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Monday, March 6th, 2017

    Time Event
    The Kiss Precise by Frederick Soddy
    For pairs of lips to kiss maybe
    Involves no trigonometry.
    This not so when four circles kiss
    Each one the other three.
    To bring this off the four must be
    As three in one or one in three.
    If one in three, beyond a doubt
    Each gets three kisses from without.
    If three in one, then is that one
    Thrice kissed internally.

    Four circles to the kissing come.
    The smaller are the benter.
    The bend is just the inverse of
    The distance form the center.
    Though their intrigue left Euclid dumb
    There's now no need for rule of thumb.
    Since zero bend's a dead straight line
    And concave bends have minus sign,
    The sum of the squares of all four bends
    Is half the square of their sum.

    To spy out spherical affairs
    An oscular surveyor
    Might find the task laborious,
    The sphere is much the gayer,
    And now besides the pair of pairs
    A fifth sphere in the kissing shares.
    Yet, signs and zero as before,
    For each to kiss the other four
    The square of the sum of all five bends
    Is thrice the sum of their squares.

    The Kiss Precise (generalized) by Thorold Gosset

    And let us not confine our cares
    To simple circles, planes and spheres,
    But rise to hyper flats and bends
    Where kissing multiple appears,
    In n-ic space the kissing pairs
    Are hyperspheres, and Truth declares,
    As n + 2 such osculate
    Each with an n + 1 fold mate
    The square of the sum of all the bends
    Is n times the sum of their squares.

    The Kiss Precise (Further Generalized) by Fred Lunnon

    How frightfully pedestrian
    My predecessors were
    To pose in space Euclidean
    Each fraternising sphere!
    Let Gauss' k squared be positive
    When space becomes elliptic,
    And conversely turn negative
    For spaces hyperbolic:
    Squared sum of bends is sum times n
    Of twice k squared plus squares of bends.

    On August 15, 1936, only a few months after Soddy's poem had been published in Nature, Thorold Gosset sent a copy of the poem to Donal Coxeter on the occasion of his wedding in the Round Church in Cambridge. Gossett enclosed in his wedding congratulations, and extension of the poem to the higher dimensions


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