March 8, 2008

  • Imaginary Numbers:

    Tilting at
    Whirligigs

    From a New York Times list

    of literary "signature passages" --

    Don Quixote -- 'wasteland and crossroad places'

    An answer:

    "The whirligig of time"
    -- Shakespeare, Twelfth Night

    and

    Log24, Twelfth Night, 2006:

    Hamilton's Whirligigs

    Hamilton's Whirligigs: The 8-element quaternion group as a subgroup of the 48-element group GL(2,3)

    Click image to enlarge.

    Related material:

    Rotation in the complex plane.

    The plane was discovered
    in the late 1700's by Wessel:

    Caspar Wessel

    by J.J. O'Connor
     and E.F. Robertson:

    "Wessel's paper [in Danish] was not noticed by the mathematical community until 1895... A French translation...
    was published in 1897 but an English translation of this most
    remarkable work was not published until 1999 (exactly 200 years after
    it was first published)....

    We have called Wessel's work remarkable, and indeed although the credit has gone to Argand, many historians of mathematics feel that Wessel's contribution was [1]:-

    ... superior to and more modern in spirit to Argand's.

    In the [1] article the approaches by Argand
    and Wessel are compared and contrasted. Of course Wessel was a surveyor
    and his paper was motivated by his surveying and cartography work:-

    Wessel's development proceeded rather directly from geometric
    problems, through geometric-intuitive reasoning, to an algebraic
    formula. Argand began with algebraic quantities and sought a geometric representation
    for them. ... Wessel's initial formulation was remarkably clear,
    direct, concise and modern    . It is regrettable that it was not
    appreciated for nearly a century and hence did not have the influence
    it merited.

    However more is claimed for Wessel's single mathematical paper than the first geometric interpretation of complex numbers. In [3]
    Crowe credits Wessel with being the first person to add vectors. Again
    this shows the depth of Wessel's thinking but again, as the paper was
    unnoticed it had no influence on mathematical development despite
    appearing in the Memoirs of the Royal Danish Academy which by any standard was a major source of publications....

    1. ... Biography in Dictionary of Scientific Biography (New York 1970-1990).

    3. M.J. Crowe, A History of Vector Analysis (Notre Dame, 1967)."