# Uriel’s stacking problem

In Scott Alexander’s kabbalistic sf story Unsong, the archangel Uriel works on a problem while other things are going on in heaven:

All the angels listened in rapt attention except Uriel, who was sort of half-paying attention while trying to balance several twelve-dimensional shapes on top of each other.

There was utter silence throughout the halls of Heaven, except a brief curse as Uriel’s hyperdimensional tower collapsed on itself and he picked up the pieces to try to rebuild it.

A great clamor arose from all the heavenly hosts, save Uriel, who took advantage of the brief lapse to conjure a parchment and pen and start working on a proof about the optimal configuration of twelve-dimensional shapes.

## A polytope on a plane

This got me thinking about the stability of stacking polytopes. That seemed complicated (I am no archangel) so I started toying with the stability of polytopes on a flat surface.

(Terminology note: I will consistently use “face” to denote the D-1 dimensional elements that bound the polytope, although “facet” is in some use.)

A face of a 3D polyhedron is stable if the polyhedron can rest on it without tipping over. This means that the projection of the center of mass onto the plane containing the face is inside the polygon. The platonic polyhedra are stable on all faces, but it is not hard to make a few faces unstable by moving a vertex far away from the center. A polyhedron has at least one stable face (if it did not, it would be a perpetual motion device: every tip will move the center of mass downwards, but there is a bound on how low it can go. A uni-stable or monostatic polyhedron has  just one stable face. It is an unsolved problem what the simplest uni-stable 3D polyhedron is, with the current record 14 faces. Also, it seems unclear whether there are monostatic simplices in dimension 9 (they exist in 10 or more dimensions, but not in 8 or fewer).

So, how many faces of a polytope will typically be unstable?

I wrote a Matlab script to generate random convex polytopes by selecting N points randomly on the surface of a D-dimensional sphere and calculating their convex hull. Using a Delaunay decomposition I can split them into simplices, which allow me to calculate the center of mass. The center of mass of a simplex is just the average of the corners $\vec{x^c_j}=\sum_{i=1}^N \vec{x}_{ij}$, and the center of mass of the polyhedron is just the sum of the simplex centers of mass weighted by their volumes: $\vec{x^p} = \sum_{j=1} V_j \vec{x^c_j}$. The volume of a simplex is $V_j=(1/D!)\mathrm{det}(X_j)$ where $X_j=[x_{1j};x_{2j};\ldots;x_{Dj}]$, the matrix made by sticking together the coordinate vectors of a simplex. Once we know this we can project the center of mass onto the plane of a face by finding its nullspace (the higher dimensional counterpart to a normal) $\vec{p}= \vec{x^p} -(\vec{x^p}\cdot \vec{n})\vec{n}$. Finally, to check whether the projection is inside the face, we can look at the matrix A where each column is the coordinates of one of the faces minus $\vec{p}$ and the final row just ones, and solve for Ax=b where b is zero except for a one in the last row (I found this neat algorithm due to elisbben on stack overflow).  If the answer vector is all positive, then the point is inside the face. Repeat for all the faces.

Whew. This math is of course really simple to do in Matlab.

The 12 dimensional case is a bit messier:

So, what is the average fraction of stable faces on a 3D polyhedron?

It tends to converge to 50%. Doing this in higher dimensions shows the same kind of convergence, although to lower fractions.

It looks like the fraction of stable faces declines exponentially with dimensionality.

Does this mean that for a sufficiently high dimension it is likely that a random polytope is unistable? The answer is no: the number of faces increases pretty exponentially (as $2^{1.7680D}$), but the number of stable faces also increases exponentially with D (as $latex 2^{0.9273 D}$).

This was based on runs with N=100. Obviously things go much faster if you select a lower N, such as 30. However, as you approach N=D the polytopes become more and more simplex-like, and simplices tend to both have fewer faces and be less stable in high dimensions, so the exponential growth stops. This actually happens far below D; for N=30 the effect is felt already in 11 dimensions. The face growth rates were also lower, with coefficients 1.1621 and 0.4730.

(There are some asymptotic formulas known for the growth of the number of faces for random convex hulls; they grow linearly with N but at an accelerating rate with D.)

Stuart Armstrong gave me a very heuristic argument for why there would be so many unstable faces. Consider building up the polytope vertex by vertex, essentially just adding together the simplices from the Delaunay decomposition. If you start from a stable state, eventually you will likely end up with an unstable face. Adding the next vertex will add a simplex to the polyhedron, and the center of mass will move in the direction of the new simplex. To have the face become stable again the shift in center of mass needs to be large enough along the directions parallel to the face to bring the projection back inside the face. But in high dimensional spaces there are many directions you can move in: the probability of a random vector being nearly parallel to another vector is very low. Hence, the next step and the following are likely to preserve the instability. So high dimensional polytopes are likely to have many unstable faces even if they are nicely inscribed in spheres.

