This theoretical paradox applies to the practical problem of measuring the lengths of coastlines... The answer you get depends greatly on the degree to which the fine details of the coastline are taken into account.

It also figures into Quantum theory by way of Feynman diagrams.

The number of "stairs" tells something similar how polygons start resembling a circle:



Your example shows well that the square root of 2 cannot be a rational number, I guess.

What about a segment with a definite length that is then made into a circle, which suddenly has an irrational number as a factor? Putting the segment into a circle shouldn't alter it's length in the least

Take a line segment of length one, then the circumference of the circle is also one: 1=pi*D means the diameter is irrational. :smile:

Or not??

I thought not. I thought it's indefinite because it's multiplied by an irrational numbers. I don't see how a length can be indefinite though. Is it legit to round it to 1?

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.

"Seems to more and more approximate . . . "

The key to dissolving the apparent paradox is to calculate the error in approximation for each tiny right triangle, then add them up. If each step has $\Delta x=\Delta y=\tfrac{1}{n}$ then the increment of error is
$\Delta \varepsilon =\left( 2-\sqrt{2} \right)\Delta x\text{ }\Rightarrow \text{ }Total\text{ }error=n\Delta \varepsilon =2-\sqrt{2}$

Each member of the sequence of functions defined on [0,1],

${{f}_{n}}(x)=\tfrac{1}{10\sqrt{n}}Sin(n\pi x),\text{ }x\in [0,1]$,

is a smooth, wiggly curve oscillating about the line segment [0,1], getting closer and closer to that line as n increases in value. The length of the nth curve is

${{L}_{n}}\gt\tfrac{\sqrt{n}}{5}\text{ }\to \text{ }\infty$,

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.

This staircase paradox is of type 2). It seems contradictory because we expect that the length of the staircase ought to approach the length of the diagonal line that it approaches. It's not really contradictory, simply because that expectation is not right. (Mathematics is filled with many paradoxes of this kind.)

If a sequence C_1, C_2, C_3, ... of curves approaches a line L in such a way that, not only do the points of the curve approach the points of the line, but also the direction of the curve approaches the direction of the line, then the lengths of the C_n's will also approach the length of L. Otherwise it just isn't necessarily true, no matter how surprising this may be.

If you were to cite any degree of magnification, I could find an n such that the approximating figures would appear to be the curve in question. :nerd:

I know, silly games.

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

Yes, but in the real world... what happens when the step length approaches the Planck length?

I tend to look at this through the lens of approximation theory which at times breaks down a process defined over an interval, to minute steps. Riemann integration theory requires such analysis, in which the area under a continuous curve is approximated by thin rectangles, then the error in approximation shrinks to zero as the number of rectangles increases without bound.

One of the celebrated theorems in this subject is Stone-Weierstrass, which says continuous functions on an interval can be approximated by polynomials.

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

Approximate(ly) a straight line can be rephrased exactly as like but not identical to a straight line. In other words, the staircase never ever actually becomes a straight line. It would've been a bona fide paradox if the staircase actually became a straight line but the word "approximate" is there for a reason.

Is this really a paradox? It's like saying Roger Federer "approximates", in appearance, to Quentin Tarantino and then saying Roger Federer is Quentin Tarantino.

Maybe the staircase times infinity equals the straight line exactly

The key to dissolving the apparent paradox is to calculate the error in approximation for each tiny right triangle, then add them up. — jgill

This is neat.

I just wrote the staircase after n refinements as a sum of 2n scaled indicator functions.

$S(n)=\sum_{i=1}^{2n} \frac{i-1}{2n} \big(\frac{i-1}{2n},\frac{i}{2n}\big)$.

Let the mapping of $S(n)$ to its arclength be $A(S(n))$. Which is always 2 for any finite n.

But $S(n)$ is discontinuous.

So the limiting procedure:

$\lim_{n\rightarrow \infty} A(S(n))$

doesn't let you take the limit inside the $S(n)$ bracket (the limit of arclength is not necessarily the arclength of the limit).

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.

There is no "more and more," although there can be be a lot of steps — tim wood

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.

The wiggly curve that uniformly converges to the line segment [0,1] is more entertaining, for its length becomes infinite.

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

Yes. These two instances rely upon point of perspective, as Mr. Wood explains. Tarski-Banach seems to depend upon the controversial Axiom of Choice, which doesn't come into play here.

What happens to all the corner points in the stairs as the number of steps increases without bound? :chin:

What happens to all the corner points in the stairs as the number of steps increases without bound? :chin: — jgill

I don't think computers can fully handle these questions because they deal with uncoutable infinities instead of countable ones

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

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.

I find no error in this.

Nice.

he number of "stairs" tells something similar how polygons start resembling a circle: — ssu

The difference is that in the case of polygons approximating a circle, with each successive step the error decreases (don't ask me for a proof - it's pretty messy, from what I remember), whereas in the case of the staircase the error stays the same throughout.

I find no error in this. — tim wood

There is no error, hence the apparent paradox.

Here is another take I just thought of. We can define two functionals, one that gives some measure of the distance between the points of the stair functions and the diagonal (e.g. the average distance), and the other that gives the length of those functions. The shape functional steadily converges to zero, but the length functional does not.

