Irrational rotations are ergodic

Posted on 1 June 2017 by John

In a blog post yesterday, I mentioned that the golden angle is an irrational portion of a circle, and so a sequence of rotations by the golden angle will not repeat itself. We can say more: rotations by an irrational portion of a circle are ergodic. Roughly speaking, this means that not only does the sequence not repeat itself, the sequence “mixes well” in a technical sense.

Ergodic functions have the property that “the time average equals the space average.” We’ll unpack what that means and illustrate it by simulation.

Suppose we pick a starting point x on the circle then repeatedly rotate it by a golden angle. Take an integrable function f on the circle and form the average of its values at the sequence of rotations. This is the time average. The space average is the integral of f over the circle, divided by the circumference of the circle. The ergodic theorem says that the time average equals the space average, except possibly for a setting of starting values of measure zero.

More generally, let X be a measure space (like the unit circle) with measure μ let T be an ergodic transformation (like rotating by a golden angle), Then for almost all starting values x we have the following:

$\lim_{n\to \infty} \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x) = \frac{1}{\mu(X)} \int_X f\, d\mu$

Let’s do a simulation to see this in practice by running the following Python script.

        from scipy import pi, cos
        from scipy.constants import golden
        from scipy.integrate import quad
        
        golden_angle = 2*pi*golden**-2
        
        def T(x):
            return (x + golden_angle) % (2*pi)
        
        def time_average(x, f, T, n):
            s = 0
            for k in range(n):
                s += f(x)
                x = T(x)
            return s/n
        
        def space_average(f):
            integral = quad(f, 0, 2*pi)[0]
            return integral / (2*pi)
        
        f = lambda x: cos(x)**2
        N = 1000000
        
        print( time_average(0, f, T, N) )
        print( space_average(f) )

In this case we get 0.49999996 for the time average, and 0.5 for the space average. They’re not the same, but we only used a finite value of n; we didn’t take a limit. We should expect the two values to be close because n is large, but we shouldn’t expect them to be equal.

Update: The code and results were updated to fix a bug pointed out in the comments below. I had written ... % 2*pi when I should have written ... % (2*pi). I assumed the modulo operator was lower precedence than multiplication, but it’s not. It was a coincidence that the buggy code was fairly accurate.

A friend of mine, a programmer with decades of experience, recently made a similar error. He’s a Clojure fan but was writing in C or some similar language. He rightfully pointed out that this kind of error simply cannot happen in Clojure. Lisps, including Clojure, don’t have operator precedence because they don’t have operators. They only have functions, and the order in which functions are called is made explicit with parentheses. The Python code x % 2*pi corresponds to (* (mod x 2) pi) in Clojure, and the Python code x % (2*pi) corresponds to (mod x (* 2 pi)).

Listening to golden angles

Posted on 31 May 2017 by John

The other day I wrote about the golden angle, a variation on the golden ratio. If φ is the golden ratio, then a golden angle is 1/φ² of a circle, approximately 137.5°, a little over a third of a circle.

Musical notes go around in a circle. After 12 half steps we’re back where we started. What would it sound like if we played intervals that went around this circle at golden angles? I’ll include audio files and the code that produced them below.

A golden interval, moving around the music circle by a golden angle, is a little more than a major third. And so a chord made of golden intervals is like an augmented major chord but stretched a bit.

An augmented major triad divides the musical circle exactly in thirds. For example, C E G#. Each note is four half steps, a major third, from the previous note. In terms of a circle, each interval is 120°. Here’s what these notes sound like in succession and as a chord.

(Download)

If we go around the musical circle in golden angles, we get something like an augmented triad but with slightly bigger intervals. In terms of a circle, each note moves 137.5° from the previous note rather than 120°. Whereas an augmented triad goes around the musical circle at 0°, 120°, and 240° degrees, a golden triad goes around 0°, 137.5°, and 275°.

A half step corresponds to 30°, so a golden angle corresponds to a little more than 4.5 half steps. If we start on C, the next note is between E and F, and the next is just a little higher than A.