The number of steps the polytope rolls over  until finding a stable face is also limited: the “drainage basin” of a stable face is a tree, with a branching degree set by D-1 (if faces are D-simplexes). So the number of steps will scale as $\log_{D-1}(2^{(1.7680 - 0.9273)D})=0.8407 D \ln(2) / \ln(D-1) \propto D/\ln(D)$. Even high-dimensional polytopes will stop flipping quickly in general. (A unistable polytope on the other hand can run through at least half of its faces, so there are some very slow ones too).

The expected minimum distance between two points on this kind of random polytope scales as $N^{-2/D}$ (if they were optimally distributed it would be $N^{-1/D}$). At the same time, if N is relatively small compared to D (the polytope is simplex-like), the average diameter (the longest edge) of each face seems to approach $\sqrt{\pi}$. Why? I think this is because  $\Gamma(1/2)$, the mean of a flipped k=2 Weibull distribution that shows up because of extreme value theory. Meanwhile the average and median cord length between random points on hyperspheres tends towards $\sqrt{2}$. Faces hence tends to be fairly wide unless N is large compared to D, but there will typically always be a few very narrow ones that are tricky to balance on.

## Stacking no-slip polytopes

What about stacking polytopes?

If you put a polytope on top of another one (assuming no slipping) at first it seems you need to use a stable face of the top polytope, but this is not enough nor necessary.

Since the underlying face is likely tilted from the horizontal, the vertical projection of the center of mass has to be within the top face. The upper polytope can be rotated, moving the projection point. The tilt angle $\theta$ (or rather, tilt angles – we are doing this in higher dimensions, remember?) generates a hypersphere of radius $d \tan(\theta)$ around the normal projection point (which is at distance d from the center of mass) where the vertical projection can intersect the face. Only parts of the hypersphere surface that are inside the face represent orientations that are stable. Even an unstable face can (sometimes) be stabilized if you turn it so that the tilted projection is inside, but for sufficiently high angles the hypersphere will be bigger than the face and it cannot be stable.

Having the top polytope stay in place is the first requirement. The second is that the bottom polytope should not become unstable. The new center of mass is moved to a point somewhere along the connecting line between the individual centers of mass of the polytopes, with exact position dependent on their volume ratio (note that turning the top polytope can move the center of mass too). This moves the projection point along the plane of the bottom face, and if it gets outside that face the assembly will tip over.

One can imagine this as adding random (D-1)-dimensional vectors of length 1/N until they reach the edge of the face. I am a bit uncertain about the properties of such random walks (all works on decreasing step size walks I have seen have been in 1D). The harmonic random walk in 1D apparently converges with probability 1, so I think the (D-1)-dimensional one also does it since the distance from the origin to the walker will be smaller than if the walker just kept to a 1D line. Since the expected distance traversed in 1D is $latex E[|X|] \approx 1.0761$ this is actually not a very extreme  shift. Given the surprisingly large diameters of the faces if $N \ll D$ the first condition might be tougher to meet than the second, but this is just a guess.

The no slipping constraint is important. If the polytopes are frictionless, then any transverse force will move them. Hence only polytopes that have some parallel top and bottom stable faces can be stacked, and the problem becomes simpler. There are still surprises there, though: even stacks of rectangular blocks can do surprising things. The block stacking problem also demonstrates that one can have 1/N overhangs (counting downwards), enabling arbitrarily large total overhangs without tipping over. With polytopes with shapes that act as counterweights the overhangs can be even larger.

## Uriel’s stacking problems

This leads to what we might call “Uriel’s stacking problem”: given a collection of no-slip convex D-dimensional polytopes, what is the tallest tower that can be constructed from them?

I suspect that this problem is NP-hard. It sounds very much like a knapsack problem, but there is a dependency on previous steps when you add a new polytope that seem to make it harder. It seems that it would not be too difficult to fool a greedy algorithm just trying to put the next polytope on the most topmost face into adding one that makes subsequent steps too unstable, forcing backtracking.

Another related problem: if the polytopes are random convex hulls of N points, what is the distribution of maximum tower heights? What if we just try random stacking?

And finally, what is the maximum overhang that can be done by stacking polytopes from a given set?

# Simple instability

As a side effect of a chat about dynamical systems models of metabolic syndrome, I came up with the following nice little toy model showing two kinds of instability: instability because of insufficient dampening, and instability because of too slow dampening.

$x'(t) = Ax(t)/N - px(t-\tau) -x(t)^3$

Where $x$ is a N-dimensional vector, A is a $N \times N$ matrix with Gaussian random numbers, and $p \geq 0, \tau \geq 0$ constants. The last term should strictly speaking be written as $||x(t)||^2 x(t)$ but I am lazy.

The first term causes chaos, as we will see below. The 1/N factor is just to compensate for the N terms. The middle term represents dampening trying to force the system to the origin, but acting with a delay $\tau$. The final term keeps the dynamics bounded: as $||x||$ becomes large this term will dominate and bring back the trajectory to the vicinity of the origin. However, it is a soft spring that has little effect close to the origin.

