What is the Staircase Line in a Unit Square?

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The discussion centers on creating a zig-zag staircase line within a unit square, illustrating that the total distance of the vertical and horizontal segments always sums to 2, regardless of how many segments are added. This phenomenon highlights a paradox where, despite the line visually approximating a diagonal, its length remains constant at 2, while the actual diagonal measures √2. The conversation touches on concepts like Minkowski's L1 distance and the Weyl Tile argument, linking them to broader implications in the philosophy of mathematics and geometric measure theory. Participants also reference external resources for further exploration of these mathematical principles. The discussion emphasizes the intriguing nature of geometric constructs and their implications in theoretical mathematics.
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Take the unit square & make a zig-zag line like a staircase from one corner to the opposite one. Then the total distance if you add up the vertical parts & & horizontal parts is 2. Even if you make trillions & trillions of 'stairs' the sum of all the vertical parts & horizontal parts is still 2 even though the graph would look more & more like a diagonal line, whose length of course is \sqrt{2}. Someone mentioned this example before & put up a link to the mathworld page on it but I couldn't find it & nothing I searched for seemed to work. :confused:
 
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Hummm... fractals?
 
The diagonal paradox, again. Use the search button. :smile:
 
Minkowski's L1 distance

Taxicab metric?
 
Weyl Tile argument

fourier jr said:
Take the unit square & make a zig-zag line like a staircase from one corner to the opposite one. Then the total distance if you add up the vertical parts & & horizontal parts is 2. Even if you make trillions & trillions of 'stairs' the sum of all the vertical parts & horizontal parts is still 2 even though the graph would look more & more like a diagonal line, whose length of course is \sqrt{2}. Someone mentioned this example before & put up a link to the mathworld page on it but I couldn't find it & nothing I searched for seemed to work. :confused:

It's related to the "Weyl Tile argument", which is discussed in some books on philosophy of mathematics, and even some web pages:
http://faculty.washington.edu/smcohen/320/atomism.htm
The argument as stated there isn't serious, but this has serious applications to why naive "quantization" of space won't work. See spin networks for a more sophisticated approach: http://math.ucr.edu/home/baez/penrose/

It's also related to a "paradox" in geometric measure theory, which is probably closer to the applications you have in mind, huh? See p. 129 of Spivak, Calculus on Manifolds.
 
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