If we keep going up in golden intervals, we do not return to the note we stared on, unlike a progression of major thirds. In fact, we never get the same note twice because a golden interval is not a rational part of a circle. Four golden angle rotations amount to 412.5°, i.e. 52.5° more than a circle. In terms of music, going up four golden intervals puts us an octave and almost a whole step higher than we stared.

Here’s what a longer progression of golden intervals sounds like. Each note keeps going but decays so you can hear both the sequence of notes and how they sound in harmony. The intention was to create something like playing the notes on a piano with the sustain pedal down.

(Download)

It sounds a little unusual but pleasant, better than I thought it would.

Here’s the Python code that produced the sound files in case you’d like to create your own. You might, for example, experiment by increasing or decreasing the decay rate. Or you might try using richer tones than just pure sine waves.

from scipy.constants import golden
import numpy as np
from scipy.io.wavfile import write

N = 44100 # samples per second

# Inputs:
# frequency in cycles per second
# duration in seconds
# decay = half-life in seconds
def tone(freq, duration, decay=0):
    t = np.arange(0, duration, 1.0/N)
    y = np.sin(freq*2*np.pi*t)
    if decay > 0:
        k = np.log(2) / decay
        y *= np.exp(-k*t)
    return y

# Scale signal to 16-bit integers,
# values between -2^15 and 2^15 - 1.
def to_integer(signal):
    signal /= max(abs(signal))
    return np.int16(signal*(2**15 - 1))

C = 262 # middle C frequency in Hz

# Play notes sequentially then as a chord
def arpeggio(n, interval):
    y = np.zeros((n+1)*N)
    for i in range(n):
        x = tone(C * interval**i, 1)
        y[i*N : (i+1)*N] = x
        y[n*N : (n+1)*N] += x
    return y

# Play notes sequentially with each sustained
def bell(n, interval, decay):
    y = np.zeros(n*N)
    for i in range(n):
        x = tone(C * interval**i, n-i, decay)
        y[i*N:] += x
    return y

major_third = 2**(1/3)
golden_interval = 2**(golden**-2)

write("augmented.wav", N, to_integer(arpeggio(3, major_third)))

write("golden_triad.wav", N, to_integer(arpeggio(3, golden_interval)))

write("bell.wav", N, to_integer(bell(9, golden_interval, 6)))

Volume of a rose-shaped torus

Posted on 18 May 2017 by John

Start with a rose, as described in the previous post:

Now spin that rose around a vertical line a distance R from the center of the rose. This makes a torus (doughnut) shape whose cross sections look like the rose above. You could think of having a cutout shaped like the rose above and extruding Play-Doh through it, then joining the ends in a loop.

Volume of rotation from rose

In case you’re curious, the image above was created with the following Mathematica command:

      RevolutionPlot3D[{4 + Cos[5 t] Cos[t], Cos[5 t] Sin[t]}, {t, 0, 2 Pi}]

What would the volume of resulting solid be?

This sounds like a horrendous calculus homework problem, but it’s actually quite easy. A theorem by Pappus of Alexandria (c. 300 AD) says that the volume is equal to the area times the circumference of the circle traversed by the centroid.

The area of a rose of the form r = cos(kθ) is simply π/2 if k is even and π/4 if k is odd. (Recall from the previous post that the number of petals is 2k if k is even and k if k is odd.)

The location of the centroid is easy: it’s at the origin by symmetry. If we rotate the rose around a vertical line x = R then the centroid travels a distance 2πR.

So putting all the pieces together, the volume is π²R if k is even and half that if k is odd. (We assume R > 1 so that the figure doesn’t intersect itself and some volume get counted twice.)

We can also find the surface area easily by another theorem of Pappus. The surface area is just the arc length of the rose times the circumference of the circle traced out by the centroid. The previous post showed how to find the arc length, so just multiply that by 2πR.

Length of a rose

Posted on 17 May 2017 by John

The polar graph of r = cos(kθ) is called a rose. If k is even, the curve will trace out 2k petals as θ runs between 0 and 2π. If k is odd, it will trace out k petals, tracing each one twice. For example, here’s a rose with k = 5.

(I rotated the graph 36° so it would be symmetric about the vertical axis rather than the horizontal axis.)

