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

Me too, I do not have the analysis fu for that.

Er, your terminology is all over the place. A continuous function has left and right limits converging to each of its points. — SophistiCat

My terminology?

I just gave you the definition twice of continuously differentiable, the derivative function is continuous.

I've made it pretty clear I'm talking about "differentiable" in the sense of differentiable.

If you tilt the X-Y axes, it will become continuous. — SophistiCat

No!

The derivative of the staircase is not continuous. I even say:

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. — boethius

Yes, keeping the staircase with vertical steps is not continuous. But no, tilting it doesn't solve the problem, only makes the problem more obvious.

So go ahead, plot the derivative of the staircase at some angle and show how it is "continuously differentiable".

The whole point of my first comment is to clarify what when he says:

since the staircase has discontinuous jumps in it — fdrake

There is only a "discontinuous jump" in the derivative, not a discontinuity in drawing the staircase. Since his comment was aimed to explain things to people that know less math, it's important to point out that "discontinuous" is often used as shorthand for "continuously differentiable" in a lot of contexts (someone who doesn't know this may wonder what is "discontinous" about the staircase).

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

No, turn the half circles anyway you like, you will always have a vertical tangent and it will not be continuously differentiable at all points.

Proof: You can move two half opposing half circles together and get a complete circle. A complete circle can be rotated at any angle you want and remains similar. At an angle, it's even worse as now the curve will retrograde and have 2 y values for some x values; so please, tell me, what's the continuous derivative function for that? it's just circles, should be easy to derive.

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

Yes, I recognize you can use a function that is continuously differential, like a sinewave and not have the discontinuous derivatives to deal with. But, as I mention, it's not exactly the same problem, and whatever answer you arrive at for a sine-wave, you cannot simply apply to the circles nor the staircase.

It should be clear that these are not necessarily part of the same class of problems.

As for solving any of them. You'll need to do so relative an axiomatic system. If it's Euclidean geometry, the questions are in some sense sensible. A Euclidean procedure can create these objects under consideration as well as procedure to make half as small staircases or circles, but the limit at infinity won't be defined as there's no such tool in Euclidean geometry.

A sinewave cannot be constructed in Eucidean geometry, which maybe an interesting aside, or maybe indication that there's important difference with your circle example.

If the goal is to investigate a different but similar looking problem, that's fine, but there must be some basis to assume conclusions about something different will hold for the something else. There maybe critical differences (such as continuous differentiability), or maybe not.

Saying "ah, we can make a different object that doesn't have that problem" does not necessarily solve the problem encountered with the first object. Yes, you can make other objects and ask a similar question; no, the answer may not be the same every time you do so. There's a lot of steps to do to arrive at such a conclusion.

However, I have not said your program won't work, just pointing out some important considerations so that, er, avoid:

your terminology is all over the place. — SophistiCat

For any continuous function like whose arclength for a <= x <= b is greater than b-a, its scaled down versions will still have the same ratio of arclength to b-a. So just about any continuous function at all that's not a constant.

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

Doesn't the "seeming straightness" of the staircase depend on the character of its visual appearance? Doesn't its visual appearance depend on facts about our visual systems and facts about the point of view we take relative to the staircase?

For instance, if you get up real close, or use a magnifying glass, you see segments of a staircase that doesn't look like a straight line anymore.

That doesn't seem so paradoxical to me. Is there another, perhaps more mathematical, paradox buried in this optical illusion?

That doesn't seem so paradoxical to me. Is there another, perhaps more mathematical, paradox buried in this optical illusion? Cabbage farmer

It's called a paradox in the literature. Note the outer corner points seem to generate a line as n increases, but is the eventual line entirely composed of a countable set of points?

You can obtain the result of the other "paradox" by drawing a symmetrical sawtooth graph on [0,1] that collapses as n increases, and whose length increases without bound. I leave this as an exercise for those interested. Is the ultimate graph differentiable anywhere?

I'm beginning to sound like a math prof. Sorry! :cool:

OK, I didn't notice that you said continuously differentiable in a later post, so sorry about that. But the staircase function is worse than not continuously differentiable - it is not differentiable at all in some points. In any case, no one has actually tried to show that not being continuously differentiable is even a sufficient criterion for the non-convergence of the length difference, and in fact it is easy to construct counterexamples. So all this arguing is kind of a waste of time.

As for solving any of them. You'll need to do so relative an axiomatic system. If it's Euclidean geometry — boethius

I am not sure why you keep talking about Euclidean geometry, which, as you admit, doesn't even have the notion of a limit. You may as well be talking about group theory. Yes, I think it's pretty obvious that we are talking in the context where limits and such are defined; real analysis will do for the purpose.