## Chaos

Let us consider the obvious fixed point $x=0$. Is it stable? If we calculate the Jacobian matrix there it becomes $J = A/N - pI$. First, consider the case where $p=0$. The eigenvalues of J will be the ones of a random Gaussian matrix with no symmetry conditions. If it had been symmetric, then Wigner’s semicircle rule implies that they would tend to be distributed as $P(\lambda)=(2/\pi)\sqrt{1-\lambda^2}$ as $N \rightarrow \infty$. However, it turns out that this is true for the non-symmetric Gaussian case too. (and might be true for any i.i.d. random numbers). This means that about half of them will have a positive real part, and that implies that the fixed point is unstable: for $p=0$ the system will be orbiting the origin in some fashion, and generically this means a chaotic attractor.

## Stability

If $p$ grows the diagonal elements of J will become more and more negative. If they are really negative then we essentially have a matrix with a negative diagonal and some tiny off-diagonal terms: the eigenvalues will almost be the diagonal ones, and they are all negative. The origin is a stable attractive fixed point in this limit.

In between, if we plot the eigenvalues as a function of $p$, we see that the semicircle just linearly moves towards the negative side and when all of it passes over, we shift from the chaotic dynamics to the fixed point. Exactly when this happens depends on the particular A we are looking at and its largest eigenvalue (which is distributed as the Tracy-Widom distribution), but it is generally pretty sharp for large N.

## Delay

But what if $\tau$ becomes large? In this case the force moving the trajectory towards the origin will no longer be based on where it is right now, but on where it was $\tau$ seconds earlier. If $p$ is small, then this is just minor noise/bias (and the dynamics is chaotic anyway). If it is large, then the trajectory will be pushed in some essentially random direction: we get instability again.

A (very slightly) more stringent way of thinking of it is to plug in $x_j(t)=c_j e^{i\lambda_j t}$ into the equation. To simplify, let’s throw away the cubic term since we want to look at behavior close to zero, and let’s use a coordinate system where the matrix is a diagonal matrix $\Lambda$. Then for $p=0$ we get $\lambda_j = \Lambda_j$, that is, the origin is a fixed point that repels or attracts trajectories depending on its eigenvalues (and we know from above that we can be pretty confident some are positive, so it is unstable overall). For $p>0$ we get $\lambda_j + pe^{-i\lambda_j \tau} = \Lambda_j$. Taylor expansion to the first order and rearranging gives us $\lambda_j \approx(\Lambda_j - p)/(1 - i p \tau)$. The  numerator means that as $p$ grows, each eigenvalue will eventually get a negative real part: that particular direction of dynamics becomes stable and attracted to the origin. But the denominator can sabotage this: it $p \tau$ gets large enough it can move the eigenvalue anywhere, causing instability.

So there you are: if you try to keep a system stable, make sure the force used is up to the task so the inherent recalcitrance cannot overwhelm it, and make sure the direction actually corresponds to the current state of the system.

# Dancing zeros

Playing with Matlab, I plotted the location of the zeros of a polynomial with normally distributed coefficients in the complex plane. It was nearly a circle:

This did not surprise me that much, since I have already toyed with the distribution of zeros of polynomials with coefficients in {-1,0,+1}, producing some neat distributions close to the unit circle (see also John Baez). A quick google found (Hughes & Nikeghbali 2008): under very general circumstances polynomial zeros tend towards the unit circle. One can heuristically motivate it in a lot of ways.

As you add more and more terms to the polynomial the zeros approach the unit circle. Each new term perturbs them a bit: at first they move around a lot as the degree goes up, but they soon stabilize into robust positions (“young” zeros move more than “old” zeros). This seems to be true regardless of whether the coefficients set in “little-endian” or “big-endian” fashion.

But then I decided to move things around: what if the coefficient on the leading term changed? How would the zeros move? I looked at the polynomial $P_\theta(z)=e^{i\theta} z^n + c_{n-1}z^{n-1}+\ldots+c_1 z + c_0$ where $c_i$ were from some suitable random sequence and $\theta$ could run around $[0,2\pi]$. Since the leading coefficient would start and end up back at 1, I knew all zeros would return to their starting position. But in between, would they jump around discontinuously or follow orderly paths?

Continuity is actually guaranteed, as shown by (Harris & Martin 1987). As you change the coefficients continuously, the zeros vary continuously too. In fact, for polynomials without multiple zeros, the zeros vary analytically with the coefficients.

As $\theta$ runs from 0 to $2\pi$ the roots move along different orbits. Some end up permuted with each other.

For low degrees, most zeros participate in a large cycle. Then more and more zeros emerge inside the unit circle and stay mostly fixed as the polynomial changes. As the degree increases they congregate towards the unit circle, while at least one large cycle wraps most of them, often making snaking detours into the zeros near the unit circle and then broad bows outside it.

In the above example, there is a 21-cycle, as well as a 2-cycle around 2 o’clock. The other zeros stay mostly put.

The real question is what determines the cycles? To understand that, we need to change not just the argument but the magnitude of $c_n$.