The arc length of a curve in polar coordinates is given by

$\int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\, d\theta$

and so we can use this find the length. The integral doesn’t have a closed form in terms of elementary functions. Instead, the result turns out to use a special function E(x), the “complete elliptic integral of the second kind,” defined by

$E(m) = \int_0^{\pi/2} \sqrt{1 - m \sin^2 x} \, dx$

Here’s the calculation for the length of a rose:

$\int_0^{2\pi} \sqrt{r^2 + (r')^2}\, d\theta &=& \int_0^{2\pi} \sqrt{cos^2 k\theta + k^2 \sin^2 k\theta} \, d\theta \\ &=& \int_0^{2\pi} \sqrt{cos^2 u + k^2 \sin^2 u} \, du \\ &=& 4 \int_0^{\pi/2} \sqrt{cos^2 u + k^2 \sin^2 u} \, du \\ &=& 4 \int_0^{\pi/2} \sqrt{1 + (k^2-1) \sin^2 u} \, du \\ &=& 4 E(-k^2 + 1)$

So the arc length of the rose r = cos(kθ) with θ running from 0 to 2π is 4 E(-k² + 1). You can calculate E in SciPy with scipy.special.ellipe.

If we compute the length of the rose at the top of the post, we get 4 E(-24) = 21.01. Does that pass the sniff test? Each petal goes from r = 0 out to r = 1 and back. If the petal were a straight line, this would have length 2. Since the petals are curved, the length of each is a little more than 2. There are five petals, so the result should be a little more than 10. But we got a little more than 20. How can that be? Since 5 is odd, the rose with k = 5 traces each petal twice, so we should expect a value of a little more than 20, which is what we got.

As k gets larger, the petals come closer to being straight lines. So we should expect that 4E(-k² + 1) approaches 4k as k gets large. The following plot of E(-k² + 1) – k provides empirical support for this conjecture by showing that the difference approaches 0, and gives an idea of the rate of convergence. It should be possible to prove that, say, that E(-k²) asymptotically approaches k, but I haven’t done this.

Related posts:

When length equals area

Posted on 15 May 2017 by John

The graph of hyperbolic cosine is called a catenary. A catenary has the following curious property: the length of a catenary between two points equals the area under the catenary between those two points.

The proof is surprisingly simple. Start with the following:

$\sqrt{1 +\left(\frac{d}{dx} \cosh(x) \right)^2} = \sqrt{1 + \sinh^2(x)} = \cosh(x)$

Now integrate the first and last expressions between two points a and b. Note that the former integral gives the arc length of cosh between a and b, and the later integral gives the area under the graph of cosh between a and b.

By the way, the most famous catenary may be the Gateway Arch in St. Louis, Missouri.

Gateway Arch at night

Solving systems of polynomial equations

Posted on 13 May 2017 by John

In a high school algebra class, you learn how to solve polynomial equations in one variable, and systems of linear equations. You might reasonably ask “So when do we combine these and learn to solve systems of polynomial equations?” The answer would be “Maybe years from now, but most likely never.” There are systematic ways to solve systems of polynomial equations, but you’re unlikely to ever see them unless you study algebraic geometry.

Here’s an example from [1]. Suppose you want to find the extreme values of x³ + 2xyz – x² on the unit sphere using Lagrange multipliers. This leads to the following system of polynomial equations where λ is the Lagrange multiplier.

$\begin{eqnarray*} 3x^2 + 2yz - 2x\lambda &=& 0 \\ 2xz - 2y\lambda &=& 0 \\ 2xy - 2z - 2z\lambda &=& 0 \\ x^2 + y^2 + z^2 - 1 &=& 0 \end{eqnarray*}$

There’s no obvious way to go about solving this system of equations. However, there is a systematic way to approach this problem using a “lexicographic Gröbner basis.” This transforms the problem from into something that looks much worse but that is actually easier to work with. And most importantly, the transformation is algorithmic. It requires some computation—there are numerous software packages for doing this—but doesn’t require a flash of insight.

