However, as the size of each step diminishes the staircase seems to more and more approximate a straight line, which has length the square root of two. — jgill
However, as the size of each step diminishes the staircase seems to more and more approximate a straight line, which has length the square root of two. — jgill
Am I missing something? For so long as there are stairs, the length is 2. Doesn't matter how many there are. There is no "more and more," although there can be be a lot of steps. The appearance of the staircase as "approaching" a straight line is in terms of the math an illusion of perception. — tim wood
However, as the size of each step diminishes the staircase seems to more and more approximate a straight line, which has length the square root of two. — jgill
The key to dissolving the apparent paradox is to calculate the error in approximation for each tiny right triangle, then add them up. — jgill
There is no "more and more," although there can be be a lot of steps — tim wood
The word "paradox" has two meanings: 1) something that is true but self-contradictory, and 2) something that is true and seems self-contradictory, but in fact isn't — Daz
What happens to all the corner points in the stairs as the number of steps increases without bound? :chin: — jgill
This I don't buy. The combined length of the line segments that represent the lengths respectively of the steps and risers that comprise the stairs is just two. There is no (other) "wiggly" line. There are only steps, however large or small, and together they cross, as steps, a distance of one horizontally and one vertically.A) As individual steps shrink in size, the inside corner point - the part of the step furthest from the imaginary limiting line if that line is visualized as above the steps - grows closer to that line. So, yes, there is "more and more". However, the total or accumulated error remains large.
More and more what? Your "seems" covered it for me. But now it seems you moved away from that. I claim without argument that the sum of the horizontal and vertical distances is always two, no matter how many steps the distance is chopped into - and I think that's immediately evident to thought.
B) The wiggly curve that uniformly converges to the line segment [0,1] is more entertaining, for its length becomes infinite. — jgill
he number of "stairs" tells something similar how polygons start resembling a circle: — ssu
I find no error in this. — tim wood
For those who don't speak math, the last bit: "letting the staircase get closer and closer to the line" doesn't entail "the length of the staircase gets closer and closer to the length of the line" since the staircase has discontinuous jumps in it. — fdrake
B) The wiggly curve that uniformly converges to the line segment [0,1] is more entertaining, for its length becomes infinite. — jgill
This I don't buy. The combined length of the line segments that represent the lengths respectively of the steps and risers that comprise the stairs is just two. There is no (other) "wiggly" line. There are only steps, however large or small, and together they cross, as steps, a distance of one horizontally and one vertically. — tim wood
Wrong paradox, Tim. The wiggly curve converges uniformly to the line segment [0,1] while its length tends to infinity. Sorry I don't have the image. — jgill
No corners here, the curve is everywhere differentiable (although the second derivative does jump around at the intersections). — SophistiCat
Non Differentiable Functions. ... We can say that f is not differentiable for any value of x where a tangent cannot 'exist' or the tangent exists but is vertical (vertical line has undefined slope, hence undefined derivative). — first web search hit
By the way, ↪fdrake and ↪boethius seemed to suggest that the key to the staircase "paradox" is in some pathology of the shape, namely its corners, where the curve is not differentiable. But this is not so — SophistiCat
My main purpose, as mentioned, was just to explain the definition of "discontinuous" and that normal calculus concepts may not apply. — boethius
You are right, it's a sufficient condition for the failure of the arc-length functional to respect the limiting procedure, not a necessary one. I believe the staircase could be approximated by some differentiable curve (replace the discontinuities with regions of sufficiently high growth, I believe polynomials would work) and cause the same issues. — fdrake
Do you know a sufficient and necessary condition that characterises this sort of pathology? Other than stating "the arc-length map of the limit of the approximating series of functions is not necessarily the limit of the arc-length map of the approximating series of functions". — fdrake
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