Examples of unexpected mathematical images

Question

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.

Answer 1

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.

Answer 2

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/abs/1101.4470v2)

Answer 3

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.

Answer 4

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 https://stefan-kohl.github.io/images/sphere1.gif.

Update: The picture has meanwhile been used on the website of the AMSI/AustMS workshop "Geometry and Analysis", Flinders University, Adelaide, September 25 – September 27, 2015.

Answer 5

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.

Answer 6

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.

Answer 7

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

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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.)

Answer 8

I think that Barnsley's 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:

Answer 9

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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Add-on (29 March 2017). It was recently found [arXiv:1702.05474] that the interior of the circle also hides interesting patterns. These are not clear in single configurations, such as the 'typical' tiling shown above, but emerge after averaging over all possible configurations. The following shows a quadrant of the square, rotated over $\pi/4$:

The uniform grey areas here correspond to the uniform coloured areas in the above picture. The bottom-left panel ($\Delta=0$) corresponds to domino tilings of the Aztec diamond. The density profile plotted contains information about average domino tilings (and configurations of a related model, see below). Two types of oscillatory patterns are clear inside the circle, one with dark bands following the circle, and one with hyperbola-like bands with asymptotes given by the two diameters of the circle that are orthogonal to the square bounding the circle (which would form an "x" in the middle of the color picture above).

In fact, domino tilings of the Aztec diamond are related to (a special one-parameter case of) the six-vertex model from statistical mechanics, with parameter $\Delta\in\mathbb{R}$, cf. this MO Q&A. At the "combinatorial" or "ice" point $\Delta=1/2$ the patterns seem almost invisible, but interestingly at the "critical point" $\Delta=-1$ there are further "higher" patterns with several saddlepoint-like features. A few weak saddle-like features can be discerned along the diagonal for the domino case $\Delta=0$ too. Especially the higher oscillatory patterns seem to be new, and beg for a mathematical understanding. From arXiv:1702.05474 (p.12):

A more quantitative understanding of these vertex-density oscillations in the temperate region, e.g. using the methods of [28] or [32], would be very interesting. In fact, similar finite-size oscillatory behaviour is known to occur for the eigenvalue distributions in random-matrix models [56], see e.g. [57]; this might shed light on the oscillations at least for $\Delta=0$, cf. [28].

Answer 10

Suppose we have a function representated as discrete Fourier series (DFS). Each DFS coefficient has an argument and modulus. But which of them is more important?

This strange question was discussed in the book "Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software" by Dennis C. Ghiglia and Mark D. Pritt. The following picture illustrates the answer:

In the first row you see Lena and Tiffany. Second row contains their DFS's absolute values and arguments.

In the last row there are LenaTiffany (absolute values from Lena, arguments from Tiffany) and TiffanyLena (absolute values from Tiffany, arguments from Lena). So it is clear that arguments are more important than absolute values!

Ghiglia and Pritt in their book crossed Einstein with Mona Lisa. This picture is taken from the talk "Control for high resolution imaging" by Oleg Soloviev (Delft University of Technology and Flexible Optical B.V.), course Control for High Resolution Imaging.

Answer 11

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?).

Answer 12

The Hofstadter butterfly

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

Answer 13

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.

Answer 14

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.

Answer 15

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.

Answer 16

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.

Answer 17

When I was plotting some parametric curves, accidentally I found this one:

$x=t\cos^3(t)$

$y=9t\sqrt{| \cos(t)|}+t\sin(\frac{t}{5})\cos(4t)$

$0<t<\frac{39\pi}{2}$

Answer 18

Abelian sandpile model:

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

Answer 19

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 $F(x,y) \in \{x^2+y^2,4x^2+y^2,x^3+x-y^2,xy\}$.

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

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

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

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

Answer 20

I was computing various (2D orthogonal) projections of the Leech polytope (the $196\,560$ points forming the smallest shell of the Leech lattice focusing on those which exhibit various symmetries (viz., eigenplanes from various elements of $\mathrm{Co}_0$, or combinations of such planes), and I stumbled upon this delightfully “snowflaky” projection:

(it comes from class 30D of $\mathrm{Co}_0$, in the ATLAS notation, starting from an eigenplane on which the latter acts as a rotation of $2\pi/10$, and shifting it ever-so-slightly toward an eigenplane on which it acts as a rotation of $2\pi/15$; to be clear, this is just a particular orthogonal projection to $\mathbb{R}^2$ of the $196\,560$ points of the first shell of the Leech lattice).

I am very much reminded of the way Conway describes the sporadic simple groups as “Christmas tree ornaments” in this Numberphile video (at 12′16″ in the video).

This Twitter thread has a number of similar images.

Answer 21

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.

Answer 22

Tupper's Self-Referential Formula is an inequality defined by:

$$\frac{1}{2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {\frac{y}{17}} \right\rfloor 2^{-17 \lfloor x \rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)},2\right)\right\rfloor$$

The best part of this is that when we plot it in certain range the $\textbf{graph is the formula itself} $

Answer 23

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

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

Answer 24

This is unexpected Voronoi diagram (see the full story). By zooming in and out of the following pictures, such a picture can be seen on the screen for a short time (less than 1 second).