What happens if we slowly increase the magnitude of the leading term, letting $c_n = re^{i\theta}$ for a r that increases from zero? It turns out that a new zero of the function zooms in from infinity towards the unit circle. A way of seeing this is to look at the polynomial as $P_n(z) = c_n z^n + P_{n-1}(z)$: the second term is nonzero and large in most places, so if $c_n$ is small the $z^n$ factor must be large (and opposite) to outweigh it and cause a zero. The exception is of course close to the zeros of $P_{n-1}(z)$, where the perturbation just moves them a tiny bit: there is a counterpart for each of the $n-1$ zeros of $P_{n-1}(z)$ among the zeros of $P_{n}(z)$. While the new root is approaching from outside, if we play with $\theta$ it will make a turn around the other zeros: it is alone in its orbit, which also encapsulates all the other zeros. Eventually it will start interacting with them, though.

If you instead start out with a large leading term, $|c_n| \gg |c_i|$, then the polynomial is essentially $P_n(z)=c_nz^n+[\mathrm{small stuff}]$ and the zeros the n-th roots of $-[\mathrm{small stuff}]/c_n$. All zeros belong to the same roughly circular orbit, moving together as $\theta$ makes a rotation. But as $|c_n|$ decreases the shared orbit develops bulges and dents, and some zeros pinch off from it into their own small circles. When does the pinching off happen? That corresponds to when two zeros coincide during the orbit: one continues on the big orbit, the other one settles down to be local. This is the one case where the analyticity of how they move depending on $c_n$ breaks down. They still move continuously, but there is a sharp turn in their movement direction. Eventually we end up in the small term case, with a single zero on a large radius orbit as $|c_n| \rightarrow 0$.

This pinching off scenario also suggests why it is rare to find shared orbits in general: they occur if two zeros coincide but with others in between them (e.g. if we number them along the orbit, $z_1=z_k$, with $z_2$ to $z_{k-1}$ separate). That requires a large pinch in the orbit, but since it is overall pretty convex and circle-like this is unlikely.

Allowing $|c_n|$ to run from $\infty$ to 0 and $\theta$ over $[0,2\pi]$ would cover the entire complex plane (except maybe the origin): for each z, there is some $c_n$ where $c_nz^n+\ldots+c_1z+c_0=0$. This is fairly obviously $c_n = f(z) = -P_{n-1}(z)/z^n$. This function has a central pole, surrounded by zeros corresponding to the zeros of $P_{n-1}(z)$. The orbits we have drawn above correspond to level sets $|f(z)|=\mathrm{const}$, and the pinching off to saddle points of this surface. To get a multi-zero orbit several zeros need to be close together enough to cause a broad valley.

There you have it, a rough theory of dancing zeros.

## References

Harris, G., & Martin, C. (1987). Shorter notes: The roots of a polynomial vary continuously as a function of the coefficients. Proceedings of the American Mathematical Society, 390-392.
Hughes, C. P., & Nikeghbali, A. (2008). The zeros of random polynomials cluster uniformly near the unit circle. Compositio Mathematica, 144(03), 734-746.

# Adding cooks to the broth

If there are $k$ key ideas needed to produce some important goal (like AI), there is a constant probability per researcher-year to come up with an idea, and the researcher works for $y$ years, what is the the probability of success? And how does it change if we add more researchers to the team?

The most obvious approach is to think of this as y Bernouilli trials with probability p of success, quickly concluding that the number of successes n at the end of y years will be distributed as $\mathrm{Pr}(n)=\binom{y}{n}p^n(1-p)^{y-n}$. Unfortunately, then the actual answer to the question will be $\mathrm{Pr}(n\geq k) = \sum_{n=k}^y \binom{y}{n}p^n(1-p)^{y-n}$ which is a real mess…

A somewhat cleaner way of thinking of the problem is to go into continuous time, treating it as a homogeneous Poisson process. There is a rate $\lambda$ of good ideas arriving to a researcher, but they can happen at any time. The time between two ideas will be exponentially distributed with parameter $\lambda$. So the time $t$ until a researcher has $k$ ideas will be the sum of $k$ exponentials, which is a random variable distributed as the Erlang distribution: $f(t; k,\lambda)=\lambda^k t^{k-1} e^{-\lambda t} / (k-1)!$.

Just like for the discrete case one can make a crude argument that we are likely to succeed if $y$ is bigger than the mean $k/\lambda$ (or $k/p$) we will have a good chance of reaching the goal. Unfortunately the variance scales as $k/\lambda^2$ – if the problems are hard, there is a significant risk of being unlucky for a long time. We have to consider the entire distribution.

Unfortunately the cumulative density function in this case is $\mathrm{Pr}(t which is again not very nice for algebraic manipulation. Still, we can plot it easily.

Before we do that, let us add extra researchers. If there are $N$ researchers, equally good, contributing to the idea generation, what is the new rate of ideas per year? Since we have assumed independence and a Poisson process, it just multiplies the rate by a factor of $N$. So we replace $\lambda$ with $\lambda N$ everywhere and get the desired answer.

This is a plot of the case $k=10, y=10$.

What we see is that for each number of scientists it is a sigmoid curve: if the discovery probability is too low, there is hardly any chance of success, when it becomes comparable to $k/N$ it rises, and sufficiently above we can be almost certain the project will succeed (the yellow plateau). Conversely, adding extra researchers has decreasing marginal returns when approaching the plateau: they make an already almost certain project even more certain. But they do have increasing marginal returns close to the dark blue “floor”: here the chances of success are small, but extra minds increase them a lot.