The transformed system looks intimidating compared to the original:

$\begin{eqnarray*} \lambda - \frac{3}{2}x - \frac{3}{2}yz - \frac{167616}{3835}z^6 - \frac{36717}{590}z^4 - \frac{134419}{7670}z^2 &=& 0 \\ x^2 + y^2 + z^2 - 1 &=& 0 \\ xy - \frac{19584}{3835}z^5 + \frac{1999}{295}z^3 - \frac{6403}{3835}z &=& 0 \\ xz + yz^2 - \frac{1152}{3835}z^5 - \frac{108}{295}z^3 + \frac{2556}{3835}z &=& 0 \\ y^3 + yz^2 - y -\frac{9216}{3835}z^5 + \frac{906}{295}z^3 - \frac{2562}{3835}z &=& 0 \\ y^2z - \frac{6912}{3835}z^5 -\frac{827}{295}z^3 -\frac{3839}{3835}z &=& 0 \\ yz^3 - yz - \frac{576}{59}z^6 + \frac{1605}{118}z^4 - \frac{453}{118}z^2 &=& 0 \\ z^7 - \frac{1763}{1152}z^5 + \frac{655}{1152}z^3 - \frac{11}{288}z &=& 0 \end{eqnarray*}$

We’ve gone from four equations to eight, from small integer coefficients to large fraction coefficients, from squares to seventh powers. And yet we’ve made progress because the four variables are less entangled in the new system.

The last equation involves only z and factors nicely:

$z\left(z^2 - \frac{4}{9}\right) \left(z^2 - \frac{11}{128}\right) = 0$

This cracks the problem wide open. We can easily find all the possible values of z, and once we substitute values for z, the rest of the equations simplify greatly and can be solved easily.

The key is that Gröbner bases transform our problem into a form that, although it appears more difficult, is easier to work with because the variables are somewhat separated. Solving one variable, z, is like pulling out a thread that then makes the rest of the threads easier to separate.

* * *

[1] David Cox et al. Applications of Computational Algebraic Geometry: American Mathematical Society Short Course January 6-7, 1997 San Diego, California (Proceedings of Symposia in Applied Mathematics)

The 3n+1 problem and Benford’s law

Posted on 3 May 2017 by John

This is the third, and last, of a series of posts on Benford’s law, this time looking at a famous open problem in computer science, the 3n + 1 problem, also known as the Collatz conjecture.

Start with a positive integer n. Compute 3n + 1 and divide by 2 repeatedly until you get an odd number. Then repeat the process. For example, suppose we start with 13. We get 3*13+1 = 40, and 40/8 = 5, so 5 is the next term in the sequence. 5*3 + 1 is 16, which is a power of 2, so we get down to 1.

Does this process always reach 1? So far nobody has found a proof or a counterexample.

If you pick a large starting number n at random, it appears that not only will the sequence terminate, the values produced by the sequence approximately follow Benford’s law (source). If you’re unfamiliar with Benford’s law, please see the first post in this series.

Here’s some Python code to play with this.

from math import log10, floor

def leading_digit(x):
    y = log10(x) % 1
    return int(floor(10**y))

# 3n+1 iteration
def iterates(seed):
    s = set()
    n = seed
    while n > 1:
        n = 3*n + 1
        while n % 2 == 0:
            n = n / 2
        s.add(n)
    return s

Let’s save the iterates starting with a large starting value:

it = iterates(378357768968665902923668054558637)

Here’s what we get and what we would expect from Benford’s law:

|---------------+----------+-----------|
| Leading digit | Observed | Predicted |
|---------------+----------+-----------|
|             1 |       46 |        53 |
|             2 |       26 |        31 |
|             3 |       29 |        22 |
|             4 |       16 |        17 |
|             5 |       24 |        14 |
|             6 |        8 |        12 |
|             7 |       12 |        10 |
|             8 |        9 |         9 |
|             9 |        7 |         8 |
|---------------+----------+-----------|

We get a chi-square of 12.88 (p = 0.116) and so we get a reasonable fit.

Here’s another run with a different starting point.

it = iterates(243963882982396137355964322146256)

which produces

|---------------+----------+-----------|
| Leading digit | Observed | Predicted |
|---------------+----------+-----------|
|             1 |       44 |        41 |
|             2 |       22 |        24 |
|             3 |       15 |        17 |
|             4 |       12 |        13 |
|             5 |       11 |        11 |
|             6 |        9 |         9 |
|             7 |       11 |         8 |
|             8 |        6 |         7 |
|             9 |        7 |         6 |
|---------------+----------+-----------|

This has a chi-square value of 2.166 (p = 0.975) which is an even better fit.

