L’escalier du Diable

Welcome one, welcome all to the Point at Infinity sideshow, where today we present a tantalizing and diabolical selection of musical and mathematical curiosities. Just watch your step; these stairs can be a bit tricky.


A few months ago, you may recall, we published two posts about the Shepard tone and the Risset rhythm, aural illusions in which a tone or rhythm seems to perpetually rise or fall in pitch or in tempo but is actually repeating the same pattern over and over again, the musical equivalents of Penrose stairs.

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Penrose stairs. (Image in the public domain.)

To accompany the posts we created some sound samples so the readers could hear the illusions themselves. A couple of weeks ago, one of these samples was used in an internet radio program on audio paradoxes released by Eat This Radio, paired with some work of Jean-Claude Risset. The entire program is really excellent, ranging from a piece by J.S. Bach to mid-twentieth century audio experiments to modern electronic music, and I encourage all of you to listen to it.


One of the pieces in the radio program is a piano étude written by György Ligeti in the late twentieth century. The étude is named L’escalier du diable, or The Devil’s Staircase, and its repeated ascents of the keyboard have a striking resonance with the never-ending ascent of the Shepard tone.


The Devil’s Staircase is also the colloquial name given to a particular mathematical function introduced by Georg Cantor in the 1880s. It is a function defined on the set of real numbers between 0 and 1 and taking values in the same interval, and it has some quite curious properties. Before we discuss it, let’s take a look at (an approximation to) the graph of the function.

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Graph of the Devil’s Staircase. By Theon, CC BY-SA 3.0.

To appreciate the strangeness of this function, let us recall some definitions regarding functions of real numbers. Very roughly speaking, a function is called continuous if it has no sudden jumps, or if its graph can be drawn without lifting the pencil from the page. Continuous functions satisfy a number of nice properties, such as the intermediate value theorem.

The derivative of a function at a given point of its domain, if it exists, measures the rate of change of the function at that point. If the x-axis measures time and the y-axis measures the position of an object along some one-dimensional track, then the derivative can be thought of as the velocity of that object. If a function is differentiable at a point (i.e., if its derivative exists there) then it must be continuous at that point, but the converse is not necessarily true. (For example, if the graph of a function has a sharp corner at a point, then the function cannot be differentiable there.)

Let’s think about what it means for a function to have a derivative of 0 at a point. It means that, at that point, the rate of change of the function has vanished. It means that, if we zoom in sufficiently close to that point, the function should look like a constant function. Its graph should look like a horizontal line. What would it mean for a function to have a derivative of 0 almost everywhere? (Here “almost everywhere” is a technical term (which I’m not going to define) and not just me being vague.) One might think that this must imply that the function is a constant function. At almost every point in its domain, the rate of change of the function is 0, so how can the value of the function change?

One will quickly discover that this is not quite right. Consider the function defined on the real numbers whose value is 0 at all negative numbers and 1 at all non-negative numbers.

step

This function has derivative 0 everywhere except at 0 itself, and yet it increases from 0 to 1. It does this quite easily by being discontinuous at 0, which, in hindsight, seems sort of like cheating. So what if we also require our function to be continuous? Now we need more exotic examples, and this is where the Devil’s Staircase comes in, for the Devil’s Staircase is a continuous function, it is differentiable almost everywhere, it has a  derivative of 0 wherever its derivative is defined, and yet it still manages to increase from 0 to 1. Wild!

What is the Devil’s Staircase exactly? I’ll give two different definitions. The first proceeds via an iterative construction. Start with the function f_0(x) = x. Its graph, between 0 and 1, is simply a straight line segment increasing from (0,0) to (1,1). Now, look at the midpoint of this increasing line segment, and draw a horizontal line segment centered there whose length is 1/3 of the horizontal line of the original increasing segment. Now connect the ends of this line segment via straight lines to (0,0) and (1,1). This new curve is the graph of a function that we call f_1. It consists of two increasing line segments with one horizontal line segment between them. Now repeat the process that took us from f_0 to f_1 on each of these increasing line segments, and let f_2 be the function whose graph is the result. Continue in this manner, constructing f_n for every natural number n.

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First three steps of the iterative construction of the Devil’s Staircase. (Image in the public domain.)

It turns out that, as n goes to infinity, the sequence of functions \langle f_n \mid n \in \mathbb{N} \rangle converges (uniformly) to a single function. This function is the Devil’s Staircase.

A more direct but also more opaque definition is as follows: Given a real number x between 0 and 1, first express x in base 3 (i.e., using only 0s, 1s, and 2s). If this base 3 representation contains a 1, then replace every digit after the first 1 with a 0. Next, replace all 2s with 1s. The result has only 0s and 1s, so we can interpret it as a binary (i.e., base 2) number, and we let f(x) be this value. Then the function f defined in this manner is the Devil’s Staircase. Play around with this definition, and you might get a feel for what it’s doing.