For any continuous function like whose arclength for a <= x <= b is greater than b-a, its scaled down versions will still have the same ratio of arclength to b-a. So just about any continuous function at all that's not a constant. — Daz

No, if the arc length decreases any faster than in the examples that have been considered so far, it will converge to the length of the diagonal, as we intuitively expect. This is easy to do with any function whose distance from the diagonal can be scaled. For example, take the half-circle function and scale the peaks down by a factor of n - it will converge like a champ.

But yes, it is evident that whether the length error is constant, converging to zero or growing without bound is pretty precarious. You have to work to make sure that you get the "intuitive" result, because a lot of the times you will get something else entirely.

You can obtain the result of the other "paradox" by drawing a symmetrical sawtooth graph on [0,1] that collapses as n increases, and whose length increases without bound. I leave this as an exercise for those interested. — jgill

Yep, if your sawtooth graph doesn't have this property out of the box, you can easily make it so by multiplying it by some uniformly increasing function of n.


The reason this looks very counterintuitive to me is because if we put aside analysis and just look at what we get in the limit, every point on the staircase converges towards a matching point on the straight line - which of course has the length of the straight line. So what gives? Well, the formal answer is that the limit towards which the sequence is converging is not an element of the sequence: the limit points do not themselves lie on any staircase curve. This is not so unusual; for example, most converging rational number sequences do not converge to rational numbers.

Still, it just looks... wrong :)

Well, the formal answer is that the limit towards which the sequence is converging is not an element of the sequence — SophistiCat

Not quite, but close. The sequence actually converges uniformly to the hypotenuse, but arc length is not necessarily preserved under uniform convergence.

Note the outer corner points seem to generate a line as n increases, but is the eventual line entirely composed of a countable set of points? How can this be? :chin:

No, if the arc length decreases any faster than in the examples that have been considered so far, it will converge to the length of the diagonal, as we intuitively expect. — SophistiCat

I wasn't explicit enough by what I meant by"scaled down versions".

First of all, I'm switching from the diagonal to just continuous functions defined on some interval of the real line. Same picture just viewed from another angle, but easier to talk about.

Now, what I mean by a "scaled down" is the graph of a continuous function y = f(x) that has been modified so that all linear measurements of the whole graph are shrunk by the same factor C > 1.

So instead of y = f(x) we're now looking at y = (1/C)*f(Cx).

This is essentially the same way the original stairstep pattern is scaled as it gets closer to the diagonal line. So if the graph is a bunch of semicircles of radius = 1, end to end resting on the x-axis, the arclength above a diameter like say 0 <= x <= 2 will be π. The ratio of arclength to the length of the interval on the x-axis is π/2.

If we shrink the whole graph down by a factor of say 100, then the semicircles now have radius = 1/100 and the arclength above a diameter, say the interval 0 <= x <= 1/50, will now be π/100 and the ratio is again π/100 / (1/50) = π/2.

I am not sure why you keep talking about Euclidean geometry, which, as you admit, doesn't even have the notion of a limit. You may as well be talking about group theory. — SophistiCat

Because the OP does not specify an axiomatic system but describes the problem essentially in Euclidean geometry. The OP doesn't say "what does real analysis say about this?".

This is also not a highly technical mathematical forum, so people unfamiliar with real analysis may wonder how the problem presents in Euclidean terms.

Furthermore, it is a fairly usual question of interest "what is the simplest system in which the problem can be described?", and the problem of making the staircase or circles smaller and smaller can be described in Euclidean geometry.

The OP doesn't even talk about limits, just this process of making things smaller.

As other posters, have noted, the process of making things smaller doesn't change anything, "zoom in" and the shape is similar. This could be a satisfactory answer for some.

The "problem" arises when we want to consider, to take your circle example, the situation where "all points" on the line are the center of a circle; that we cannot zoom in and see circles.

I don't have time to fully investigate the characteristics of such an object, cause there's a pandemic, but considering the lack of intellectual rigour is essentially the cause of building a system so fragile to the pandemic in the first place as well as failing to contain it "because stocks might go down", I feel I need to be even more vigilant in these troubling times on all fronts!

Yes, you can smooth out corners to make a curve continuously differentiable; no, whatever conclusions are drawn from doing that don't automatically apply to the corner case, maybe we're interested in investigating the corners and want to deal with what happens when, trying to take the limit of shrinkifying the stair lengths, essentially every point becomes non-differentiable (that the object is "only corners", or at least all the rational points are defined as corners or some kind of scheme like this; may or may not be of interest to people here).