We can for example plot the ratio of success probability for $\lambda=0.09$ to the one researcher case as we add researchers:

Even with 10 researchers the success probability is just 40%, but clearly the benefit of adding extra researchers is positive. The curve is not quite exponential; it slackens off and will eventually become a big sigmoid. But the overall lesson seems to hold: if the project is a longshot, adding extra brains makes it roughly exponentially more likely to succeed.

It is also worth recognizing that in this model time is on par with discovery rate and number of researchers: what matters is the product $\lambda y N$ and how it compares to $k$.

This all assumes that ideas arrive independently, and that there are no overheads for having a large team. In reality these things are far more complex. For example, sometimes you need to have idea 1 or 2 before idea 3 becomes possible: that makes the time $t_3$ of that idea distributed as an exponential plus the distribution of $\mathrm{min}(t_1,t_2)$. If the first two ideas are independent and exponential with rates $\lambda, \mu$, then the minimum is distributed as an exponential with rate $\lambda+\mu$. If they instead require each other, we get a non-exponential distribution (the pdf is $\lambda e^{-\lambda t} + \mu e^{-\mu t} - (\lambda+\mu)e^{-(\lambda+\mu)t}$). Some discoveries or bureaucratic scalings may change the rates. One can construct complex trees of intellectual pathways, unfortunately quickly making the distributions impossible to write out (but still easy to run Monte Carlo on). However, as long as the probabilities and the induced correlations small, I think we can linearise and keep the overall guess that extra minds are exponentially better.

In short: if the cooks are unlikely to succeed at making the broth, adding more is a good idea. If they already have a good chance, consider managing them better.

# Julia lace gaskets

While surfing the web I came across a neat Julia set, defined by iterating $z_{n+1}=f(z_n)=z_n^2+c/z_n^3$ for some complex constant c. Here are some typical pictures, and two animations: one moving around a circle in the c-plane, one moving slowly down from c=1 to c=0.

# The points behind the set

What is going on?

The first step in analysing fractals like this is to find the fixed points and their preimages. $z=\infty$ is clearly mapped to itself. The $z^2$ term will tend to make large magnitude iterates approach infinity, so it is an attractive fixed point.

$z=0$ is a preimage of infinity: iterates falling on zero will be mapped onto infinity. Nearby points will also end up attracted to infinity, so we have a basin of attraction to infinity around the origin. Preimages of the origin will be mapped to infinity in two steps: $0=z^2+c/z^3$ has the solutions $z=(-c)^{1/5}$ – this is where the pentagonal symmetry comes from, since these five points are symmetric. Their preimages and so on will also be mapped to infinity, so we have a hierarchy of basins of attraction sending points away forming some gasket-like structure. The Julia set consists of the points that never gets mapped away, the boundary of this hierarchy of basins.

The other fixed points are defined by $z=z^2+c/z^3$, which can be rearranged into $z^5-z^4+c=0$. They don’t have any neat expression and actually do not affect the big picture dynamics as much. The main reason seems to be that they are unstable. However, their location and the derivative close to them affect the shapes in the Julia set as we will see. Their preimages will be surrounded by the same structures (scaled and rotated) as they have.

Below are examples with preimages of zero marked as white circles, fixed points as red crosses, and critical points as black squares.

# The set behind the points

A simple way of mapping the dynamics is to look at the (generalized) Mandelbrot set for the function, taking a suitable starting point $z_0=(3c/2)^{1/5}$ and mapping out its fate in the c-plane. Why that particular point? Because it is one of the critical point where $f'(z)=0$, and a theorem by Julia and Fatou tells us that its fate indicates whether the Julia set is filled or dust-like: bounded orbits of the critical points of a map imply a connected Julia set. When c is in the Mandelbrot set the Julia image has “thick” regions with finite area that do not escape to infinity. When c is outside, then most points end up at infinity, and what remains is either dust or a thin gasket with no area.

The set is much smaller than the vanilla $f(z)=z^2+c$ Mandelbrot, with a cuspy main body surrounded by a net reminiscent of the gaskets in the Julia set. It also has satellite vanilla Mandelbrots, which is not surprising at all: the square term tends to dominate in many locations. As one zooms into the region near the origin a long spar covered in Mandelbrot sets runs towards the origin, surrounded by lacework.

One surprising thing is that the spar does not reach the origin – it stops at $c=4.5 \cdot 10^{-5}$. Looking at the dynamics, above this point the iterates of the critical point jump around in the interval [0,1], forming a typical Feigenbaum cascade of period doubling as you go out along the spar (just like on the spar of the vanilla Mandelbrot set). But at this location points now are mapped outside the interval, running off to infinity: one of the critical points breaches a basin boundary, causing iterates to run off and the earlier separate basins to merge. Below this point the dynamics is almost completely dominated by the squaring, turning the Julia set into a product of a Cantor set and a circle (a bit wobbly for higher c; it is all very similar to KAM torii). The empty spaces correspond to the regions where preimages of zero throw points to infinity, while along the wobbly circles points get their argument angles multiplied by two for every iteration by the dominant quadratic term: they are basically shift maps. For c=0 it is just the filled unit disk.

