On: The Manhattan Grid extended to every point on Eart...

[stouset]

> Why do the avenue lines meet at an antipode? In Manhattan, they are parallel, they don't converge.

Sure they do. Manhattan isn't a plane; it's a region on the surface of a sphere. From Wikipedia[1], "In the spherical plane, all geodesics are great circles. ...all great circles intersect each other."

Clearly, it's the non-intersecting streets that are at fault here. They should intersect, but don't.

[1]: http://en.wikipedia.org/wiki/Parallel_(geometry)#Spherical

[T-hawk → stouset]

That the avenues converge is your assumption. The avenues may not be geodesics. They may be parallel small circles. Manhattan is of course a region on the surface of a sphere, but the avenues could behave either way.

Throwing together that promised trigonometry, with some approximations (using round numbers, ignoring the oblateness of the sphere, ignoring the skew of Manhattan's grid from due north-south):

  40° = approx latitude of Manhattan's south end
  40.27° = approx latitude of Manhattan's other end 30 km north
  40000 km * cos(40°) = circumference of the 40° parallel
  40000 km * cos(40.27°) = circumference of the 40.27° parallel
  0.996 = ratio of the distance between avenues at the north end compared to the south
  300 m = assumed approximate distance between avenues at the south end
  298.8 m = expected distance between avenues at the north end

If the avenues are geodesics, they should be 1.2m closer to each other at the north end of Manhattan than the south end. Unfortunately a difference that small is probably essentially noise and below statistical significance to actually measure; the width of a sidewalk or a bicycle lane.

[hrldcpr → stouset]

You're right about the avenues, but as for the streets I decided to make them concentric circles, completely analogous to lines of latitude. So they never intersect.