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My intuition is that the layout of Burning Man's city "grid" optimizes for the smallest sum of all distances between any two points on the map.

Am I correct? Is the proof obvious? Or is there another way to lay out a city to minimize the distance between any two arbitrary points?

The constraints are that a city "block" is a fixed size, there is an "empty" central public space of a certain size and you must travel only over the roads between blocks when moving between two points (any path through the public space is acceptable though.) Feel free to assume the streets on the map below completely encircle the public space, that's not really germane to the practical problem. Also we can ignore what's marked on the map as Centre Camp/Rod's Road.

Edit: here's the picture linked to:

Burning Man's city grid

Willie Wong
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Robert Atkins
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  • I removed the ‘PDF’ notation: the link is actually to a JPG. – Brian M. Scott Sep 10 '11 at 07:28
  • I think we want to maintain the semi-annulus shape and total area. Each point is either a road or a habitat, and the road-area and habitat-area are also constant. The goal is to minimize the expected length of a shortest path (not intersecting any habitat) between any two uniformly selected points on a road. Is this a correct interpretation of the problem? – Dan Brumleve Sep 10 '11 at 10:09
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    Following my previous comment, the best case would be a straight-line distance, which can be approximated by making the roads ever thinner and the habitats ever smaller. For this to be a sensible question, we need more constraints. Must there be a certain number of habitats? Do the roads have a minimum width? Do the habitats have a minimum area? – Dan Brumleve Sep 10 '11 at 10:18
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    Concerning the tags - if this is trigonometry, it isn't graph theory, and if it's graph theory, it isn't trigonometry. Maybe someone who understands the problem better than I do can re-tag. I'll add optimization. – Gerry Myerson Sep 10 '11 at 10:25
  • Is there a reason why a simple disk wouldn't work? – Gerben Sep 10 '11 at 10:26
  • I'm not a mathematician so I wasn't sure which tag might be appropriate and both the above sounded like they were in the ball park. Please feel free to re-tag. I've updated the question to include some constraints. – Robert Atkins Sep 10 '11 at 18:04
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    @Dan: you're close, the question is more like if we drop the constraint of the semi-annular shape, keeping the habitat area constant and adding whatever length of road necessary, is there a shape that gives us the smallest expected distance between any two given habitats. – Robert Atkins Sep 10 '11 at 18:16
  • Robert, it is a disc, as Gerben says. – Dan Brumleve Sep 14 '11 at 12:54
  • As I edited the question to say, the section taken out of the actual city map is not germane to the question—it is effectively a disc already. What you're saying is you agree with my intuition, I'm really looking for a (reference to a?) proof that the intuition is correct. – Robert Atkins Sep 17 '11 at 09:25

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