[T-hawk]

Very cool of course.

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

I understand the difference between latitude and longitude lines. A meridian of longitude is a great circle centered at the Earth's center; a line of latitude is a small circle (the analogue of a chord in a 2D circle on a plane) whose center lies north or south of the Earth's center in three dimensions. Longitude lines divide a sphere like slices of an orange, converging at poles; latitude lines divide a sphere like a tomato slicer and do not converge.

There's actually two "poles"; aside from the one in Uzbekistan that everyone is seeing, there's another in the South Pacific Ocean at the antipodal point from Uzbekistan. So the avenues are being treated as meridian lines; great circles. Would it be more accurate to extrapolate avenues as parallel small circles?

We could test this theory by inspecting whether Manhattan's actual grid respects the curvature of the Earth. If the avenues are closer together at the northern end of the island, then the avenues actually do behave as meridians. If not, then the extrapolated avenue lines should be small circles and would not converge. You'd still have a pole in Uzbekistan, where the last street becomes an arbitrarily small circle, but just one avenue line through it. (I gotta run at the moment but will throw some trigonometry at this later.)

[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]

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.

[kilovoltaire]

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.

[kilovoltaire]

Haha interesting point about measuring the avenues to see which method is more accurate.

I was primarily inspired by lines of latitude (streets) and longitude (avenues), but I'd definitely be interested to see what it looks like when both are treated as concentric small circles.