Happy Quaternions Day


“Here as he walked by [in Dublin] on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i2 = j2 = k2 = ijk = -1 & cut it on a stone of this bridge.”

It’s the 175th anniversary.

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  1. Lee Rudolph says

    FROBENIUS, A SESQUILOGUE

     

    (FROBENIUS, exhausted and delirious after proving the uniqueness of the Quaternions, reflects on the philosophy, life, and works of his mentor SIR WILLIAM ROWAN HAMILTON, and thus invokes SIR WILLIAM’s ghost.)

     

    Quod erat demonstrandum, Q.E.D.—                          [1]
    SIR WILLIAM would have been so proud of me!
    Now his Quaternions are proved unique,
    And though some scornful men may cry, “A freak,”      [2]
    I know that they are good, and useful, too!
    To state your theorem, struggle, prove it true:
    What then’s more glorious? He said, “Alone,               [3]
    Beauty stands naked. After one has shown
    The use of a new theory, tailored clothes
    For Beauty—not to cover her, but to disclose
    New charms beside the old—then one is worth
    The name of Mathematician.”
                                                    Scorn and mirth
    Do them ill credit; for they know I’ve tried,
    Tried and succeeded. My work is applied,
    No doubt about it! Can’t they let me be?                     [4]
    SIR WILLIAM would have been so proud of me!
     
    Calm, now; these months of work have warped my rnind,
    Or bent my judgment. Can I really find
    Justification for a year spent so,                                  [5]
    Fourteen months squandered on one proof? Say no,
    SIR WILLIAM: “FROBENIUS, you’ve put the frosting  [6]
    On cake that needed none. One year, exhausting
    Yourself night after night; and, after all,
    What would it matter if it should befall
    That my i, j, and k were not unique?
    Would that stop you from using them? You seem to seek
    A strange monopoly.”
                                     Pure mathematics seems
    At times, alas, the fleetingest of dreams,
    One it is my damnation to pursue.
    SIR WILLIAM, are you damned? No news of you,   [7]
    Nor NEWTON, nor the others, comes this way.
    Feh! you are dead; there’s nothing more to say;
    How shall we judge you, we who are alive,
    But by your works?
                                      So, then: when you were five,    [8]
    Hebrew and Latin and Greek; when you were ten,
    Sanskrit, scrawled Arabic, and Persian; then,
    At thirteen, that language which transcends ail time.  [9]
    CLAIRAUT’s Algèbre, lacking rhythm, rhyme,
    And meter, moved you more than HOMER could.
     
    Far less than midway through your life, a wood
    As dark as DANTE’s, older than his Creation,
    Closed you in: and through it lay salvation,
    And through it you set off, blazing new trails.
    And, oh! the stories of you! I’ve read tales—
    At seventeen you gave LAPLACE the lie,
    Corrected his figures. Eighteen: who’d deny,
    By then, you were the first of Ireland’s minds?
    Your Optics—not since NEWTON (his name winds
    No longer, broader, better marked a course)
    Has so much light been shed by one lone source.
    At twenty-two, professor; knight at thirty.
    No work for your hands; if the nails were dirty,
    That was just ink.
                                        But one idea stuck         [10]
    Fast in your mind; for fifteen years, no luck
    Nor furious genius could dislodge the thought
    Or solve the problem. (And I said my lot
    Was hard? One year? Oh, fie, FROBENIUS, fie!)
    “Papà, have you learnt yet how to multiply
    Your ‘triplets’?” “No, son, I can only add,          [11]
    Add and subtract’em.”
                                             Did they think you mad,
    Mad for your fifteen years of “wasted time”?
    Perhaps. I’m certain that they think that I‘m       [12]
    Mad as a hatter.
                                 It’s the hatter’s trade
    That drives him mad; do they think I am made
    Of sterner stuff than hatters? In the felt
    He makes hats from, are poisons; I have dealt
    With stronger. It’s calomel (I think) they use      [13]
    To keep the felt from rotting; if they lose
    A hatter now and then, because the rot
    Turned to his mind and kidneys—well, they’ve got
    Another, hatters come cheap.
                                                   And calomel?
    Dug from the earth. Mathematics comes from HELL! [14]

        *  *  *

    “Indeed? Then is this Hell, where I have dreamed      [15]
    These years that I have slept? What always seemed
    To me the hellish waste in mathematics
    Was ‘purity’. Why, you’ve Dynamics, Statics,
    Optics and Hydraulics—bridges to build,
    If it comes to that. Why have you got to gild
    The lily with ‘purity’? A waste of life!
    You worry me.”                                                          [16]
                             SIR WILLIAM, once your wife          [17]
    Worried for you fifteen years, unceasing—
    Each day your hopes and prospects were decreasing
    Until it seemed they could decrease no further,
    And your dear HELEN told you, “BILL, it’s murther,
    Yer murtherin’ yirself.”
                                            You didn’t eat
    Unless she brought you food; sheet after sheet
    Of foolscap heaped up on your desk each night:
    But your equations never worked out right—          [18]
    She knew it, for you had them burnt each morning.
     
