Sunday, September 22, 2013

A Song about Pi

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, Calif., 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, 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 AABA, where the A theme is repeated, followed by a contrasting B theme, then a return to the original A theme.

Early on, Kaplansky noticed that certain songs have a more subtle, complex structure. This alternative form can be described as AA'BAA'(B/2)A", where A is a four-bar phrase, A' and A" are variants, and 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.


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.


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.

Original version posted July 12, 1999


Albers, D.J., G.L. Alexanderson, and C. Reid. 1990. Irving Kaplansky. In More Mathematical People: ContemporaryConversations. Academic Press.

Kaplansky, I. 1992. The deep young man. Mathematical Intelligencer 14(No. 4):62.

An online video of Irving's Kaplansky's lecture on "Fun with mathematics: Some thoughts from seven decades" is available at

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.