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    Re: The flat earth notion
    From: Bill Noyce
    Date: 2003 Nov 5, 12:12 -0500

    > A loxodrome only reaches either pole after
    > _infinite_ time, gradually spiralling in towards the pole but never
    > quite getting there.
    Although a loxodrome spirals around the pole an infinite number of
    times, it has only a finite length.  That's one of the funny things
    about it.  To see this intuitively, imagine it has gotten close enough
    to the pole that the earth can be considered to be flat.  In this
    case, the loxodrome degenerates to a planar logarithmic spiral --
    a spiral that cuts each radius at the same angle.  The logarithmic
    spiral has the property that there is a constant factor f between
    the length of one 360-degree segment and the next.  Therefore, the
    total length of the inward spiral, starting from where the path is
    one mile long, is 1 + f + f^2 + f^3 + ..., which is finite as long
    as f<1.  (The familiar case where f=0.5 adds up to 2.)

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