So when we allow c to move around a circle as in the animations, the part that passes through the Mandelbrot set has thick regions that thin as we approach the edge of the set. Since the edge is highly convoluted the passage can be quite complex (especially if the circle is “tangent” to it) and the regions undergo complex twisting and implosions/explosions. During the rest of the orbit the preimages just quietly rotate, forming a fractal gasket. A gasket that sometimes looks like a model of the hyperbolic plane, since each preimage has five other preimages, naturally forming an exponential hierarchy that has to be squeezed into a finite roughly circular space.

# A prime minimal surface

A short while ago I mentioned lacunary functions, complex functions that are analytic inside a disc but cannot be continued outside it. Then I remembered the wonderful Weierstrass-Enneper representation formula, which assigns a minimal surface to (nearly) any pair of complex functions. What happens when you make a minimal surface from a lacunary function?

I was not first with this idea. In fact, F.F. de Brito used this back in 1992 to demonstrate that there exist complete embedded minimal surfaces in 3-space that are contained between two planes.

Here is the surface defined by the function $g(z)=\sum_{p \mathrm{is prime}} z^p$, the Taylor series that only includes all prime powers, combined with $f(z)=1$.

Close to zero, the surface is flat. Away from zero it begins to wobble as increasingly high powers in the series begin to dominate. It behaves very much like a higher-degree Enneper surface, but with a wobble that is composed of smaller wobbles. It is cool to consider that this apparently irregular pattern corresponds to the apparently irregular pattern of all primes.

# Some math for Epiphany

## Analytic functions helping you out

Recently I chatted with a mathematician friend about generating functions in combinatorics. Normally they are treated as a neat symbolic trick: you have a sequence $a_n$ (typically how many there are of some kind of object of size $n$), you formally define a function $f(z)=\sum_{n=0}^\infty a_n z^n$, you derive some constraints on the function, and from this you get a formula for the $a_n$ or other useful data. Convergence does not matter, since this is purely symbolic. We used this in our paper counting tie knots. It is a delightful way of solving recurrence relations or bundle up moments of probability distributions.

I innocently wondered if the function (especially its zeroes and poles) held any interesting information. My friend told me that there was analytic combinatorics: you can actually take $f(z)$ seriously as a (complex) function and use the powerful machinery of complex analysis to calculate asymptotic behavior for the $a_n$ from the location and type of the “dominant” singularities. He pointed me at the excellent course notes from a course at Princeton linked to the textbook by Philippe Flajolet and Robert Sedgewick. They show a procedure for taking combinatorial objects, converting them symbolically into generating functions, and then get their asymptotic behavior from the properties of the functions. This is extraordinarily neat, both in terms of efficiency and in linking different branches of math.

In our case, one can show nearly by inspection that the number of Fink-Mao tie knots grow with the number of moves as $\sim 2^n$, while single tuck tie knots grow as $\sim \sqrt{6}^n$.

## Analytic functions behaving badly

The second piece of math I found this weekend was about random Taylor series and lacunary functions.

If $f(z)=\sum_{n=0}^\infty X_n z^n$ where $X_n$ are independent random numbers, what kind of functions do we get? Trying it with complex Gaussian $X_n$ produces a disk of convergence with some nondescript function on the inside.

Replacing the complex Gaussian with a real one, or uniform random numbers, or even power-law numbers gives the same behavior. They all seem to have radius 1. This is not just a vanilla disk of convergence (where an analytic function reaches a pole or singularity somewhere on the boundary but is otherwise fine and continuable), but a natural boundary – that is, a boundary so dense with poles or singularities that continuation beyond it is not possible at all.

The locus classicus about random Taylor series is apparently Kahane, J.-P. (1985), Some Random Series of Functions. 2nd ed., Cambridge University Press, Cambridge.

A naive handwave argument is that for $|z|<1$ we have an exponentially decaying sequence of $z^n$, so if the $X_n$ have some finite average size $E(X)$ and not too divergent variance we should expect convergence, while outside the unit circle any nonzero $E(X)$ will allow it to diverge. We can even invoke the Markov inequality $P(X>t) \leq E(X)/t$ to argue that a series $\sum X_n f(n)$ would converge if $\sum f(n)/n$ converges. However, this is not correct enough for proper mathematics. One entirely possible Gaussian outcome is $X_n=1,10,100,1000,\ldots$ or worse. We need to speak of probabilistic convergence.

Andrés E. Caicedo has a good post about how to approach it properly. The “trick” is the awesome Kolmogorov zero-one law that implies that since the radius of convergence depends on the entire series X_n rather than any finite subset (and they are all independent) it will be a constant.