And now, on our way out, some musical addenda. An encore, if you will. First, after making the Risset rhythms for the aforementioned post, I did some further coding and wrote a little program that can take any short audio snippet and make a Risset rhythm out of it. Here’s an example, first accelerating and then decelerating, using a bit from a Schubert piano trio.

You may recognize the sample from the soundtrack to Barry Lyndon.

Finally, I can’t help but include here one of my favorite pieces by Ligeti, Poema sinfónico para 100 Metrónomos.


Cover image: Devil’s Staircase Wilderness, Oregon, USA

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Hippasus and the Infinite Descent

Today, dear Reader, we bring you a story. A story of Mathematics and Music, of Reason and Passion, of Drama and Irony. It is the story of Hippasus of Metapontum, of his remarkable life and his equally remarkable death. Before we begin, a note of warning. Not everything presented here is true, but all of it is meaningful.


Ippaso_di_Metaponto
Hippasus of Metapontum

A Greek philosopher and mathematician from the 5th century BCE, Hippasus was a follower of Pythagoras. The Pythagoreans believed in the transmigration of souls, subscribed to the belief that All is Number (where “Number” is, of course, “Whole Number”), made great strides in the study of musical harmony, and eschewed the eating of beans. Our hero was a particularly illustrious Pythagorean. He performed experiments linking the sizes of metal discs to the tones they emit upon being struck, developed a theory of the musical scale and a theory of proportions, and showed how to inscribe a regular dodecahedron in a sphere. The regular dodecahedron is a twelve-sided solid whose faces are regular pentagons, shapes which were dear to the Pythagoreans and central to our story. The pentagram, the five-sided star formed by extending the sides of a regular pentagon, and whose tips themselves form a regular pentagon, was a religious symbol of the Pythagoreans and a mark of recognition amongst themselves.

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Ancient Greek coin with a pentagram

A corollary to the Pythagorean doctrine that All is Number was a belief, at the time, that any two lengths are commensurable, i.e., that given any two lengths, they are both whole number multiples of some fixed smaller length. In modern language, this amounts to the assertion that, given any two lengths, their ratio is a rational number, i.e., can be expressed as the ratio of whole numbers. It is only fitting that the first evidence to the contrary would come from the pentagram.

It happened when Hippasus was stargazing. He saw five stars forming a perfect regular pentagon, and inside this regular pentagon he formed a pentagram, at the center of which lay another regular pentagon, into which he formed another pentagram, at the center of which lay another regular pentagon, into which he formed another pentagram. An infinite web of similar triangles was woven through his mind and, in a flash of insight, he realized something terrible: the lengths of one side of the regular pentagon and one side of the pentagram found inside it are incommensurable.

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The lengths of one side of a regular pentagon and one side of the inscribed pentagram are incommensurable. In other words, if s denotes the length of the black line segment AB and t denotes the length of the blue line segment AC, then t/s (and therefore also s/t) is irrational.

The next day, he told his fellow Pythagoreans of his discovery, and they were horrified. They could not have this knowledge, which struck at the core of their belief system, getting out into the wider world. So they took Hippasus far out to sea, and they threw him overboard.


Infinite descent is a proof technique that morally dates back to the ancient Greeks but really came into its own in the work of Pierre de Fermat in the 17th century. The idea behind it is simple and immediately appealing. Suppose we want to prove that there are no positive integers satisfying a particular property. One way to prove this would be to show that, given any positive integer n satisfying this property, we could always find a smaller positive integer n' satisfying the same property. Repeating the argument with n' in place of n, we could find a still smaller positive integer n'' satisfying the same property. Continuing, we would construct a decreasing sequence of positive integers, n > n' > n'' > n''' > \ldots , the aforementioned “infinite descent”. But of course there can be no infinite decreasing sequences of positive integers (if a decreasing sequence starts with n, it can of course have at most n elements). One thus reaches a contradiction and concludes that there are no positive integers with the given property.

Fermat made great use of the method of infinite descent in his work on number theory. One particularly striking application came in the proof of a special case of his famous Last Theorem: there are no positive integers a,b,c such that a^4 + b^4 = c^4. Fermat showed that, if a_0,b_0,c_0 are positive integers such that a_0^4 + b_0^4 = c_0^4, then we can construct positive integers a_1, b_1, c_1 such that a_1^4 + b_1^4 = c_1^4 and c_1 < c_0. One can then continue, with a_1, b_1, c_1 in place of a_0, b_0, c_0, and obtain a_2, b_2, c_2 with a_2^4 + b_2^4 = c_2^4 and c_2 < c_1. This of course leads to the infinite descent c_0 > c_1 > c_2 > \ldots and a contradiction.