My intuition for these kinds of problems tells me my intuition may or may not be correct for these kinds of problems. Making the circles or the corners or the sinewave periods "dense" could have surprising results, which may not be the same for each case, or maybe even different ways to construct these kinds of object within the same system, not to mention different systems, with different results.

This isn't a criticism of your approach. It's totally valid to consider the case of the circle as a closely related problem to the stairs, and the case of the sinewave as closely related to the circles. It's not valid to assume conclusions automatically propagate backwards to all these cases, an argument is needed for why this would be so (lot's of theorems have "only necessarily valid for continuously differentiable functions" warning attached); nor is it automatic that those previous harder problems are not interesting because a simpler problem has been found. It happens all the time that very subtle differences make an approach work for one problem and not another and that harder problems are chosen over simpler ones. It's all I'm pointing out, I'm sure you agree now that my position is totally clear.

The deeper problem concerns the fact that in Cartesian co-ordinates the notion of vertical and horizontal Euclidean lengths is incommensurate with the notion of diagonal Euclidean lengths; hence the reason why a hypotenuse that is diagonal to the Cartesian axis is assigned an irrational number such as Sqrt(2), which of course isn't a quantity but a non-terminating algorithm for generating a Cauchy sequence.

Still, it just looks... wrong — SophistiCat

Looked at this some more. A general explanation seems to be "the arc length functional is not a continuous functional in the infinite norm (uniform convergence norm) on the space of continuous functions". We can treat the staircase as an example of this fact (after rotating it to make it continuous).

Interesting take on the situation. (For the uninitiated, functionals map functions to numbers)

the notion of vertical and horizontal Euclidean lengths is incommensurate with the notion of diagonal Euclidean lengths — sime

What does that mean, to say that the notion of vertical and horizontal lengths is incommensurate with the notion of diagonal lengths?

It is somewhat subtle to define for which subsets of the plane the concept of length makes sense, and to define exactly what the length is of such a subset. But in the standard definition, there is no distinction between diagonal lengths and horizontal or vertical ones.

Putting more symbols on it.

Let $C$ be the set of continuous functions over $\mathbb{R}$, then define the function $d : C \times C \rightarrow \mathbb{R}^+$ by $d(f_1,f_2) = \sup_{x \in \mathbb{R}} |f_1 (x) - f_2 (x)|$. $d$ is a metric, so $(C,d)$ can be thought of as a topological space.

We want to consider functionals over $C$, which are mappings $C \rightarrow \mathbb{R}$. Consider some subset $M$ of $C$ and a function $f_0 \in M$. Call a functional $L$ continuous at a function $f_0 \in M$ when and only when for all $\epsilon \gt 0$ there exists an open $U \subset M$ with $f_0 \in U$ such that $| L(f) - L(f_0) | \lt \epsilon$.

(Edit: an open ball with radius R centred at a function f (the collection of which generates this topology) seem to be the collection of functions whose supnorm is less than R; so can be thought of as the set of continuous functions which converge uniformly to f with R as the "minimal" least upper bound which can be used in a uniform convergence proof to f for all the ball elements.)

Stipulate that continuous functions have an arc length, and let the approximating staircase with $2n$ steps be $S(n)$. If we consider $f_0$ as the straight line, $S(n)$ has arclength 2 for arbitrarily small $d$-balls around $f_0$, which contain an $S(n)$ by uniform convergence.

The arclength functional is probably nowhere continuous in that sense? There will always be a series of approximating curves you can construct that have any desired arclength that uniformly converge to the desired function.

Because the OP does not specify an axiomatic system but describes the problem essentially in Euclidean geometry. — boethius

Well, no, it doesn't, because there isn't any problem so long as we stay with Euclidean geometry (as rightly noted). The apparent problem only arises when we introduce the notion of a limit, and perhaps other implicit assumptions.

Note the outer corner points seem to generate a line as n increases, but is the eventual line entirely composed of a countable set of points? How can this be? — jgill

maybe we're interested in investigating the corners and want to deal with what happens when, trying to take the limit of shrinkifying the stair lengths, essentially every point becomes non-differentiable (that the object is "only corners", or at least all the rational points are defined as corners or some kind of scheme like this; may or may not be of interest to people here). — boethius

"Almost none" of the limit points on the diagonal (let's just call it that for brevity) is a corner point, for the simple reason that there is only a countable number of them. Also, keep in mind that the diagonal (which we interpret as the limit point of the sequence of curves) is not itself part of the sequence and does not have the same properties. Every member of the sequence is piecewise-differentiable, while the diagonal is, of course, everywhere differential.

