TaxiCab - Discrete vs. Continuous (taxicab geometry) •...

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Unformatted text preview: Discrete vs. Continuous (taxicab geometry) • There is a strong tradition within academia (and science in particular) to assume that most of the time, it is OK to use a continuous model of a system, even if the system at issue is inherently discrete. The typical “justiFcation” is that “in the limit, discrete becomes continuous . . .” • What follows here is sort of a joke, but contains ideas worth thinking about . . . • Let’s consider “distance” -- but in a special case. The “geometry” here is often called “taxicab geometry,” using “taxicab distance.” • Consider “distance” in a city, where we must follow streets. We want to go from A to B: A B 1 1 • The sides of the square are length 1, and we can either go this way: A B 1 1 • Or this way: A B 1 1 • But either way, the “taxicab distance” from A to B is 2: A B 1 1 • Now suppose there is “redevelopment,” and the city block is subdivided: A B 1/2 1/2 1/2 1/2 A B 1/2 1/2 1/2 1/2 • We can go from A to B in various ways: • We can go from A to B in various ways: A B 1/2 1/2 1/2 1/2 • We can go from A to B in various ways: A B 1/2 1/2 1/2 1/2 • We can go from A to B in various ways: A B 1/2 1/2 1/2 1/2 • We can go from A to B in various ways: A B 1/2 1/2 1/2 1/2 • We can go from A to B in various ways: A B 1/2 1/2 1/2 1/2 • We can go from A to B in various ways: A B 1/2 1/2 1/2 1/2 • But the “taxicab distance” from A to B is still exactly 2: A B 1/2 1/2 1/2 1/2 • Suppose we “subdivide” more: A B 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4 • Again, there are various ways we can go from A to B: A B 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4 • Again, there are various ways we can go from A to B: A B 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4 • Again, there are various ways we can go from A to B: A B 1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4 •...
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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TaxiCab - Discrete vs. Continuous (taxicab geometry) •...

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