LEIBNITZ’S BOWLING PINS AND INFINITY

STEVEN STROGATZ WRITES ABOUT  “BOWLING FOR NOBELS,” The New York Times, June 30, 2025. One of his topics: triangular numbers as exemplified by objects arranged in perfect equilateral triangles.

Familiar Triangular Numbers. The simplest triangular number is 3; the next possible array has six objects.

These and the following images by Jens Mortensen for The New York Times.

Then there’s the familiar array of ten bowling pins; the next, fifteen pool balls: 1, 3, 6, 10, 15. And so on.

Notice the pattern: Start with 1; then 1+2=3; 1+2+3=6; 1+2+3+4=10; 1+2+3+4+5=15. Hmm….

An Infinitude of Triangular Numbers. Now think about these in triangular arrays: Each successive array has a new bottom row containing one more element than the preceding bottom row.

Thus, triangular numbers go on forever: Take any triangular array 1+2+3+…..n. Add a bottom row of n+1 more elements. And the new array is also triangular.

Gottfried Wilhelm Leibnitz, 1646–1716, German polymath, independent inventor (as was Sir Isaac Newton) in developing calculus, the mathematics of continuous change.  

Enter Leibnitz. Strogatz recounts that triangular numbers “have been studied for centuries, but perhaps their most glorious day occurs in 1672. Gottfried Wilhelm Leibniz, 26 years old, is an aspiring polymath eager to learn what the best mathematicians in Europe are working on. He turns to Christiaan Huygens, an established superstar, who gives him a problem that has come up in Huygens’s work on games of chance.”

Summing an Infinite Series. “The question,” Strogatz says, “is how to add up the reciprocals of all the triangular numbers: 1 + 1/3 + 1/6 + 1/10 + 1/15 + …. At first glance, it seems impossible–the sum is endless. Yet Leibniz spots a remarkable pattern: Each number he wants to add is related to the difference of two consecutive fractions 1/1, 1/2, 1/3, 1/4, …,  like this:

1     = 2( 1/1 – 1/2)

1/3   =2( 1/2 – 1/3)

1/6   =2( 1/3 – 1/4)

1/10  =2( 1/4 – 1/5)

1/15  =2( 1/5 – 1/6)

and so on. 

His Hands Never Leave His Wrists…. Then Leibniz looks at the stack of equations above — infinitely many! — and adds them from top to bottom. On the left, he gets 1 + 1/3 + 1/6 + 1/10 + …, the infinite series to be summed.

Summing at right, looking at pairs of entries, he observes that each – 1/n above cancels with 1/n in the term below. What results on the right side is 2 x (1/1) = 2. 

Thus, tada, the infinite sum of all triangular numbers 1 + 1/3 + 1/6 + 1/10 + 1/15 + … = 2.

A Personal Comment. This demonstrates why Leibnitz is a mathematician; I am a mere dabbler in dynamical systems theory. ds

© Dennis Simanaitis, SimanaitisSays.com, 2025