It illustrates not the Voronoi diagram but a "brittle view of the internal logic of the X86 FPU". The same page contains some examples which are almost the same as pictures from the answer of joro but much more funny:

Such pictures were discussed in Russian "Kvant" magazine.

Answer 25

Having browsed all these fantastic images several times, only now did I remember that I also have something really unexpected to show. Many years ago Winfried Bruns suggested certain question about generic $n$-vectors (homogeneous elements in exterior algebras). I could not contribute much, and still return to his question from time to time. One really surprising feature we encountered there is some fractal-like pattern out of the blue.

For some "generic" field $k$, take a generic element $a\in\Lambda^n(k^{3n})$. Exterior multiplication by $a$ induces a linear operator $a\wedge\_:\Lambda^n(k^{3n})\to\Lambda^{2n}(k^{3n})$, which in the standard bases may be viewed as a (square) $\binom{3n}n\times\binom{3n}{2n}$-matrix. Plotting just nonzero entries of this matrix reveals something that was really unexpected for me. Here is an example with $n=5$ (a $3003\times3003$-matrix, since $\binom{15}5=\binom{15}{10}=3003$):

Answer 26

The first sequence without triple in arithmetic progression
MathOverflow post - A229037 - Numberphile video (from 3:36 to 7:41)

The following colored plot of 16 million terms is due to reddit user garnet420 (horizontal divisions are 1000000; vertical divisions are 25000).

A variation by Richard Stanley: we only exclude the triple in weakly increasing arithmetic progression
MathOverflow post (last part) - Comment - A309890 - Listen it (marimba, incredibly pleasant!)

Answer 27

The following shows a plane section of the $E_8$ lattice along a random translate of a Coxeter plane (plane of symmetry of order $30$). The gray scale indicates the (squared) distance to the the closest lattice point (with pure black indicating the distance $0$ and pure white indicating the distance $1$; so the light lines essentially show the Voronoi diagram of $E_8$ intersected along this plane):

I find it fascinating how approximate order $30$ symmetry appears in various places in this image.

It is even more interesting in video, where we can see the Voronoi cells appear and disappear as we translate the plane of section:

• Section along a random plane

• Section along a plane parallel to a Coxeter plane

• Section along a plane parallel to a plane with symmetry of order $24$

(In each case, the plane section is translated uniformly along a random axis perpendicular to it; in the latter two videos, the plane encounters a lattice point at the exact middle of the video and in the center of the image: this reveals a spectacular perfect symmetry at this point, while the symmetry is only approximate at other times.)

I also experimented with other kinds of coloring (defining the color of a point by the projection of the nearest lattice point along three perpendicular axes, see here), but they don't seem as visually interesting.

Answer 28

What do numbers look like? ( John Williamson) https://johnhw.github.io/umap_primes/index.md.html

Very interesting visualizations have been obtained relatively recently. Just the integers from 1 to 1 mio have been considered, each integer converted to a vector corresponding to its prime decomposition, thus obtained a dataset in higher-dimensional space, and final step is applying dimensional reduction technique (UMAP (see e.g. John Baez's blog for discussion)) to get 2-dimensional matrix which is visualized.

Other sequences can be processed in a similar way to get more beautiful pictures (see the link above).

PS

Imho, the outcome is very unexpected, it might happen that certain patterns more likely due to UMAP, rather due to hidden patterns in integer numbers.

Answer 29

A classical exercise in a first calculus course is to find a function $f:[0,1]\to \mathbb{R}$ whose discontinuity points are precisely the rational points. An example is the "popcorn function": $$ f(x)=\begin{cases} 1/q \text{ if } x=p/q\in \mathbb{Q} \text{ (in reduced form)} \\ 0 \text{ if } x\notin \mathbb{Q} \end{cases} $$

I was certain that this function is the type designed purely for counter-example purposes, and not the one you would come across "in nature"; up until a year ago when I tried to understand the space of characters of the discrete Heisenberg group. It happens to be a bundle over a circle, where the fiber of a point $x$ is a torus of size $f(x)$:

The meaning of the term character I use here is in the sense of Thoma, as explained in this book. It extends the notion of characters of abelian groups in the Pontryagin sense and of finite groups in the Frobenius sense, and thus provides a way of doing harmonic analysis on arbitrary groups.

In a recent paper, Bader and I study dynamics and random walks on the space above (as well as dual spaces of other nilpotent groups), for the sake of understanding arithmetic groups.

Answer 30

The Hasse diagram of a (partially) ordered set is often used to graphically represent an order relation; one may think of it as the minimal directed acyclic graph inducing this relation (its transitive reduction), with all arcs drawn from top to bottom.

The partitions of an integer $n$ (non-increasing sequences of integers of sum $n$) are ordered by the dominance order. A similar order may be defined on the different ways to write an integer $n$ as a sum of powers of integer $b$. Both have the lattice structure, and have a striking self-similar structure.

This self-similar structure was first noticed by observing the following drawings of Hasse diagrams.

The first one is defined over the integer partitions of $40$ and its self-similarity was studied in this paper.

                             

The second one is defined over the partitions of $80$ into powers of $2$ and its self-similarity was studied in this paper.

They are also mentioned in this discussion.