There's a nice example of intransitive geometry in the latest Pussycat Dolls video ("Hush hush").
No, really, there is. It's the bit where the girls are on four staircases attached to the sides of a cube, that each have a different local direction of "down". The "stairwell" section of the video starts at about 58 seconds in and goes on until about a minute thirty. While you're waiting for it to start you'll have to put up with the sight of Nicole Scherzinger nekked in a bathtub making "ooo, yeah" noises for nearly a minute, though. Sometimes doing research for this blog is really tough.
The video seems to be inspired by the famous "Relativity" lithograph by M. C. Escher, which had three intersecting sets of stairs and platforms set into three perpendicular walls, as a piece of "impossible" architecture (physically you could build it, but you wouldn't be able to walk on all the surfaces as the people do in the illustration).Escher's illustration was incredibly influential, and as well as the Pussycat Dolls video (!), there are some more literal tributes online, including Andrew Lipson's recreation of the scene using Lego, part of the 1986 movie Labyrinth, and a funny short video called Relativity 2.0, that has people trapped in a nightmarish Escherian shopping mall.
If you know of any other especially good ones, please add them to the end of this post as a comment!
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