"Almost none" of the limit points on the diagonal (let's just call it that for brevity) is a corner point, for the simple reason that there is only a countable number of them. Also, keep in mind that the diagonal (which we interpret as the limit point of the sequence of curves) is not itself part of the sequence and does not have the same properties. Every member of the sequence is piecewise-differentiable, while the diagonal is, of course, everywhere differential. — SophistiCat

Do you think it's the case that in the limit the corner points become dense in the straight line, despite remaining countable? Edit: it seems either that is true, or the notion of corner breaks down in the limit (and so the corner points are dense in the line by virtue of every line point being a corner point).

Do you think it's the case that in the limit the corner points become dense in the straight line, despite remaining countable? — fdrake

Aren't the corner points just the dyadic rationals of the form $\frac{a}{2^n}$? In which case they're dense.

Can you create a kind of "staircase" that converges uniformly to the hypotenuse and preserves arc length, so no "paradox?" :chin:

Aren't the corner points just the dyadic rationals — fishfry

Ah yes! You can see them as the end points of the interval expressions here, should've seen that sooner.

The corner points correspond indeed to to rational dyadics, as seen on rulers. The denominators being powers of two. The paradox here is that the stepped diagonal has length two, while appearing to have a length of the square-root of two.

It's indeed somehow similar to the use of Feynman diagrams in quantum field theory, each higher order diagram giving increasingly smaller and increasingly more contributions to an interaction process. The first stair represents the first order process. One right angle on the line. (inside the square). The second order process, introducing two extra vertices (which in a Feynman diagram can be put in in a variety of ways though), is corresponding to the second stair. Again two vertices are added for a third order diagram, corresponding to virtual particles. These four extra vertices can already be added in a lot of ways, each diagram contributing to the scattering process.On the staircase though, four new angles are added. Not two. The two more vertices are added again, giving rise to a new spectrum of diagrams. Ad infinitum! All contributing with a fastly decreasing weight. Normally second order contributions will do. I have done these calculations, but it's boring! The similarity with this staircase is enlightening! In any book on diagrammatics, this enormity can be seen.See, for a small example, here:

https://i.pinimg.com/736x/27/ad/3b/27ad3b1f4776c5d591a4b84e889433aa--feynman-diagram-science-art.jpg


To make a line infinite without adding a second dense dimension is impossible. A Peano curve (Giuseppe Peano already made this in the end of the 19th century) is a one dimensional infinite line that's fit in a square (like a 2-meter DNA string is fit into a small nuclei, though the two ends of DNA don't identify with infinity). This curve is a predecessor of fractals, which would occupy a fraction of the square.

The old switcheroo!

Seems like all the verticals and horizontals shrink to zero leaving the points of a line. It's an iteration of triangles placed on the diagonal. Each time the number of triangles is twice as big while their is sqrt2 smaller. Leading eventually to the shape of the line. If we look through a super microscope, the length still 2 though. A long line?

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

Except the sum of the chord side lengths on the polygons you've shown do approach the circumference of the circle. If I remember correctly, this is how pi was first estimated.

Oddities that math people explain in different ways. But intuitively it sure seems like a paradox. — jgill

It's not a paradox. The diagonal line represents the distance I travel going from the bottom to the top - √2. It doesn't matter whether I use stairs or a smooth diagonal, that's the distance I'm travelling.

Isn't the distance traveled on the stairs always 2? You not go in a straight line on the stairs. Only on a flat slope, if not slippery. The paradox is that the length of the stairs seems sqrt2 but is 2.

Isn't that just calculus?

How is an infinite line between 0 and 1 constructed?

Isn't the distance traveled on the stairs always 2? You not go in a straight line on the stairs. Only on a flat slope, if not slippery. The paradox is that the length of the stairs seems sqrt2 but is 2. — Raymond

If I walk a mile on a flat surface, how far did I walk? A mile. I don't count the distance my feet moved when they went up and down as I walked. Why would I do that on the stairs? My feet don't follow the discontinuous path of the stair.

I feel myself always hopping on a stair. My center of mass seems not to go in a straight line. Maybe you walk the stairs while your CoM floats linea recta.

How is an infinite line between 0 and 1 constructed? — Raymond

A Short Note: Extending the Diagonal Paradox

It's not a paradox — T Clark

:sad:

My feet don't follow the discontinuous path of the stair. — T Clark

The path is continuous. Sounds like you just float up. :gasp: