# Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an unexpected turn, or let me have a moment of enlightenment.

For example, a hidden symmetry is revealed or a connection to another field becomes apparent.

Question: Give an example of a picture from your research, description on how it was generated, and what insight it gave.

I am especially interested in what techniques people use to make images, this is something that I find a bit lacking in most research articles. From answers to this question; hope to learn some "standard" tricks/transformations one can do on data, to reveal hidden structure.

As an example, a couple of years ago, I studied asymptotics of (generalized) eigenvalues of non-square Toeplitz matrices. The following two pictures revealed a hidden connection to orthogonal polynomials in several variables, and a connection to Schur polynomials and representation theory. Without these hints, I have no idea what would have happened. Explanation: The deltoid picture is a 2-dimensional subspace of $\mathbb{C}^2$ where certain generalized eigenvalues for a simple, but large Toeplitz matrix appeared, so this is essentially solutions to a highly degenerate system of polynomial equations. Using a certain map, these roots could be lifted to the hexagonal region, revealing a very structured pattern. This gave insight in how the limit density of the roots is. This is essentially roots of a 2d-analogue of Chebyshev polynomials, but I did not know that at the time. The subspace in $\mathbb{C}^2$ where the deltoid lives is quite special, and we could not explain this. A subsequent paper by a different author answered this question, which lead to an analogue of being Hermitian for rectangular Toeplitz matrices.

Perhaps you do not have a single picture; then you might want to illustrate a transformation that you test on data you generate. For example, every polynomial defines a coamoeba, by mapping roots $z_i$ to $\arg z_i$. This transformation sometimes reveal interesting structure, and it partially did in the example above.

If you don't generate pictures in your research, you can still participate in the discussion, by submitting a (historical) picture you think had a similar impact (with motivation). Examples I think that can appear here might be the first picture of the Mandelbrot set, the first bifurcation diagram, or perhaps roots of polynomials with integer coefficients.

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This should maybe be community wiki... – Per Alexandersson Aug 9 '14 at 7:01
The appearance of Apollonian circle packing in questions related to the scaling limit of the abelian sandpile model and integer superharmonic functions was quite unexpected. See the papers arxiv.org/abs/1208.4839 and arxiv.org/abs/1309.3267. As I understand it, this observation was made by computing some explicit examples and noticing the fractal pattern. – Sam Hopkins Aug 9 '14 at 7:27
@SamHopkins: This should be an answer! I have seen sandpile-models, and Apollonian gaskets, but never expected a connection! – Per Alexandersson Aug 9 '14 at 7:35
Would <experimental-mathematics> or <visualization> be relevant tags for this question? – J W Aug 11 '14 at 10:39
also see eg Phase Plots of Complex Functions: a Journey in Illustration / Wegert, used to visualize the Riemann zeta fn & related ones – vzn Aug 14 '14 at 15:29

The third image below was certainly unexpected for my soon-to-be-collaborators, Emmanuel Candes and Justin Romberg. They started with a standard image in signal processing, the Logan-Shepp phantom:

They took a sparse set of Fourier measurements of this image along 22 radial lines (simulating a crude MRI scan). Conventional wisdom was that this was a very lossy set of measurements, losing most of the original data. Indeed, if one tried to use the standard least squares method to reconstruct the image from this data, one got terrible results:

However, Emmanuel and Justin were experimenting with a different method, in which one minimised the total variation norm rather than the least squares norm subject to the given measurements, and were hoping to get a somewhat better reconstruction. What they actually got was this:

Unbelievably, using only about 2% of the available Fourier coefficients, they had managed to reconstruct the original Logan-Shepp phantom so perfectly that the differences were invisible to the naked eye.

When Emmanuel told me this result, I couldn't believe it either, and tried to write down a theoretical proof that such perfect reconstruction was impossible from so little data. Much to my surprise, I found instead that random matrix theory could be used to guarantee exact reconstruction from a remarkably small number of measurements. We then worked together to optimise and streamline the results; this led to some of the pioneering work in the area now known as compressed sensing.

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This is amazing! Here's the paper reference ("Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information"). home.ustc.edu.cn/~zhanghan/cs/Candes%20et%20al.06.pdf – Alex R. Aug 10 '14 at 5:08

The Histogram of all OEIS sequences shows an unexpected gap known as sloane's gap. The plot shows how cultural factors influence mathematics. (http://arxiv.org/pdf/1101.4470v2.pdf)

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This is really interesting! – Per Alexandersson Aug 10 '14 at 21:05
$N(n)$ is the number of times an integer $n$ occurs in the database. (This wasn't clear to me from the plot.) – Kirill Aug 11 '14 at 10:00
From the article: "[...] the series of absent numbers was found to comprise 11630, 12067, 12407, 12887, 13258...". What about an OEIS sequence made up of numbers that aren't members of any OEIS sequence? :) – Emanuele Tron Aug 26 '14 at 11:51
11630 is in five OEIS sequences, for example, oeis.org/A163279, which was posted in 2009. 12067 is in 9 sequences, including oeis.org/A163675, from 2009. 12407 is in two sequences, 12887 in seven, 13258 in three. – Gerry Myerson Jun 22 '15 at 23:54
It is better to link to arxiv abstract pages and not the PDF directly – Mariano Suárez-Alvarez Mar 13 at 1:50

Some years ago I was pleasantly surprised when an idea of Jan Mycielski led me to find a very explicit Banach-Tarski paradox in the hyperbolic plane, H^2. H^2 can be decomposed into three simple sets such that each is a third of the space, but also each is a half of the space.

In fact, I found recently how to this even a little more simply, but I like this picture. THe second image is just a viewpoint shift of the first, but makes evident how the blue and green together are congruent to the red.

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One can obtain a nice picture showing somewhat unexpected patterns by marking all rational points on the unit sphere whose coordinates have denominator less than some upper bound, and projecting this to one of the coordinate planes (cf. this answer of mine to another question). The following picture shows such projection of one octant of the sphere:

This picture in resolution 2048 x 2048 pixels can be found at http://www.gap-system.org/DevelopersPages/StefanKohl/images/sphere1.gif.

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Really a nice picture! Got people interested on MSE, me included. – MvG Sep 1 '14 at 14:06
It would look even nicer under stereographic projection, since all of the patterns would be circular rather than elliptical. – Adam P. Goucher Jun 22 '15 at 23:48
@AdamP.Goucher: You mean like in the second image here? – Stefan Kohl Jun 23 '15 at 8:55
@StefanKohl Yes, precisely! – Adam P. Goucher Jun 24 '15 at 19:35

John Baez explains here how plotting the roots of polynomials with integer coefficients led to patterns ressembling well known fractals, and how some people figured out ways to explain the unexpected connection.

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This image, from the MO question "Gaussian prime spirals," was certainly unexpected:

But the main question I raised,

Q1. Does the spiral always form a cycle?

seems out of reach (as per François Brunault's comment) under "current technology." (Stan Wagon found a cycle of length 3,900,404.)

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That is really a nice example! – Per Alexandersson Aug 9 '14 at 12:27
Joseph, link to the image appears broken to me. – joro Jan 12 '15 at 11:44
@joro: Looks like a server is down. Several of my images have gone missing. – Joseph O'Rourke Jan 12 '15 at 12:28
OK, just to let you know. You have an option to upload them via the user interface on SE under CC license if you wish. – joro Jan 12 '15 at 12:30

I'm guessing that no one expected uniformly random Aztec diamonds (and similar lozenge/domino tilings) to exhibit circular limit shapes with frozen regions outside. .

The colors in the image are determined from a certain combinatorial object called a height function.

There are a number of ways of generating these images, but the most useful one is via the domino shuffling algorithm. Essentially one builds successively larger uniform tilings by taking a uniformly random $n\times n$ tiling and then expanding it followed by filling in the blanks thus getting a uniformly random $n+1$ tiling.

A nice summary can be found here.

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I think that Barnsleys Fern is a really surprising image, that such complex shapes can be encoded in four very simple affine transformations.

If you allow for a larger class of functions (stochastic, $\mathbb{R}^3 \to \mathbb{R}^3$, and introduce a log-density plot and color each point according to orbit history, the possibilities are endless (image created by Silvia C.):

The most common applications of the latter algorithm seems to be producing abstract book covers for books about the universe:

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This image shows the boundary of the space of "stabilizable matrices", which in some precise sense dictate the behavior of the scaling limit of the abelian sandpile model on the grid $\mathbb{Z}^2$:

This image is taken from http://arxiv.org/abs/1208.4839. See also http://arxiv.org/abs/1309.3267. The appearance of Apollonian circle packing in questions related to the scaling limit of the abelian sandpile model and integer superharmonic functions was quite unexpected. As I understand it, the connection was made by computing the above image numerically (and maybe it was even the case that the authors thought they had a mistake in their code when these fractal patterns emerged?).

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The spiral of prime numbers (white dots) the "pattern" is amazing, for an explanation of the picture you can take a look to this short youtube video.

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Looking at this again I can't help but think of Ramsey Theory. – Jp McCarthy Aug 21 '15 at 12:49

Hardly research level, though visually interesting.

These might show some relations between the discrete and the continuous.

For integer $n$, let $M$ be $n$ by $n$ matrix. For some function $F$, define $M_{x,y}=F(x,y) \mod n$.

Map $M_{x,y}$ to shadows of grey where smaller is darker and larger is closer to white.

Here are some examples for $F(x,y) \in \{x^2+y^2,4x^2+y^2,x^3+x-y^2,xy\}$.

$$x^2+y^2, n=503$$

$$4x^2+y^2, n=503$$

$$x^3+x-y^2, n=503$$

$$xy, n=1961=37\cdot 53$$

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What is your $n$ ? – darij grinberg Aug 9 '14 at 10:59
These types of images I found very rare. Thanks for uploading it. – user56917 Aug 9 '14 at 11:02
@joro: This is quite nice, I remember experimenting a lot on my graph calculator with this type of patterns when I was half my current age. – Per Alexandersson Aug 9 '14 at 11:35
@Wolfgang The lines appear artifacts of scaling. On a bigger plot they disappear: s12.postimg.org/fpi9upxgt/x_2_y_2_2.png – joro Aug 9 '14 at 13:15
Imagine drawing mod n, mod 2n, mod 3n,... the picture will be essentially the same, just the period between repeating "colors" (or shadows of grey) changes. Likewise between mod n and mod (n+1) etc. So, as @Per also says, the "discretisation" doesn't reveal many more details. You might as well define a continuous (say, periodic) color spectrum. – Wolfgang Aug 9 '14 at 16:39

plots, as a function of the $y$-coordinate, the spectrum of the almost Mathieu operator $H^y:l^2(\mathbb Z)\to l^2(\mathbb Z)$ $$H^y(f)(n)=f(n+1)+f(n-1)+2\cos(2\pi ny)f(n).$$

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there are many aspects of the Collatz conjecture that lend themselves to visualization to the point that significant research insights not found elsewhere can be found in basic graphs of its properties, and a visualization-based/-centric approach can constitute the base of a major "attack" on the problem. one might state that it is an entirely new form of mathematical exploration when combined with computational experiments. with a few caveats on this notoriously difficult problem that even top experts like Erdos are quite wary of:

• note the literature on Collatz is quite sizeable and not highly detailed anywhere (although there are good high-level surveys/ overviews by Lagarias).
• many visualizations of it only look very random, so a lot of ingenuity is required but also rewarded.

here is one such striking example that apparently has not been published (outside of cyberspace).

this visualization shows the function/graph/tree $f'^n(x)$ where $n$ is the $n$th iteration of the Collatz function working in reverse. ie the function starts at 1 and based on the conjecture, visits all integers. the $x$ axis is logarithmic scale. a $2n$ operation moves upward to the right, a $(n-1)/3$ operation moves up to the left. there are two inset details of line intersection "closeups" that show the fractal quality, somewhat reminiscent of the rings of Saturn.

the insight is that this shows the dichotomy/ juxtaposition of order (macroscopic) vs randomness (microscopic) in the problem and leads to other ideas/ strategies about how to approach further analysis.

plots were generated with Ruby/Gnuplot. more details on generation and other visualizations on this page.

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A long time ago, while attempting to classify certain two-dimensional rational conformal field theories (these are certain quantum field theories which enjoy a particular high level of mathematical rigorosity), I found an interesting image which is related to the modular group $\mathbb{P}\mathrm{SL}(2,\mathbb{Z})$, i.e. the matrices $$M = \begin{pmatrix}a & b\\ c& d\end{pmatrix}\,,\ \ \ \ a,b,c,d\in\mathbb{Z}\,, \mathrm{det}\,M = +1\,.$$ Leaving out the details of my classification attempts, I discovered a set of certain conformal field theories characterized by two real parameters $x$ and $y$. They turned out to be rational if and only if $x=a/d$ and $y=b/c$ are both rational numbers with the additional condition that $ad-bc=1$. The connection to the elements of the modular group should be clear from my suggestive notation.

Now, within the classification of conformal field theories, it was natural to look at the $x$-$y$-plane and plot all the points which belong to the set of the rational theories -- which produces the following type of image (showing the first quadrant, and only points with $x<1$ and $y<1$), which I called "modular chaos"

As one might guess, this is only a crude approximation as only points up to a (rather small) maximal denominator are plotted. In fact, one can show that the emerging pattern is dense in $\mathbb{R}^2$, but it is also apparent that it has some fractal-like structure. (Actually, to be more precise, to have points in all four quadrants of the $x$-$y$-plane, one has to consider the weaker condition $ad - bc = \pm 1$.

The above image is not yet particular beautiful, but one can consider the whole plane and use a Poincare map to squeeze it into a unit disk. To enumerate and plot the valid points, one generates the modular group by the two matrices $$S = \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}\,,\ \ \ \ T = \begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}\,,$$ keeping in mind the relations $S^2=(ST)^3=\mathbb{1}$. It is relatively easy to generate the group in terms of words in $S$ and $T$ up to length $40$ as a binary tree. Encoding by color the length of the word, one finds the following much nicer image .

If you are interested in the connection to conformal field theories, see my two works arxiv:hep-th/9312097 and arxiv:hep-th/9207019. The one from 1993 contains my "proof" that the set is dense in the $\mathbb{R}^2$. I apologize to the mathematicians for the lack of rigor, I am a mere theoretical physicist.

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A recent blog post from google shows what happens if you enhance the parts of an image that triggers image recognition (using neural networks) of certain features.

The results are quite spooky, and reveal some hidden structure on what the neural network actually look for when recognizing certain features.

This is the text about the image below:

Instead of exactly prescribing which feature we want the network to amplify, we can also let the network make that decision. In this case we simply feed the network an arbitrary image or photo and let the network analyze the picture. We then pick a layer and ask the network to enhance whatever it detected. Each layer of the network deals with features at a different level of abstraction, so the complexity of features we generate depends on which layer we choose to enhance. For example, lower layers tend to produce strokes or simple ornament-like patterns, because those layers are sensitive to basic features such as edges and their orientations.

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ok but neural network are also quite random, and if they do the job or not is quite random too. so you enhanced some random features of the image :) – user1952009 Mar 11 at 18:47

The discovery of the special nature of Costa's minimal surface has been made on a visualization.

Generally visualization seems to play an important role in the study of minimal surfaces.

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This image shows the behavior of a certain function (basically, the "inverse temperature modulo a timescale") associated to various "greedily refined" Markov partitions for the geodesic flow on a g-torus as a function of the (log-) number of partition refinements. When I saw that not only the limit but also even the oscillatory behavior was essentially identical for different genera, I was convinced that there was true physical relevance for this very abstract quantity. There was no reason (other than physics!) to expect such uniformity. Details are in http://arxiv.org/abs/1009.2127

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Students learning about polar coordinates for the first time may investigate the "roses"

Maybe they will even discover "greatest common divisor" from these.

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This is beautiful, but badly lacking context... – darij grinberg Jan 12 '15 at 16:46

When Thierry Gallay and I introduced the Numerical measure of a matrix, we encountered the following simulation of the measure density (here a $3\times3$ matrix). This lead us to conjecture, and eventually prove, that the density is constant (in general, a polynomial of degree $n-3$) in the curved triangle. We eventually made a link with the theory of lacunae of hyperbolic differential operators.

The lines are level lines of the density. The outer line, where the density vanishes, is the boundary of the Numerical range. It is convex, according to the Toeplitz-Hausdorf Theorem.

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Since the oeis has added the feature of having sequences displayed graphically, it has become so much easier to get a quick impression of their behaviour, particularly for many self similar sequences.

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Do you know any instances where one of these pictures have lead to an insight, or unexpected development? – Per Alexandersson Aug 9 '14 at 8:15
@PerAlexandersson I don't know if these pictures have already raised unexpected developments, but obviously they allow deeper insights, e.g. comparing the toothpick sequence oeis.org/A188346/graph with this one oeis.org/A187210/graph. – Wolfgang Aug 9 '14 at 8:44

The picture is taken from The Amazing, Autotuning Sandpile by Jordan Ellenberg.

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Abelian sandpile model on a square grid, to be more precise. See other grids at math.cmu.edu/~wes/sandgallery.html . – darij grinberg Aug 20 '15 at 13:30

We were highly impressed how very similar growth rules can form a mushroom shape. In the model, each point the growing fungi (network of the thin threads) generates some abstract scalar field. The tips turn towards preferred value of the field and branch when the field drops below some threshold. Add the preferred 45 degree orientation in the earth gravity field - and this is enough to make the system to grow into almost perfect shape of the most primitive mushrooms. Complete description and equations can be found in the published articles, referenced from the neighbour sensing model entry in Wikipedia.

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Sorry for the duplicate with @paw 's entry , but I'm the discoverer of Sloane's gap and I don't have enough reputation here yet to comment his/her answer...

So here is another explanation of this phenomenon I propose in http://www.drgoulu.com/2009/04/18/nombres-mineralises/ with this figure:

• primes are shown in red,
• perfect powers are shown in green
• highly composite numbers are shown in yellow

Since these numbers cover most of the "high zone" of Sloane's gap, I suspect the gap arises from the fact that many OEIS properties are derived from a few very basic ones like the 3 listed above. So numbers in these sequences will very likely also be present in many other sequences.

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About 6 percent of men (many less women) are blindcolour. Please be nice for these disabled persons and don't use green and red spots in the same figure. – Denis Serre Mar 11 at 15:12
What do the axes mean? And what are the blue dots? – darij grinberg Mar 12 at 23:33
the plot shows for each integer n=1..10000 the number of sequences in OEIS in which n appears. The unexpected thing is the white strip in the points cloud, which shows there are numbers more "interesting" (above) than others (below). The blue dots are numbers which do not belong to the 3 "base" properties (primes, perfect powers, HCN) – Dr. Goulu Mar 14 at 6:22

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This is neat, but probably needs a couple of qualifications: (1) The apparent randomness in the picture is due to the centers of the circles being chosen at random (there is no deterministic chaos here), and (2) the concentric circles are (as far as I can tell) just an artistic way to make the various regions easier to distinguish from each other. – darij grinberg Aug 20 '15 at 13:26

For better visualizing and understanding fractals like the Mandelbrot set, the idea of color cycling is a great invention.