Golden angle

Posted on 1 May 2017 by John

The golden angle is related to the golden ratio, but it is not as well known. And the relationship is not quite what you might think at first.

The golden ratio φ is (1 + √5)/2. A golden rectangle is one in which the ratio of the longer side to the shorter side is φ. Credit cards, for example, are typically golden rectangles.

You might guess that a golden angle is 1/φ of a circle, but it’s actually 1/φ² of a circle. Let a be the length of an arc cut out of a circle by a golden angle and b be the length of its complement. Then by definition the ratio of b to a is φ. In other words, the golden angle is defined in terms of the ratio of its complementary arc, not of the entire circle. [1]

The video below has many references to the golden angle. It says that the golden angle is 137.5 degrees, which is fine given the context of a popular video. But this doesn’t explain where the angle comes from or give its exact value of 360/φ² degrees.

[1] Why does this work out to 1/φ²? The ratio b/a equals φ, by definition. So the ratio of a to the whole circle is

a/(a + b) = a/(a + φa) = 1/(1 + φ) = 1/φ²

since φ satisfies the quadratic equation 1 + φ = φ².

Flying through a 3D fractal

Posted on 18 April 2017 by John

A Menger sponge is created by starting with a cube a recursively removing chunks of it. Draw a 3×3 grid on one face of the cube, then remove the middle square, all the way through the cube. Then do the same for each of the eight remaining squares. Repeat this process over and over, and do it for each face.

The holes are all rectangular, so it’s surprising that the geometry is so varied when you slice open a Menger sponge. For example, when you cut it on the diagonal, you can see stars! (I wrote about this here.)

I mentioned this blog post to a friend at Go 3D Now, a company that does 3D scanning and animation, and he created the video below. The video starts out by taking you through the sponge, then at about the half way part the sponge splits apart.

Computing harmonic numbers

Posted on 18 April 2017 by John

The harmonic numbers are defined by

Harmonic numbers are sort of a discrete analog of logarithms since

$\log n = \int_1^n \frac{1}{x} \, dx$

As n goes to infinity, the difference between H_n and log n is Euler’s constant γ = 0.57721… [1]

How would you compute H_n? For small n, simply use the definition. But if n is very large, there’s a way to approximate H_n without having to do a large sum.

Since in the limit H_n – log n goes to γ, a crude approximation would be

$H_n \approx \log n + \gamma$

But we could do much better by adding a couple terms to the approximation above. [2] That is,

$H_n \approx \log n + \gamma + \frac{1}{2n} - \frac{1}{12n^2}$

The error in the approximation above is between 0 and 1/120n⁴.

So if you used this to compute the 1000th harmonic number, the error would be less than one part in 120,000,000,000,000. Said another way, for n = 1000 the approximation differs from the exact value in the 15th significant digit, approximately the resolution of floating point numbers (i.e. IEEE 754 double precision).

And the formula is even more accurate for larger n. If we wanted to compute the millionth harmonic number, the error in our approximation would be somewhere around the 26th decimal place.

* * *

[1] See Julian Havil’s excellent Gamma: Exploring Euler’s Constant. It’s popular-level book, but more sophisticated than most such books.

[2] There’s a sequence of increasingly accurate approximations that keep adding reciprocals of even powers of n, based on truncating an asymptotic series. See Concrete Mathematics for details.

Uniform distribution of powers mod 1

Posted on 2 April 2017 by John

A few days ago I wrote about how powers of the golden ratio are nearly integers but powers of π are not. This post is similar but takes a little different perspective. Instead of looking at how close powers are to the nearest integers, we’ll look at how close they are to their floor, the largest integer below. Put another way, we’ll throw away the integer parts and look at the decimal parts.

First a theorem:

For almost all x > 1, the sequence (xⁿ) for n = 1, 2, 3, … is u.d. mod 1. [1]

Here “almost all” is a technical term meaning that the set of x‘s for which the statement above does not hold has Lebesgue measure zero. The abbreviation “u.d.” stands for “uniformly distributed.” A sequence uniformly distributed mod 1 if the fractional parts of the sequence are distributed like uniform random variables.

