Will limit of discrete steps give Pythagoras theorem?

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SUMMARY

The discussion centers on the relationship between discrete steps and the Pythagorean theorem in the context of right triangles. It asserts that moving from one corner of a right triangle to another using only vertical and horizontal steps results in a distance equal to the sum of the two sides, not the hypotenuse. The conversation highlights the mathematical principle that the limit of the lengths of curves does not equate to the length of the limit curve, emphasizing the necessity of smoothness for such equivalence. This argument serves as a critique of the concept of discrete space.

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  • Understanding of the Pythagorean theorem
  • Familiarity with limits in calculus
  • Basic knowledge of geometry, particularly right triangles
  • Concept of smoothness in mathematical analysis
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krishna mohan
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Hi...

It is an easy to see fact that, instead of moving along the hypotenuse of a right triangle, one starts from the lower corner and reach the upper corner moving only along the directions of the other two sides, i.e only vertically and horizontally and not diagonally...the distance moved is just the sum of the other two sides...even if we take a limit of the steps being infinitesimal...
Somewhere I remember having read of this being an argument against discreteness of space...

Can someone throw some light on this topic??
 
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I don't understand how that works. The problem is that the limit of lengths of curves isn't necessarily the length of the limit curve (I think you need smoothness for this). To illustrate, in your example, the limit of your "curves" is indeed the hypotenuse, but all of the intermediate curves will have length (base x height), so the limit of the lengths aren't equal to the length of the limit curve. Hope that makes sense!
 

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