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

For those that don't speak half-math, the staircase has corners where the tangent to the corner is not defined. A tangent, in 2D, is a line that is "stuck" up against the object you're interested in, intersecting one point but no other points, at least "locally", but, crucially, in only one way. Ex. if you stick a line against a circle it can only be in one way, forming a T with the radius line; a line with a different angle to the radius will intersect more points in the circle, no way to escape it.

Discontinuous above refers to points with undefined tangents, not an actual cut in the line that draws the staircase, which is a continuous line in the normal sense of being able to draw it continuously without lifting the pencil off the paper.

It is in trying to draw the derivative, all the tangent values, where the discontinuities appear. The tangent to the vertical part of the staircase is also just a vertical line, then meets a point without a undefined tangent at the corner, and then suddenly switches to the tangent being just a horizontal line following the horizontal part of the staircase; of course best to use a coordinate system at some angle to the staircase as otherwise the vertical lines are at the same x coordinate as the corners and it's not so clear what the jumping is.

A corner has no unique tangent line, which means has no derivative, which means what seems similar and doable in other calculus contexts does not work with the staircase, basically.

A better paradox, I think, is accept it just stays the same length, but we allow the staircase to get such small steps that it "occupies" the same area as the line, in the sense of not allowing the staircase steps to cross some definition of the "closest" lines adjacent to the approximating line, i.e. building the staircase to stay in a lane of area zero, but still insist it has length 2.

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.


Back to the Diagonal:

Imagine a straight line that rests on the outer stair corners, from the bottom most step to the topmost step. As the steps shrink that line more closely resembles the hypotenuse of the large triangle of length square root of two. Now, what can you say about all those outer corner points, which lie on this evolving line, as the number of steps becomes infinite?

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

What paradox is this? I tried to look it up but only found the staircase paradox itself. Is there a path that converges uniformly to the unit interval but whose length is infinite?

It's in the second half of .

There must be some neat identity for elliptical functions at work here, because otherwise I wouldn't know how to calculate such a limit.


By the way, and 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. Consider a similar example, where in place of straight lines there are smooth curves. I'll use half-circle arcs for simplicity:



No corners here, the curve is everywhere differentiable (although the second derivative does jump around at the intersections). As with the staircase, the amplitude of the wave tends to zero as the number of crests increases without bound. But as with the staircase, the length of the curve does not approach the length of the diagonal. We don't even have to do the calculation to see that the length of the curve does not depend on the number of crests (this is because the length of each half-circle is proportional to its diameter, and the total length of all diameters is the length of the diagonal). And so the length of the wavy curve is always $\frac{\pi}{\sqrt{2}}$.

What's more, with a simple modification we can make the length of the wavy curve increase without bound, just as in @jgill's example. Just replace half-circles with ellipses whose major axis is perpendicular to the diagonal and scaled by a factor of $\sqrt{n}$.

No corners here, the curve is everywhere differentiable (although the second derivative does jump around at the intersections). — SophistiCat

These jumps are what I was referring to as not everywhere differentialable.

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

However, I'm not saying that differentiability is the end of it, only that non-differentiability is a good indication that limits may not apply as we might intuitively expect from cases of continuously differentiable functions, and my main purpose was to explain what discontinuous meant in the context.

Non-diferentiable points can be approached in various ways that may or may not make sense given the context of the problem. Physicists, for instance, may approach these points with a important theorem called "whatever, I don't care, we'll figure it out later, or someone will eventually I'm sure".

To "solve" these problems we'd need to choose an axiomatic system. If we view these problems in Euclidean geometry, there's no axiom of infinity, so the question doesn't really make sense (you can keep making smaller staircases or smaller circles, but you can't say you have done so "infinitely many times" and each staircase or circle is indistinguishable, in some sense, from a single point along a approximate line).

The half-circle wave is smooth though, i.e. the tangent (first derivative) exists everywhere. I used it for simplicity, but if we want an honest to goodness infinitely differentiable curve, we can use the sine function as in John's other example - just scale it by an additional factor of $\frac{1}{\sqrt{n}}$ to make it behave. (I think this will work...)

Yes, that's why I said "non-differentiability is not the end to it".

But it's a good litmus test (that intuitions may not apply).

Without the non-differentiable points that are perpendicular to the axis of the line, however, it's not exactly the same problem though.

But you need to select axioms, re-define the problem as sets for instance, and so on, to then "prove" an answer relative those axioms. Without doing that my response will be "maybe".

My main purpose, as mentioned, was just to explain the definition of "discontinuous" and that normal calculus concepts may not apply. I do not have a problem with a system where the limit in these questions does not make sense, is just undefined, nor a problem with a system where the limit does make sense (and even different answers in different systems would not bother me).

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

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.

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".

My main purpose, as mentioned, was just to explain the definition of "discontinuous" and that normal calculus concepts may not apply. — boethius

Er, your terminology is all over the place. A continuous function has left and right limits converging to each of its points. The staircase function is, strictly speaking, discontinuous as it is pictured in the OP, but that is just an artefact of the coordiate system. If you tilt the X-Y axes, it will become continuous.

A differentiable (or smooth) function has the first derivative at each point; the half-circle function is differentiable (again, modulo axis orientation).

An infinitely differentiable function has all derivatives; the sine function is infinitely differentiable.

There are also piecewise- versions of all these (piecewise-continuous, etc.).

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

Yes, that's just what I did with the half-circle curve, and I think the sine curve (with proper scaling) would work as well.

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

Interesting question, but beyond my modest pay grade, I am afraid :)