Points outside the fractal are colored according to the number of iterations when a threshold assuring divergence ("bail out") is reached. Imagining the fractal bearing en electrical charge or a temperature, the points of same color, i.e. of same rate of divergence, form "equipotential lines" around it. Of course, those lines become more and more intricate as one comes close to the fractal.
So far, this is only static, but now cycling in time through the colors of the (periodic) color palette, either towards the fractal or outward, reveals so much more about its hard-to-see structures. E.g. for the Mandelbrot set, knowing that it is simply connected, cycling helps particularly in regions with spiral-like patterns to get an idea "where it is connected".
Just google for the terms fractal color cycling and you'll find tons of more or less hallucinating videos.

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This wasn't exactly research, but I have a couple animations I made using a modified version of Melinda Green's Buddhabrot method to render the Mandelbrot set, and what came out was definitely unexpected and pretty shocking to me. I don't think I've ever seen this particular method anywhere else. I've been hoping to get some proper mathematicians to look at the process and give me some insight into why such wild objects seem to form.

This is the first one I made.

Then I tried to make a higher definition animation with different inputs.

You can turn up the quality to see the detail a bit better before watching them. It defaults to 480p, but can be changed to 720p.

To create these, I first began with Melinda's method, which is still explained at her site. It's basically a heat map of how many points in each pixel escaped to "infinity" under the action of the complex seed function. To create motion I decided I would take the coefficients of the function, which was a generalized Mandelbrot-type equation like this:

$$z(w) = aw^3 + bw^2 + cw + C$$

Where $w$ is the complex conjugate of the previous value of the function.

And I would treat those coefficients like a 3-vector (a, b, c). To create motion, I rotated that vector just as if it was a spatial vector rotating through space. The animations are built up of individual images created by slightly transforming the coefficients little by little.

I would really enjoy hearing any insights people have as to why such incredible structures seem to come alive in these visualizations. It is almost eerie. You can see there is a smoke-like effect that gathers around the extended "arms" of the object as it moves, and it almost acts like it is responding to some kind of attractive force (which is mystifying considering what we're looking at). It also has these little three-pointed sparks that fly off the tips, but eventually look like creases in fabric rather than little stars. There are even biological looking structures that appear when the sparks come together and seem to annihilate each other.

On a simpler level, it shocked me that it actually looks like a very distorted physical rotation of some object rotating in higher dimensions, even though it is only a rotation in coefficient space, and not a an actual rotation of spatial coordinates. About halfway through each video, you can see that it really is a rotational transformation, because it comes back around and repeats the entire rotation once more as the vector comes back through its initial position, which was something like (1, 0, 0). In fact, in the first video you can see the exact moment it repeats because the numbers didn't come back around exactly right due to rounding errors that I fixed in the second video.

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These images are the graphs of simple functions using the sinus. You can see them, animated with a function tracer in Flash here: graph of two unexpected functions

$$a=a+3 \\ b=b+10 \times cos(a)\\ \begin{cases} x=a \times cos(a)+b \times cos(b)\\ y=b \times sin(b)+a \times sin(a)\end{cases}$$ $$a=a+\pi/3 \\ b=b+a \times sin(1/a)+a\times cos(1/a)\\da=da+0.0001\\ \begin{cases} x=0.02 \times 1/a \times cos(b\times da)+a \times cos(b\times da)\\ y=0.02 \times 1/a \times sin(b\times da)+a \times sin(b\times da)\end{cases}$$

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Is this part of your research? – Per Alexandersson Aug 14 '14 at 20:38
@Per Alexandersson Yes, I coded the tracer with actionscript and tested functions... – helloflash Aug 14 '14 at 22:12
And so? What mathematical insights did this give rise to? – Todd Trimble Aug 17 '14 at 14:00
@Todd Trimble This specific research requires no superior knowledge, but who says that it was its pretension? It's an interesting way to create patterns and find textures. – helloflash Aug 17 '14 at 19:35
Sorry for not responding earlier. My comment was in reference to the wording of the OP, which asks specifically what mathematical insights did the image give rise to. I too take aesthetic pleasure in the pictures derived from applying the tracer to your parametric equations, but my reading is that the OP is interested specifically in examples which produced a mathematical insight, in order to be considered on-topic for MO. – Todd Trimble Aug 27 '14 at 19:36

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