Off-topic circumnavigating the earth in a mile or less [was Re: Significant digits in a float?]

[Chris Angelico]

On Wed, Apr 30, 2014 at 11:14 PM, Ethan Furman <ethan at stoneleaf.us> wrote:
>> Any point where the mile east takes you an exact number of times
>> around the globe. So, anywhere exactly one mile north of that, which
>> is a number of circles not far from the south pole.
>
>
> It is my contention, completely unbacked by actual research, that if you
> find such a spot (heading a mile east takes you an integral number of times
> around the pole), that there is not enough Earth left to walk a mile north
> so that you could then turn-around a walk a mile south to get back to such a
> location.

The circle where the distance is exactly one mile will be fairly near
the south pole. There should be plenty of planet a mile to the north
of that.

If the earth were a perfect sphere, the place we're looking for is the
place where cutting across the sphere is 1/π miles. The radius of the
earth is approximately 4000 miles (give or take). So we're looking for
the place where the chord across a radius 4000 circle is 1/π; that
means the triangle formed by a radius of the earth and half of 1/π and
an unknown side (the distance from the centre of the earth to the
point where the chord meets it - a smidge less than 4000, but the
exact distance is immaterial) is a right triangle. Trig functions to
the rescue! We want latitude 90°-(asin 1/8000π). It's practically at
the south pole: 89.9977° south (89°59'52").

Are my calculations correct?

ChrisA