Science

# Celebrate Hamilton Day, a Better Mathematical Holiday

## Pi Day may be fun, but this October holiday celebrates mathematics as a human endeavor.

Lists of math holidays invariably omit a worthy one. Observed in Ireland every Oct. 16, Hamilton Day commemorates—no, not a bastard, orphan, son of a whore and a Scotsman—the eureka moment William Rowan Hamilton (1805–1865) experienced while walking along Dublin’s Royal Canal in 1843. It celebrates a mathematical breakthrough, one that not only enables space exploration and virtual reality but expanded the realm of mathematical possibility.

Consider Hamilton Day a ready-made alternative to America’s current favorite math holiday, Pi Day. Pi Day received congressional recognition in 2009, and annual festivities have school children and pie lovers paying at least lip service to the discipline. But the pi(e) parties’ typically scant mathematical content has set naysayers ranting in recent years, griping that revelers, reciting digits and gorging on desserts, have forgotten (or perhaps never knew?) the “reason for the season.”

The solution? Stop the hand-wringing, hush the pi versus tau debates, and instead embrace a wholly different opportunity to rejoice in the marvels of mathematics: Hamilton Day. To do so would encourage a deeper dive into mathematics and cultivate an enhanced appreciation of it.

So what revelation dawned on Hamilton as he strolled? (And was it hip-hop musical material?)

Hamilton’s epiphany involved an unprecedented take on multiplication. In the multiplication familiar from grade school, what’s called the commutative property holds: Order doesn’t matter. Three times five equals five times three, a×b = b×a for any numbers a and b. For most of human history, multiplication commuted. And that was that.

Hamilton did not set out to topple this age-old norm. His challenge to it arose as an unintended consequence of his quest to extend “number couples” to three dimensions. Known today as complex or imaginary numbers, number couples have the form a+bi where i is defined by the property i2 = -1. A number couple a+bi can be represented as the point (a,b) on the coordinate plane, and addition and multiplication of number couples correspond, respectively, to translations and rotations in the plane. Hamilton hoped that motion in three dimensions could be similarly captured by arithmetic operations on number triples. Defining these operations, however, turned out to be tricky.

Hamilton wanted his number triples to be closed under multiplication. A set of numbers is closed under an operation if performing the operation on numbers in the set always produces a number in the set. Multiply an integer by another integer, for example, and the result is also an integer. Multiply two complex numbers together, and you get a complex number. Hamilton wanted to define multiplication on number triples such that they too had this property. (To see why this might not be straightforward, try multiplying the number triples a+bi+cj and d+ei+fj, with i2 =j2 = -1. Ask yourself what would have to be true for this product to be a triple of the same form.)

Hamilton also expected number triples to satisfy what he called the law of the moduli (which, though we needn’t get into the details, says that the product of the lengths equals the length of the product), and thus complicated the matter further. When he defined multiplication such that the number triples remained closed, the law of the moduli failed to hold; when he rejiggered his definition to satisfy the law of the moduli, he lost closure.

Hamilton was stuck. And for good reason—the task Hamilton had set for himself has since been shown to be impossible.

Morning after morning in early October 1843, Hamilton’s children asked whether he could multiply triples yet.

“Whereto I was always obliged to reply,” he later recalled to his son Archibald, “with a sad shake of the head, ‘No, I can only add and subtract them.’ ”

That changed on Oct. 16. One moment Hamilton was walking the canal with his wife. The next: “An electric current seemed to close,” he later wrote, “and a spark flashed forth.”

Hamilton realized that if he admitted a fourth dimension—if he exchanged number triples for number quadruples of the form a+bi+cj+dk and defined multiplication in a particular way—then all would work out as he wished. Hamilton would go on to spend years and write volumes exploring the effects and import of his four-dimensional answer to the three-space puzzle, but he knew on Oct. 16 that he had found the key.

Flush with his insight, Hamilton could not resist the impulse to record it. Then and there, he took a knife to the Broome Bridge, carving into its stone his multiplication formula: i² = j² = k² = ijk = -1.

So this act of 19th-century vandalism is worthy of celebration why?

Quaternions—that’s what Hamilton called his number quadruples—have proven extremely useful, for one thing. Eclipsed for much of the 20th century by such tools as vector analysis and matrix algebra, quaternions have more recently enjoyed a renaissance. Today’s programmers and engineers harness their ability to describe three-dimensional rotations to simplify aerospace navigation, enhance the realism of flight simulators and video games, and power eye-tracking and face-recognition technologies. We have quaternions to thank for Tomb Raider’s Lara Croft, for the eye-popping special effects in The Matrix Reloaded, and for Curiosity’s 2012 landing on Mars.

The importance of quaternions hinges as much on their nonconformity as their myriad applications, however. Hamilton devised a multiplication that met all his requirements, sure, but what he ended up with was … not commutative. If you derive a quaternion multiplication table from the fundamental formula Hamilton carved into Broome Bridge, you’ll find that ij = k, but ji = -k. (Matrix multiplication doesn’t commute either, but that came later.)

This break with the commutativity convention earned Hamilton the title “liberator of algebra,” Maynooth University mathematician Fiacre O’Cairbre explains. O’Cairbre, who leads an annual walk retracing Hamilton’s path along the canal, credits his countryman with freeing mathematicians’ minds to conceive of new number systems unshackled by the rules of ordinary arithmetic. Indeed, mere months after Hamilton’s quaternion brain wave, his friend John T. Graves debuted “octaves,” a kind of double quaternion now called octonions. Octonion multiplication is not only not commutative but also not associative. (An operation is associative if (a×bc = a×(b×c).) Post-Hamilton mathematics took a lot less for granted.

So there’s much to celebrate on Hamilton Day. Beyond its utility, we can appreciate mathematics as a human endeavor, with struggles and setbacks and triumphs. We can highlight the opportunity math affords for daring, creativity, and out-of-the-box thinking.

And you needn’t tag along on O’Cairbre’s walk or attend the Royal Irish Academy’s public maths lecture to mark the day.

Observance could be as simple as remembering a delightful idea you’ve had, recalling a time when sustained cogitation led to an eventual aha!, or writing a math-related poem. (Friends with both William Wordsworth and Samuel Taylor Coleridge, Hamilton saw a deep kinship between science and poetry. There’s also a poetic form called the quaternion, though four-ness is all it shares with Hamilton’s quadruples.)