In the paper "Explorations into Euclidian and Taxi Cab Minimum Distances" by Hudson Cooke he explains our size problem and gives proof and reason for why distances between two points

*are*the shortest. He explains that you can simply find the distance between two points on an open plain using the Pythagorean theorem. He then says that in a Taxi Cab plain, (where you can't draw lines diagonally, but instead have to follow lines all with 90 degree angles) if both point A and B are at opposite corners of a plain then there is only one distance that can take you from one point to another. It doesn't matter which specific route you take it will end up being the same distance as any other route.

Then I realized that if you draw your route and make unnecessary turns along the way then it will obviously take much longer and be a longer line drawn from point A to point B.

I'm probably just completely missing the point that Hudson is making but if anyone else has something to say about this please do, because if we figure this out then I think we will have a definite answer to the size problem.

Here's the link: https://dl.dropbox.com/u/3450194/Hudson%20final.pdf

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