­at­ure, he pa­tiently worked through math­em­at­ic­al ar­gu­ments, proved the­or­ems, and pre­pared pa­pers for pub­lic­a­tion. His re­mark­ably wide-ran­ging ef­forts be­lied the oft-re­peated no­tion that math­em­aticians are most pro­duct­ive when they are young.

A dis­tin­guished math­em­atician who made ma­jor con­tri­bu­tions to al­gebra and oth­er fields, Ka­plansky was born in Toronto, Ontario, sev­er­al years after his par­ents had emig­rated from Po­land. In the be­gin­ning, his par­ents thought that he was go­ing to be­come a con­cert pi­an­ist. By the time he was 5 years old, he was tak­ing pi­ano les­sons. That las­ted for about 11 years, un­til he fi­nally real­ized that he was nev­er go­ing to be a pi­an­ist of dis­tinc­tion.

Non­ethe­less, Ka­plansky loved play­ing the pi­ano, and mu­sic re­mained a lifelong hobby. “I some­times say that God in­ten­ded me to be the per­fect ac­com­pan­ist—the per­fect re­hears­al pi­an­ist might be a bet­ter way of say­ing it,” he said. “I play loud, I play in time, but I don’t play very well.”

While in high school, Ka­plansky star­ted to play in dance bands. Dur­ing his gradu­ate stud­ies at Har­vard, he was a mem­ber of a small combo that per­formed in loc­al night clubs. For a while, he hos­ted a reg­u­lar ra­dio pro­gram, where he played im­it­a­tions of pop­u­lar artists of the day and com­men­ted on their mu­sic.

A little later, when Ka­plansky be­came a math in­struct­or at Har­vard, one of his stu­dents was Tom Lehr­er [­pe­­er], later to be­come fam­ous for his witty dit­ties about sci­ence and math (see “Tom Lehr­er’s De­riv­at­ive Dit­ties” for sev­er­al ex­amples). In 1945, Ka­plansky moved to the Uni­versity of Chica­go, where he re­mained un­til 1984, when he re­tired, then be­came MSRI dir­ect­or.

Songs had al­ways in­ter­ested him, par­tic­u­larly those of the peri­od from 1920 to 1950. These songs ten­ded to have a par­tic­u­lar struc­ture: the form \( \mathrm{AABA} \), where the \( \mathrm{A} \) theme is re­peated, fol­lowed by a con­trast­ing \( \mathrm{B} \) theme, then a re­turn to the ori­gin­al \( \mathrm{A} \) theme.

Early on, Ka­plansky no­ticed that cer­tain songs have a more subtle, com­plex struc­ture. This al­tern­at­ive form can be de­scribed as \( \mathrm{AA^{\prime}BAA^{\prime}(B/2)A^{\prime\prime}} \), where \( \mathrm{A} \) is a four-bar phrase, \( \mathrm{A^{\prime}} \) and \( \mathrm{A^{\prime\prime}} \) are vari­ants, and \( \mathrm{B} \) is a con­trast­ing eight-bar phrase. “I don’t think any­one had no­ticed that be­fore,” he said.

Ka­plansky’s dis­cov­ery is noted in a book about the Amer­ic­an mu­sic­al by the late Chica­go film schol­ar Ger­ald Mast.

Ka­plansky ar­gued that the second struc­ture is really a su­per­i­or form for songs. To demon­strate his point, he once used it to turn an un­prom­ising source of them­at­ic ma­ter­i­al — the first 14 decim­al di­gits of pi — in­to a pass­able tune. In es­sence, each note of the song’s chor­us cor­res­ponds to a par­tic­u­lar decim­al di­git.

When Chica­go col­league En­id Rieser heard the melody at Ka­plansky’s de­but lec­ture on the sub­ject in 1971, she was in­spired to write lyr­ics for the chor­us.

A Song about Pi

Through all the by­gone ages,
Philo­soph­ers and sages
Have med­it­ated on the circle’s mys­ter­ies.
From Eu­c­lid to Py­thagoras,
From Gauss to Anaxag’ras,
Their thoughts have filled the libr’ies bul­ging his­tor­ies.
And yet there was ela­tion
Throughout the whole Greek na­tion
When Archimedes did his mighty com­pu­ta­tion!
He said:

3 1 41 Oh (5) my (9), here’s (2) a (6) song (5) to (3) sing (5) about (8,9) pi (7).
Not a sigma or mu but a well-known Greek let­ter too.
You can have your al­phas and your great phi-bates, and omeg­as for a friend,
But that’s just what a circle doesn’t have — the be­gin­ning or an end.
3 1 4 1 5 9 is a ra­tio we don’t define;
Two pi times radii gives cir­cumf’rence you can rely;
If you square the ra­di­us times the pi, you will get the circle’s space.
Here’s my song about pi, fit for a math­em­atician’s em­brace.

The chor­us is in the key of C ma­jor, and the mu­sic­al note C cor­res­ponds to 1, D to 2, and so on, in the decim­al di­gits of pi.

You can hear a per­form­ance of the song by sing­er-song­writer Lucy Ka­plansky (Irving Ka­plansky’s daugh­ter) on You­Tube. A club head­liner, re­cord­ing artist, and former psy­cho­lo­gist, Lucy Ka­plansky has her own dis­tinct­ive style but doesn’t mind oc­ca­sion­ally show­cas­ing her fath­er’s old-fash­ioned tun­e­man­ship.

In 1993, Irving Ka­plansky wrote new lyr­ics for the ven­er­able song “That’s En­ter­tain­ment!” to cel­eb­rate his en­thu­si­asm for math­em­at­ics. He ded­ic­ated the verses to Tom Lehr­er.

That’s Math­em­at­ics

The fun when two par­al­lels meet
Or a group with an ac­tion dis­crete
Or the thrill when some decim­als re­peat,
That’s math­em­at­ics.
A nova, in­cred­ibly bright,
Or the speed of a photon of light,
An­drew Wiles, prov­ing Fer­mat was right,
That’s math­em­at­ics.
The odds of a bet when you’re rolling two dice,
The mar­velous fact that four col­ors suf­fice,
Slick soft­ware set­ting a price,
And the square on the hy­po­tenuse
Will bring us a lot o’ news.
In genes a double helix we see
And we cheer when an al­gebra’s free
And in fact life’s a big PDE.
We’ll be on the go
When we learn to grow with math­em­at­ics.

With Lag­range every­one of us swears
That all things are the sums of four squares,
Like as not, three will do but who cares.
That’s math­em­at­ics.
Sporad­ic groups are the ul­ti­mate bricks,
Find­ing them took some dev­il­ish tricks,
Now we know — there are just 26.
That’s math­em­at­ics.
The func­tion of Riemann is look­ing just fine,
It may have its zer­os on one spe­cial line.
This thought is yours and it’s mine.
We may soon learn about it
But some­how I doubt it.
Don’t waste time ask­ing wheth­er or why
A good the­or­em is worth a real try,
Go ahead — prove tran­scend­ence of pi;
Of sci­ence the queen
We’re all of us keen on math­em­at­ics.

An on­line video of Irving Ka­plansky’s lec­ture on “Fun with math­em­at­ics: Some thoughts from sev­en dec­ades” is avail­able at­tails/lec­ture12090. [Ed­it­or’s note: Link up­dated on 17 Janu­ary 2017.]

In the in­terest of full dis­clos­ure, I should note that I went to the same high school (Har­bord Col­legi­ate in Toronto) as Ka­plansky and also at­ten­ded the Uni­versity of Toronto, though my school­ing oc­curred a gen­er­a­tion later.