This kind of natural boundary disk of convergence may look odd to beginning students of complex analysis: after all, none of the functions we normally encounter behave like this. Except that this is of course selection bias. If you look at the example series for lacunary functions they all look like fairly reasonable sparse Taylor series like $z+z^4+z^8+z^16+^32+\lddots$. In calculus we are used to worrying that the coefficients in front of the z-terms of a series don’t diminish fast enough: having fewer nonzero terms seems entirely innocuous. But as Hadamard showed, it is enough that the size of the gaps grow geometrically for the function to get a natural boundary (in fact, even denser series do this – for example having just prime powers). The same is true for Fourier series. Weierstrass’ famous continuous but nowhere differentiable function is lacunary (in his 1880 paper on analytic continuation he gives the example $\sum a_n z^{b^n}$ of an uncontinuable function). In fact, as Emile Borel found and Steinhardt eventually proved in a stricter sense, in general (“almost surely”) a Taylor series isn’t continuable because of boundaries.

One could of course try to combine the analytic combinatorics with the lacunary stuff. In a sense a lacunary generating function is a worst case scenario for the singularity-measuring methods used in analytical combinatorics since you get an infinite number of them at a finite and equal distance, and now have to average them together somehow. Intuitively this case seems to correspond to counting something that becomes rarer at a geometric rate or faster. But the Borel-Steinhardt results suggest that even objects that do not become rare could have nasty natural boundaries – if the number $a_n$ were due to something close enough to random we should expect estimating asymptotics to be hard. The funniest example I can think of is the number of roots of Chaitin-style Diophantine equations where for each $n$ it is an independent arithmetic fact whether there are any: this is hardcore random, and presumably the exact asymptotic growth rate will be uncomputable both practically and theoretically.

# Did amphetamines help Erdős?

During my work on the Paris talk I began to wonder whether Paul Erdős (who I used as an example of a respected academic who used cognitive enhancers) could actually have been shown to have benefited from his amphetamine use, which began in 1971 according to Hill (2004). One way of investigating is his publication record: how many papers did he produce per year before or after 1971? Here is a plot, based on Jerrold Grossman’s 2010 bibliography:

The green dashed line is the start of amphetamine use, and the red dashed life is the date of death. Yes, there is a fairly significant posthumous tail: old mathematicians never die, they just asymptote towards zero. Overall, the later part is more productive per year than the early part (before 1971 the mean and standard deviation was 14.6±7.5, after 24.4±16.1; a Kruskal-Wallis test rejects that they are the same distribution, p=2.2e-10).

This does not prove anything. After all, his academic network was growing and he moved from topic to topic, so we cannot prove any causal effect of the amphetamine: for all we know, it might have been holding him back.

One possible argument might be that he did not do his best work on amphetamine. To check this, I took the Wikipedia article that lists things named after Erdős, and tried to find years for the discovery/conjecture. These are marked with red crosses in the diagram, slightly jittered. We can see a few clusters that may correspond to creative periods: one in 35-41, one in 46-51, one in 56-60. After 1970 the distribution was more even and sparse. 76% of the most famous results were done before 1971; given that this is 60% of the entire career it does not look that unlikely to be due to chance (a binomial test gives p=0.06).

Again this does not prove anything. Maybe mathematics really is a young man’s game, and we should expect key results early. There may also have been more time to recognize and name results from the earlier career.

In the end, this is merely a statistical anecdote. It does show that one can be a productive, well-renowned (if eccentric) academic while on enhancers for a long time. But given the N=1, firm conclusions or advice are hard to draw.

Erdős’s friends worried about his drug use, and in 1979 Graham bet Erdős \$500 that he couldn’t stop taking amphetamines for a month. Erdős accepted, and went cold turkey for a complete month. Erdős’s comment at the end of the month was “You’ve showed me I’m not an addict. But I didn’t get any work done. I’d get up in the morning and stare at a blank piece of paper. I’d have no ideas, just like an ordinary person. You’ve set mathematics back a month.” He then immediately started taking amphetamines again. (Hill 2004)

# Packing my circles

One of the first fractals I ever saw was the Apollonian gasket, the shape that emerges if you draw the circle internally tangent to three other tangent circles. It is somewhat similar to the Sierpinski triangle, but has a more organic flair. I can still remember opening my copy of Mandelbrot’s The Fractal Geometry of Nature and encountering this amazing shape. There is a lot of interesting things going on here.

Here is a simple algorithm for generating related circle packings, trading recursion for flexibility:

1. Start with a domain and calculate the distance to the border for all interior points.
2. Place a circle of radius $\alpha d^*$ at the point with maximal distance $d^*=\max d(x,y)$ from the border.
3. Recalculate the distances, treating the new circle as a part of the border.
4. Repeat (2-3) until the radius becomes smaller than some tolerance.

This is easily implemented in Matlab if we discretize the domain and use an array of distances $d(x,y)$, which is then updated $d(x,y) \leftarrow \min(d(x,y), D(x,y))$ where $D(x,y)$ is the distance to the circle. This trades exactness for some discretization error, but it can easily handle nearly arbitrary shapes.

It is interesting to note that the topology is Apollonian nearly everywhere: as soon as three circles form a curvilinear triangle the interior will be a standard gasket if $\alpha=1$.

In the above pictures the first circle tends to dominate. In fact, the size distribution of circles is a power law: the number of circles larger than r grows as $N(r)\propto r^-\delta$ as we approach zero, with $\delta \approx 1.3$. This is unsurprising: given a generic curved triangle, the inscribed circle will be a fraction of the radii of the bordering circles. If one looks at integral circle packings it is possible to see that the curvatures of subsequent circles grow quadratically along each “horn”, but different “horns” have different growths. Because of the curvature the self-similarity is nontrivial: there is actually, as far as I know, still no analytic expression of the fractal dimension of the gasket. Still, one can show that the packing exponent $\delta$ is the Hausdorff dimension of the gasket.

Anyway, to make the first circle less dominant we can either place a non-optimal circle somewhere, or use lower $\alpha$.

If we place a circle in the centre of a square with a radius smaller than the distance to the edge, it gets surrounded by larger circles.

If the circle is misaligned, it is no problem for the tiling: any discrepancy can be filled with sufficiently small circles. There is however room for arbitrariness: when a bow-tie-shaped region shows up there are often two possible ways of placing a maximal circle in it, and whichever gets selected breaks the symmetry, typically producing more arbitrary bow-ties. For “neat” arrangements with the right relationships between circle curvatures and positions this does not happen (they have circle chains corresponding to various integer curvature relationships), but the generic case is a mess. If we move the seed circle around, the rest of the arrangement both show random jitter and occasional large-scale reorganizations.

When we let $\alpha<1$ we get sponge-like fractals: these are relatives to the Menger sponge and the Cantor set. The domain gets an infinity of circles punched out of itself, with a total area approaching the area of the domain, so the total measure goes to zero.

That these images have an organic look is not surprising. Vascular systems likely grow by finding the locations furthest away from existing vascularization, then filling in the gaps recursively (OK, things are a bit more complex).

# How small is the wiki?

Recently I encountered a specialist Wiki. I pressed “random page” a few times, and got a repeat page after 5 tries. How many pages should I expect this small wiki to have?

We can compare this to the German tank problem. Note that it is different; in the tank problem we have a maximum sample (maybe like the web pages on the site were numbered), while here we have number of samples before repetition.

We can of course use Bayes theorem for this. If I get a repeat after $k$ random samples, the posterior distribution of $N$, the number of pages, is $P(N|k) = P(k|N)P(N)/P(k)$.

If I randomly sample from $N$ pages, the probability of getting a repeat on my second try is $1/N$, on my third try $2/N$, and so on: $P(k|N)=(k-1)/N$. Of course, there has to be more pages than $k-1$, otherwise a repeat must have happened before step $k$, so this is valid for $k \leq N+1$. Otherwise, $P(k|N)=0$ for $k>N+1$.

The prior $P(N)$ needs to be decided. One approach is to assume that websites have a power-law distributed number of pages. The majority are tiny, and then there are huge ones like Wikipedia; the exponent is close to 1. This gives us $P(N) = N^{-\alpha}/\zeta(\alpha)$. Note the appearance of the Riemann zeta function as a normalisation factor.

We can calculate $P(k)$ by summing over the different possible $N$: $P(k)=\sum_{N=1}^\infty P(k|N)P(N) = \frac{k-1}{\zeta(\alpha)}\sum_{N=k-1}^\infty N^{-(\alpha+1)}$ $=\frac{k-1}{\zeta(\alpha)}(\zeta(\alpha+1)-\sum_{i=1}^{k-2}i^{-(\alpha+1)})$.

Putting it all together we get $P(N|k)=N^{-(\alpha+1)}/(\zeta(\alpha+1) -\sum_{i=1}^{k-2}i^{-(\alpha+1)})$ for $N\geq k-1$. The posterior distribution of number of pages is another power-law. Note that the dependency on $k$ is rather subtle: it is in the support of the distribution, and the upper limit of the partial sum.

What about the expected number of pages in the wiki? $E(N|k)=\sum_{N=1}^\infty N P(N|k) = \sum_{N=k-1}^\infty N^{-\alpha}/(\zeta(\alpha+1) -\sum_{i=1}^{k-2}i^{-(\alpha+1)})$ $=\frac{\zeta(\alpha)-\sum_{i=1}^{k-2} i^{-\alpha}}{\zeta(\alpha+1)-\sum_{i=1}^{k-2}i^{-(\alpha+1)}}$. The expectation is the ratio of the zeta functions of $\alpha$ and $\alpha+1$, minus the first $k-2$ terms of their series.

So, what does this tell us about the wiki I started with? Assuming $\alpha=1.1$ (close to the behavior of big websites), it predicts $E(N|k)\approx 21.28$. If one assumes a higher $\alpha=2$ the number of pages would be 7 (which was close to the size of the wiki when I looked at it last night – it has grown enough today for k to equal 13 when I tried it today).

So, can we derive a useful rule of thumb for the expected number of pages? Dividing by $k$ shows that $E(N|k)$ approaches proportionality, especially for larger $\alpha$:

So a good rule of thumb is that if you get $k$ pages before a repeat, expect between $2k$ and $4k$ pages on the site. However, remember that we are dealing with power-laws, so the variance can be surprisingly high.