Even though the statement holds for almost all x, it’s hard to prove for particular values of x. And it’s easy to find particular values of x for which the theorem does not hold.

From [1]:

… it is interesting to note that one does not know whether sequences such as (eⁿ), (πⁿ), or even ((3/2)ⁿ) are u.d. mod 1 or not.

Obviously powers of integers are not u.d. mod 1 because their fractional parts are all 0. And we’ve shown before that powers of the golden ratio and powers of the plastic constant are near integers, i.e. their fractional parts cluster near 0 and 1.

The curious part about the quote above is that it’s not clear whether powers of 3/2 are uniformly distributed mod 1. I wouldn’t expect powers of any rational number to be u.d. mod 1. Either my intuition was wrong, or it’s right but hasn’t been proved, at least not when [1] was written.

The next post will look at powers of 3/2 mod 1 and whether they appear to be uniformly distributed.

* * *

[1] Kuipers and Niederreiter, Uniform Distribution of Sequences

Natural transformations

Posted on 16 March 2017 by John

The ladder of abstractions in category theory starts with categories, then functors, then natural transformations. Unfortunately, natural transformations don’t seem very natural when you first see the definition. This is ironic since the original motivation for developing category theory was to formalize the intuitive notion of a transformation being “natural.” Historically, functors were defined in order to define natural transformations, and categories were defined in order to define functors, just the opposite of the order in which they are introduced now.

A category is a collection of objects and arrows between objects. Usually these “arrows” are functions, but in general they don’t have to be.

A functor maps a category to another category. Since a category consists of objects and arrows, a functor maps objects to objects and arrows to arrows.

A natural transformation maps functors to functors. Sounds reasonable, but what does that mean?

You can think of a functor as a way to create a picture of one category inside another. Suppose you have some category and pick out two objects in that category, A and B, and suppose there is an arrow f between A and B. Then a functor F would take A and B and give you objects FA and FB in another category, and an arrow Ff between FA and FB. You could do the same with another functor G. So the objects A and B and the arrow between them in the first category have counterparts under the functors F and G in the new category as in the two diagrams below.

A natural transformation α between F and G is something that connects these two diagrams into one diagram that commutes.

The natural transformation α is a collection of arrows in the new category, one for every object in the original category. So we have an arrow α_A for the object A and another arrow α_B for the object B. These arrows are called the components of α at A and B respectively.

Note that the components of α depend on the objects A and B but not on the arrow f. If f represents any other arrow from A to B in the original category, the same arrows α_A and α_B fill in the diagram.

Natural transformations are meant to capture the idea that a transformation is “natural” in the sense of not depending on any arbitrary choices. If a transformation does depend on arbitrary choices, the arrows α_A and α_B would not be reusable but would have to change when f changes.

The next post will discuss the canonical examples of natural and unnatural transformations.

Related: Applied category theory

Eight Ramanujan posts

Posted on 14 March 2017 by John

Srinivasa Ramanujan

Eight short, accessible blog posts based on the work of the intriguing mathematician Srinivasa Ramanujan:

Complex analysis image quilt

Posted on 10 March 2017 by John

A blog post by Evelyn Lamb yesterday introduced Thomas Baruchel’s web site by of images from complex analysis.

I wondered what a collage of these images would look like, so I used the ImageQuilts software by Edward Tufte and Adam Schwartz to create the image below.

Image quilt from complex analysis graphs

Related: Applied complex analysis

How areas of math are connected

Posted on 9 March 2017 by John

In my previous post, I discussed how number theory and topology relate to other areas of math. Part of that was to show a couple diagrams from Jean Dieudonné’s book Panorama of Pure Mathematics, as seen by N. Bourbaki. That book has only small star-shaped diagrams considering one area of math at a time. I’ve created a diagram that pastes these local views into one grand diagram. Along the way I’ve done a little editing because the original diagrams were not entirely consistent.

Here’s a condensed view of the graph. You can find the full image here.

