A geographical existence proof

If you spend time in the Mission in San Francisco, you think of Mission Street and Van Ness Avenue as both running north-south, with Mission parallel to and slightly to the west of Van Ness.

But north of there, Mission is one of the major streets downtown, and Van Ness runs through neighborhoods to the west of downtown. That is, Mission is now east of Van Ness.

Therefore, if you assume that each street exists only in one piece, they must cross each other. A sketch of a proof, which works because the streets aren’t too curvy: any line of latitude within the part of the city in question intersects each of Van Ness and Mission, exactly once. Take the difference between the longitude (west of Greenwich, because why not?) at which that line intersects Van Ness and the longitude at which it intersects Mission. At the latitude of, say, 24th Street, this is negative (Van Ness is east of Mission, so has smaller numerical longitude) and at the latitude of, say, Geary, this is positive. By the intermediate value theorem it must be zero at some point, the latitude of the intersection.

(Inspired by being caught in a traffic jam a few days ago, near the intersection of Van Ness and Mission, which I had previously not recognized existed, despite being familiar with both streets on both sides of the intersection.)

4 thoughts on “A geographical existence proof

  1. Alternatively, you can consider South Van Ness to be an extension of Howard, not of Van Ness, and pretend the two blocks between Market and Division don’t have “south” on them 🙂

  2. Apparently South Van Ness was originally called Howard, according to someone who lived there. (As far as I can tell, it wasn’t “South Howard” — South Van Ness is so-called because numbering on Van Ness starts with 1 at Market and goes north.) That really makes more sense – that way it lines up with the continuations of Mission, Folsom, Harrison, and Bryant southward from SoMa into the Mission. But then why isn’t Potrero an extension of Brannan?

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