What is the Staircase Line in a Unit Square?

[fourier jr]

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.

 

[Werg22]

Hummm... fractals?

 

[radou]

The diagonal paradox, again. Use the search button.

 

[fopc]

Minkowski's L1 distance

Taxicab metric?

 

[Chris Hillman]

Weyl Tile argument

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.

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.

 

[fourier jr]

aha i found what I was looking for:
https://www.physicsforums.com/showthread.php?t=80796
http://mathworld.wolfram.com/DiagonalParadox.html
thx everybody