The graph is so dense that it’s hard to tell which areas have the most or least connections. Here are some tables to clarify that. First, counting how many areas an particular area contributes to, i.e. number of outgoing arrows.

|-------------------------------------+---------------|
| Area                                | Contributions |
|-------------------------------------+---------------|
| Homological algebra                 |            12 |
| Lie groups                          |            11 |
| Algebraic and differential topology |            10 |
| Categories and sheaves              |             9 |
| Commutative algebra                 |             9 |
| Commutative harmonic analysis       |             9 |
| Algebraic geometry                  |             8 |
| Differential geometry and manifolds |             8 |
| Integration                         |             8 |
| Partial differential equations      |             8 |
| General algebra                     |             7 |
| Noncommutative harmonic analysis    |             6 |
| Ordinary differential equations     |             6 |
| Spectral theory of operators        |             6 |
| Analytic geometry                   |             5 |
| Automorphic and modular forms       |             5 |
| Classical analysis                  |             5 |
| Mathematical logic                  |             5 |
| Abstract groups                     |             4 |
| Ergodic theory                      |             4 |
| Probability theory                  |             4 |
| Topological vector spaces           |             4 |
| General topology                    |             3 |
| Number theory                       |             3 |
| Von Neumann algebras                |             2 |
| Set theory                          |             1 |
|-------------------------------------+---------------|

Next, counting the sources each area draws on, i.e. counting incoming arrows.

|-------------------------------------+---------|
| Area                                | Sources |
|-------------------------------------+---------|
| Algebraic geometry                  |      13 |
| Number theory                       |      12 |
| Lie groups                          |      11 |
| Noncommutative harmonic analysis    |      11 |
| Algebraic and differential topology |      10 |
| Analytic geometry                   |      10 |
| Automorphic and modular forms       |      10 |
| Ordinary differential equations     |      10 |
| Ergodic theory                      |       9 |
| Partial differential equations      |       9 |
| Abstract groups                     |       8 |
| Differential geometry and manifolds |       8 |
| Commutative algebra                 |       6 |
| Commutative harmonic analysis       |       6 |
| Probability theory                  |       5 |
| Categories and sheaves              |       4 |
| Homological algebra                 |       4 |
| Spectral theory of operators        |       4 |
| Von Neumann algebras                |       4 |
| General algebra                     |       2 |
| Mathematical logic                  |       1 |
| Set theory                          |       1 |
| Classical analysis                  |       0 |
| General topology                    |       0 |
| Integration                         |       0 |
| Topological vector spaces           |       0 |
|-------------------------------------+---------|

Finally, connectedness, counting incoming and outgoing arrows.

|-------------------------------------+-------------|
| Area                                | Connections |
|-------------------------------------+-------------|
| Lie groups                          |          22 |
| Algebraic geometry                  |          21 |
| Algebraic and differential topology |          20 |
| Noncommutative harmonic analysis    |          17 |
| Partial differential equations      |          17 |
| Differential geometry and manifolds |          16 |
| Homological algebra                 |          16 |
| Ordinary differential equations     |          16 |
| Analytic geometry                   |          15 |
| Automorphic and modular forms       |          15 |
| Commutative algebra                 |          15 |
| Commutative harmonic analysis       |          15 |
| Number theory                       |          15 |
| Categories and sheaves              |          13 |
| Ergodic theory                      |          13 |
| Abstract groups                     |          12 |
| General algebra                     |          10 |
| Spectral theory of operators        |          10 |
| Probability theory                  |           9 |
| Integration                         |           8 |
| Mathematical logic                  |           6 |
| Von Neumann algebras                |           6 |
| Classical analysis                  |           5 |
| Topological vector spaces           |           4 |
| General topology                    |           3 |
| Set theory                          |           2 |
|-------------------------------------+-------------|

There are some real quirks here. The most foundational areas get short shrift. Set theory contributes to only one area of math?! Topological vector spaces don’t depend on anything, not even topology?!

I suspect Dieudonné had in mind fairly high-level contributions. Topological vector spaces, for example, obviously depend on topology, but not deeply. You could do research in the area while seldom drawing on more than an introductory topology course. Elementary logic and set theory are used everywhere, but most mathematicians have no need for advanced logic or set theory.

More math diagrams: