What is the Staircase Line in a Unit Square?

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In summary, the conversation discusses the concept of the "diagonal paradox" where, when constructing a zig-zag line from one corner to the opposite corner of a unit square, the total distance of the vertical and horizontal parts is always 2, regardless of the number of "stairs" added. This idea is related to the Weyl Tile argument and has applications in philosophy of mathematics and geometric measure theory. A link to the MathWorld page on the topic is mentioned and further resources are provided.
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fourier jr
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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 [tex]\sqrt{2}[/tex]. 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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  • #2
Hummm... fractals?
 
  • #3
The diagonal paradox, again. Use the search button. :smile:
 
  • #4
Minkowski's L1 distance

Taxicab metric?
 
  • #5
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 [tex]\sqrt{2}[/tex]. 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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What is the Staircase Line in a Unit Square?

The Staircase Line in a Unit Square refers to the graphed line that connects the points where the x and y coordinates are both integers within a unit square, or a square with sides of length 1. It is also known as the "Jump Function" or "Step Function".

What does the Staircase Line look like?

The Staircase Line appears as a series of horizontal and vertical lines within a unit square, creating a staircase-like pattern. It is made up of multiple line segments, each of which has a slope of either 0 or undefined.

What is the equation for the Staircase Line?

There is no single equation for the Staircase Line, as it is made up of multiple line segments. However, each segment can be described with a piecewise function, with each segment having a different slope and y-intercept.

What is the purpose of the Staircase Line?

The Staircase Line is often used in mathematics, specifically in introductory calculus, to help students understand the concept of limits and continuity. It is also used in computer graphics to create pixelated images and in statistics to represent data in a discrete form.

Can the Staircase Line be extended beyond a unit square?

Yes, the Staircase Line can be extended beyond a unit square, as long as the x and y coordinates are still integers. This will result in a larger staircase pattern with more line segments. However, it is typically only graphed within a unit square for simplicity and to illustrate the concept of a step function.

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