Proof That Pi = 2: Intuitively Wrong?

  • Context: Graduate 
  • Thread starter Thread starter SpY]
  • Start date Start date
  • Tags Tags
    Pi
Click For Summary

Discussion Overview

The discussion revolves around a proof that suggests pi equals 2, which participants find intuitively incorrect. The conversation explores various mathematical approaches and reasoning related to arc lengths, limits, and the properties of paths in geometry.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that the proof's final step incorrectly concludes that the sum of arc lengths equals the length of the rod.
  • Others present alternative methods of calculating distances using piecewise linear paths, asserting that these methods also yield a total distance of 2.
  • A participant questions the meaning of "necessarily" in the context of convergence of arc lengths to limit paths, suggesting that it may be true in some cases but not universally.
  • Concerns are raised about the nature of the limiting process and the differentiability of the resulting curves, with some asserting that the broken arc line is not continuously differentiable.
  • Participants discuss the implications of using infinitesimal segments in approximating distances, with some arguing that this leads to misconceptions about limits and ratios in triangles.
  • There is a contention regarding the evaluation of limits involving the hypotenuse and the lengths of the other sides of triangles, with differing views on the correctness of these evaluations.

Areas of Agreement / Disagreement

Participants do not reach consensus on the validity of the original proof or the interpretations of limits and distances. Multiple competing views remain regarding the nature of convergence and the properties of the paths discussed.

Contextual Notes

Limitations in the discussion include unresolved definitions of limit paths, the nature of convergence in piecewise differentiable paths, and the assumptions underlying the use of infinitesimals in distance calculations.

  • #31
Of course that's not true, and I never said that.
 
Mathematics news on Phys.org
  • #32
adriank said:
Of course that's not true, and I never said that.

I edited my post.
 
  • #33
Right, and it's precisely because \gamma_n'(t) doesn't converge that the lengths don't converge to the length of the limit. But every length involved exists.

This is a classic example where some behaviour that is constant during some limiting process is not preserved once you take the actual limit. (In other words, the "length of a path function" is not "continuous".)
 
  • #34
I was simply arguing that the lengths do not converge at all, not just to that particular value.
 
  • #35
The derivatives don't converge, but the lengths certainly do, since they're the constant \sqrt2! They don't converge to 1, that's all.
 

Similar threads

Undergrad Is e^pi a unique transcendental, or perhaps not transcendental?
  • · Replies 1 ·
Replies
1
Views
2K
Area surface of revolution (rotating this astroid curve around the x-axis)
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Possible simple formula for ellipse circumference
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Determine ## \beta ## as a function of ##\theta## linkage
  • · Replies 2 ·
Replies
2
Views
2K
MHB Angular and Linear Speed....Part 1
  • · Replies 4 ·
Replies
4
Views
2K
Circle approximation of a wave pulse on a string or spring
  • · Replies 29 ·
Replies
29
Views
2K
MHB DeMoivre's Theorem express (sqrt(2)/2 + sqrt(2)/2 i)^8 in a+bi form
  • · Replies 2 ·
Replies
2
Views
2K
Solving Semi-circle Inside Triangle Geometry Problem
  • · Replies 13 ·
Replies
13
Views
4K
Undergrad Simple thought experiment with Stefan-Boltzmann law: energy
  • · Replies 7 ·
Replies
7
Views
965
MHB Find the exact shaded area of the region in 4 overlapping circles
  • · Replies 1 ·
Replies
1
Views
3K