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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: The flat earth notion
From: Bill Noyce
Date: 2003 Nov 5, 12:12 -0500
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.)