We will use the method of infinite descent to prove the result of Hippasus mentioned above. This will not be exactly the way the ancient Greeks would have presented the proof, but it is very similar in spirit, and we make no apologies for the anachronism. Let’s get started.


Suppose, for the sake of an eventual contradiction, that we have a regular pentagon whose side length is commensurable with the side length of the inscribed pentagram. Let s denote the side length of the pentagon and t denote the side length of the pentagram (i.e., the diagonal of the pentagon). In modern language, our assumption is that \frac{t}{s} is rational, i.e., that there are positive integers p and q such that \frac{t}{s} = \frac{p}{q}. By scaling the pentagon, we can in fact assume that t = p and s = q. So let this be our starting assumption: there is a regular pentagon whose side length s and diagonal length t are both positive integers.

Let us label the vertices of the pentagon by the letters A,B,C,D,E. The center of the pentagram forms another regular pentagon, whose vertices we shall call a,b,c,d,e. This is shown in the diagram below. Note that s is the length of the line segment connecting A and B (we will denote this length by |AB|), and t is |AC|. Let s' denote the side length of the inner pentagon, i.e., |ab|, and let t' denote the length of the diagonal of the inner pentagon, i.e., |ac|.

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Figure 1

We now make use of the wealth of congruent triangles in the diagram. We first observe that, whenever a pentagram is inscribed into a regular pentagon, the diagonals that form the pentagram exactly trisect the angles of the pentagon. (Exercise: Prove this!) Therefore, the triangle formed by A, E and D is congruent to that formed by A, e, and D, on account of their sharing a side and having the same angles on either end of that side. In particular, we have |Ae| = |AE| = s.

Next, consider the triangle formed by A, b, and d. By our observation at the start of the previous paragraph, the angle at b in this triangle has the same measure as the angle at A. But then this triangle is isosceles, so we have |Ad| = |db| = t'. Of course, we clearly have |Ad| = |Bd| = |Be| = |Ce| =…, so all of these lengths are equal to t'.

proof_diagram
Figure 1 again

Let’s now see what we have. Consider first |AC|. By definition, we have |AC| = t. But also |AC| = |Ae| + |Ce|, and we saw previously that |Ae| = s and |Ce| = t'. All together, we have t = |AC| = |Ae| + |Ce| = s + t', or t' = t-s.

Next, consider |Ae|. We have already seen that |Ae| = s. But we also have |Ae| = |Ad| + |de|. We saw previously that |Ad| = t', and by definition we have |de| = s'. All together, we have s = |Ae| = |Ad| + |de| = t' + s', or s' = s - t'. Since we already know that t' = t-s, this yields s' = 2s - t.

This might not seem like much, but we’re actually almost done now! The key observation is that, since s and t are integers, and since s' = 2s - t and t' = t-s, it follows that s' and t' are also (positive) integers. We started with a regular pentagon whose side length s and diagonal length t were integers and produced a smaller pentagon whose side length s' and diagonal length t' are also integers. But now we can continue this process, producing smaller and smaller regular pentagons and producing infinite descending sequences of positive integers, s > s' > s'' > \ldots and t > t' > t'' > \ldots.

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The infinite descent

This is a contradiction, and we have thus shown that \frac{t}{s} is irrational. But what is this ratio exactly? Well, it turns out to be a fascinating number, easily #2 on the list of Most Famous Ratios of All Time: \phi, a.k.a. the golden ratio! But that is a story for another time…


As a short addendum to today’s story, here’s a little-known fact about the end of Hippasus’ life. It turns out that, after being thrown overboard by his fellow Pythagoreans, he has not and will never in fact reach the seabed! For to get there, he would first have to travel halfway down, and then he would have to travel half of the remaining distance, and then half of the still remaining distance, and so on and so forth, completing an endless sequence of tasks, and thus he remains to this date in the midst of an infinite descent of his own.


Notes:

(1) \sqrt{2} is more commonly cited as being the first irrational number discovered by the Pythagoreans, and it is almost always the first number proven to be irrational in classrooms today. However, the proof of the irrationality of \sqrt{2} possessed by the Greeks is not the simple number-theoretic proof used today and is in fact a rather complex elaboration of the proof of the incommensurability of the side and diagonal lengths of a regular pentagon. Given this fact and the centrality of the pentagram in Pythagorean intellectual life, some scholars have suggested that perhaps this was in fact the first proof of the existence of incommensurability and that the proof of the irrationality of \sqrt{2} came later. We have adopted this hypothesis for the purpose of our story today.

(2) This story is derived from legends that significantly post-date the death of Hippasus. It seems unlikely actually to have happened as presented here.

(3) This post was in part inspired by the episode “Drowned at Sea”, from the excellent podcast, Hi-Phi Nation, by philosopher Barry Lam. Check it out!


Cover image: “Rainstorm Over the Sea” by John Constable