# Folded Corners

I was recently reminded of a Richard Feynman anecdote I once read in James Gleick’s biography, *Genius: The Life and Science of Richard Feynman*. I had given the book to my Dad as a gift almost 20 years ago, and — as is tradition for me when I give books to family — I borrowed and read the book immediately.

Yesterday, I borrowed the book from my Dad again to see if I could find the passage. I vaguely knew where the anecdote was, but it was going to require some flipping and scanning, as the book is over 500 pages long with dense type.

To my delight, the corner of the page I was seeking was folded over! It was the only corner folded in the whole book. Apparently, I had been so struck by that very anecdote when I first read the book 20 years ago, and I had folded the corner to hold the place.

I stopped folding corners years ago (reverting instead to Post-Its), and I barely even read paper books anymore, thanks to my Kindle. This discovery was a nice, visceral reminder of the joys of paper artifacts that you can touch and feel and fold.

For those of you curious about the anecdote that has stuck with me all these years, it was about an informal lecture Feynman gave to his peers while he was at Los Alamos working on the Manhattan Project. Here’s the passage, from pages 181-182 of the original hardback edition:

Meanwhile, under the influence of this primal dissection of mathematics, Feynman retreated from pragmatic engineering long enough to put together a public lecture on “Some Interesting Properties of Numbers.” It was a stunning exercise in arithmetic, logic, and — though he would never have used the word — philosophy. He invited his distinguished audience (“all the might minds,” he wrote his mother a few days later) to discard all knowledge of mathematics and begin from first principles — specifically, from a child’s knowledge of counting in units. He defined addition,

a+b, as the operation of countingbunits from a starting point,a. He defined multiplication (countingbtimes). He defined exponentiation (multiplyingbtimes). He derived the simple laws of the kinda+b=b+aand (a+b) +c=a+ (b+c), laws that were usually assumed unconsciously, though quantum mechanics itself had shown how crucially some mathematical operations did depend on their ordering. Still taking nothing for granted, Feynman showed how pure logic made it necessary to conceive of inverse operations: subtraction, division, and the taking of logarithms. He could always ask a new question that perforce required a new arithmetical invention. Thus he broadened the class of objects represented by his lettersa,b, andcand the class of rules by which he was manipulating them. By his original definition, negative numbers meant nothing. Fractions, fractional exponents, imaginary roots of negative numbers — these had no immediate connection to counting, but Feynman continued pulling them from his silvery logical engine. He turned to irrational numbers and complex numbers and complex powers of complex numbers — these came inexorably as soon as one from facing up to the question: What number,i, when multiplied by itself, equals negative one? He reminded his audience how to compute a logarithm from scratch and showed how the numbers converged as he took successive square roots of ten and thus, as an inevitable by-product, derived the “natural base”e, that ubiquitous fundamental constant. He was recapitulating centuries of mathematical history — yet not quite recapitulating, because only a modern shift of perspective made it possible to see the fabric whole. Having conceived of complex powers, he began to compute complex powers. He made a table of his results and showed how they oscillated, swinging from one to zero to negative one and back again in a wave that he drew for his audience, though they knew perfectly well what a sine wave looked like. He had arrived at trigonometric functions. Now he posed one more question, as fundamental as all the others, yet encompassing them all in the round recursive net he had been spinning for a mere hour: To what power mustebe raised to reachi? (They already knew the answer, thateandiandπwere conjoined as if by an invisible membrane, but as he told his mother, “I went pretty fast & didn’t give them a hell of a lot of time to work out the reason for one fact before I was showing them another still more amazing.”) He now repeated the assertion he had written elatedly in his notebook at the age of fourteen, that the oddly polyglot statemente^{πi}+ 1 = 0 was the most remarkable formula in mathematics. Algebra and geometry, their distinct languages notwithstanding, were one and the same, a bit of child’s arithmetic abstracted and generalized by a few minutes of the purest logic. “Well,” he wrote, “all the mighty minds were mightily impressed by my little feats of arithmetic.”

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