Estimation quiz!

[Thomas Okken]

(08-07-2020 06:57 AM)EdS2 Wrote:  

(08-04-2020 12:33 PM)Thomas Okken Wrote:  \( { \sqrt{6 - \sqrt{2}} \over 2 } \pi R \)

An interesting formula! (Not an estimation, as such... but what does it tell us? From pole to pole on the surface is 12000 miles more or less (or 20k km) and this formula gives us about 13500 miles - not a great deal more. That surprises me, a little. More on the loxodrome here.)

My reasoning is that while traveling at a certain compass heading, the ratio between the total distance traveled and the distance traveled north is constant (although it now looks to me like the value of that constant that I figured, sqrt(6-sqrt(2))/2, is wrong, and the actual value is slightly greater). And since NNW is only 22.5° from N, that ratio is not much greater than one.

The path traveled itself is interesting because it spirals around the poles infinitely many times without ever reaching them, yet nonetheless has a finite length. You can visualize it by extending a Mercator projection infinitely far north and south; a loxodrome is then just a straight line that wraps around the side edges infinitely many times.