Next message: Bruce Stark: "Re: The flat earth notion"
> 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.)