    Then, one day, it struck you without warning,         [19]
    As you and she were walking. In the stone
    Of Brougham Bridge you carved it—not alone:
    Names of half Dublin’s lovers must entwine
    With that one short, sweet, and immortal line
              i²=j²=k²=ijk=–1                                              [20]
    Which, written once, can never be erased,
    Though love and stone shall crumble.
                                                    All the waste
    Was hers—her worry. Don’t you worry, now.
      
    “No waste of my time, GEORG. We’ll both allow       [21]
    That I am dead; there’s nothing more to say.
    Scant news of you, you youngsters, comes this way:
    How shall we judge you, who are still alive,
    But by your works?”
                                       Works? Don’t you think that I’ve    [22]
    Worked hard? We’ve all worked, WILLIAM—sometimes well,
    More often not; and we’ve all gone through Hell
    Trying to follow you. We cannot catch you.
    Yes, some of us, although they could not match you,
    Ran far, and reached the gates of Paradise;
    And others (happy men!)—they heard the price
    CHARON asked for crossing, knew they could not pay,
    And stopped; but I—I would go all the way,
    I thought. And we who crossed—how much it cost us!
    If you played DANTE, WILLIAM, I play FAUSTUS.
     
    “Rather a petty one, GEORG; you have sold               [23]
    Your soul for a pile of faëry gold
    That turns to ashes in the light of day,
    Not true treasure; and you’ve lit your way
    With ignis fatuus; and it will fade                                  [24]
    Sooner than you think, and leave you in the shade.”
        *  *  *
    Your shade, SIR WILLIAM. All of you block our light,  [25]
    So that we cannot see, and cannot fight
    Shadows, not shadows dancing on the wall.
    What would it matter if it should befall
    That your—my i, j, k are no damned use?
    Would that make them less beautiful? I’ll seduce
    Beauty from your workshop; we shall play
    In the fresh, pure air, in the clean light of day
    (Damn this dim gaslight!); and she shall go bare,
    Naked as the newborn—she’ll not wear
    Mechanic’s coveralls. We’ll live and love
    Far from this world, never thinking of
    Utility. I’ll follow and let Beauty lead.
    Knight, if you sleep, you sleep in Hell, indeed.    [26]

                            FINIS

    =======================================================
    NOTES.
    I wrote this poem over 50 years ago, as an exercise in heroic couplets; its age (or rather my age when I wrote it) shows badly in some places. Thirty years later it was published in The Mathematical Intelligencer, with minor editorial interferenceschanges that I have reverted. Ideally the text of the poem would be typeset in one column (not two as in the Intelligencer; the numbered notes below would (again ideally) be unnumbered, set in smaller type, and placed in a parallel column (in the manner of Coleridge in his Rime of the Ancient Mariner).
    The alleged facts of Hamilton’s life, if not entirely accurate, are (I think) at least up to the level of, say, Eric Temple Bell. The alleged facts of Frobenius’s life—other than the fact that he did, indeed, prove that the quaternions are the only (associative) real division algebra other than the reals themselves the the complex numbers—are, as far as I know, base slanders invented by me with considerable poetic license; in particular, I have no reason to believe either than he was mad or that Hamilton was actually his mentor. The structural equations of the quaternions are what they are.
     
    [1] FROBENIUS, his proof completed at last, [2] but fearful of his detractors, [3] remembers the late SIR WILLIAM ROWAN HAMILTON’S poor opinion of Pure Mathematics, [4] and tries to justify himself, [5] without success. [6] He imagines his erstwhile mentor scolding him, [7] then passes into a revery in which he reviews the life and works of the dead Irish knight: [7] his childhood; [8] his adolescence; [9] his young manhood; [10] his long hears of searching for an algebra of vectors [11] in physical space of three dimensions. [12] FROBENIUS grows frenzied, [13] his frenzy increases, [14] and he collapses with a shriek. [15] SIR WILLIAM’S ghost appears and speaks, [16] continuing the earlier scolding. [17] FROBENIUS replies, reminding the ghost of its own quest, [18] fruitless so long, [19] which was so suddenly and unexpectedly successful. [20] The QUATERNIONS! [21] The ghost is not impressed. [22] FROBENIUS contends with the ghost. [23] The ghost, scornful to the end, [24] fades away, [25] leaving FROBENIUS [26] to have the last word.

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