# 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.

• 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.

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)

• 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
• It is better to link to arxiv abstract pages and not the PDF directly – Mariano Suárez-Álvarez Mar 13 '16 at 1:50
• @paw, 14972 is in oeis.org/A272544, "Number of active (ON,black) cells at stage $2^n-1$ of the two-dimensional cellular automaton defined by `Rule 493', based on the 5-celled von Neumann neighborhood." The entry was made on 2 May 2016. – Gerry Myerson Jun 9 '16 at 22:58

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.

• Is this a troll? Look at the center of the circle. The top image has 3 lines through it, and the bottom one doesn't. They're not the same image... – Clark Gaebel Feb 27 '17 at 2:27
• The author didn't mention it, but the images are not standard Euclidean 2D plane, but rather are projections of hyperbolic plane using a Poincare disk projection. It has a fisheye-like effect: Straight lines appear as circular arcs meeting the bounding circle at right angles. The outer circle is the projection of all the points at infinity. The projection can be re-centered and thus altered yet describe the same ideal image. – Jonathan Lidbeck Feb 27 '17 at 6:57
• Gee, I've never been called a troll before. Lidbeck has it perfectly right. It is like the viewer is in an airplane and moves from being over one point (where 3 lines intersect) to being over another point (the center of image #2, where there are no intersections. I have a demo of this that actually shows the motion: it is at: demonstrations.wolfram.com/TheBanachTarskiParadox – stan wagon Feb 28 '17 at 2:51

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.

• 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
• What would the fourier transform of that look like? – Rudi_Birnbaum Jul 6 '16 at 21:55

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.

• Reminds me of the popular complex plane fractal "inside" colouring algorithm "Triangular Inequality Average" with some of the more angular fractals like the Burning Ship, and Nova. – alan2here Mar 12 '17 at 4:16

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.)

• 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

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.

• 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 '16 at 15:12
• What do the axes mean? And what are the blue dots? – darij grinberg Mar 12 '16 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 '16 at 6:22
• I made a similar plot back in 2006 by querying Google for each number and using the purported number of search results as a value. Sloane's gap appears here too, separating primes from ordinary numbers: imgur.com/a/edAhR – Dan Brumleve Feb 26 '17 at 21:05

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:

• By sheer coincidence, I've been listening to Bloom Ascension by Steve Roach while reading this thread, and the album cover looks very similar to the Sylvia C image: steveroach.bandcamp.com/album/bloom-ascension. – Todd Trimble Aug 24 '19 at 0:07
• @ToddTrimble Ah, yes the cover image is also generated by the flame fractal algorithm. The picture you refer to is a "bloom" style fractal - image googling for "bloom fractal" gives a bunch of more examples :) – Per Alexandersson Aug 24 '19 at 5:19

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.

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].

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?).

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.

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

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

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.

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.

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.

• 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 :) – reuns Mar 11 '16 at 18:47
• @user1952009 These patterns are not random at all. These are the result of edge detecting filters and other features useful for image recognition. – Houshalter Feb 26 '17 at 17:55

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.

• There's a really nice video with some fascinating visualizations of this here, they kind of blow away the traditional visualizations. – Jason C Apr 30 '17 at 23:42

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

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

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

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

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

• What is your $n$ ? – darij grinberg Aug 9 '14 at 10:59
• @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
• Such pictures were discussed in "Kvant" magazine, see kvant.ras.ru/1987/11/pti.htm – Alexey Ustinov Aug 23 '16 at 11:33

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 $\textbf{graph is the formula itself}$

• Why is this surprising? What images can't you encode by choosing some n? It looks like that n might contain more information that the image. – Douglas Zare Nov 13 '16 at 0:11
• That’s right: the formula is just a gimmick for displaying an arbitrary 17-row bitmap encoded in the binary value of n. Since n is not part of the output, there’s no self reference—in fact, Tupper himself never called the formula “self-referential”. For details, see my Quora answer at qr.ae/TklUCI. – Anders Kaseorg Feb 26 '17 at 22:36

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

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.

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:

(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.

This is unexpected Voronoi diagram (see the full story).

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.

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.

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

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

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 $\binom{15}5=\binom{15}{10}=3003$):

Coefficients of cyclotomic polynomials with composite number:

The picture is taken from slides of the talk Cyclotomic Numerical Semigroups-2 given by Pieter Moree at International meeting on numerical semigroups with applications.

More images here

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 is very nice! – Per Alexandersson Aug 13 '19 at 20:18

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

$x=tcos^3(t)$

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

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

• What an accident! It reminds my favourite joke on ships in bottles. – Fedor Petrov Jun 10 '16 at 22:00
• Amusing as it is, I don't think this picture really captures the question - it does not hint about some deeper mathematics that is to be discovered, or further research.... – Per Alexandersson Jun 10 '16 at 22:31
• @FedorPetrov Do you have a reference? :) – მამუკა ჯიბლაძე Feb 26 '17 at 6:36
• @მამუკაჯიბლაძე bash.im/quote/392487 (if you undesrtand Russian) How to make ships in bottles? You put some glue and other stuff in a bottle and shake it. You get something inside a bottle, sometimes ships. – Fedor Petrov Feb 26 '17 at 10:50
• @FedorPetrov Thanks, great! Especially other stuff (yes I understand Russian :)) – მამუკა ჯიბლაძე Feb 26 '17 at 14:11

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.

• 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 very top of the large cardinal hierarchy was an unlikely place to look for computer generated mathematical images.

The $n$-th classical Laver table is the unique algebra $A_{n}=(\{1,...,2^{n}\},*_{n})$ where

1. $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$, and

2. $x*_{n}1=x+1\mod 2^{n}$.

Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. The critical points of non-trivial elementary embeddings $j\in\mathcal{E}_{\lambda}$ are known as rank-into-rank cardinals and the rank-into-rank cardinals are among the largest of the local large cardinals and the axiom positing the existence of a rank-into-rank cardinal is one of the strongest large cardinal axioms.

Define an operation $*$ on $\mathcal{E}_{\lambda}$ by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. For each limit ordinal $\gamma<\lambda$, let $\equiv^{\gamma}$ be the equivalence relation on $\mathcal{E}_{\lambda}$ where $j\equiv^{\gamma}k$ iff $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for each $x\in V_{\gamma}$. Then for all $j\in\mathcal{E}_{\lambda}$ and limit ordinals $\gamma<\lambda$, there is some $n$ where $(\langle j\rangle/\equiv^{\gamma})\simeq A_{n}$.

Let $L_{n}:\{0,...,2^{n}-1\}\rightarrow\{0,...,2^{n}-1\}$ be the mapping where $L_{n}(x)$ is the number obtained by reversing the digits in the binary expansion of $x$. In other words, $L_{n}(\sum_{k=0}^{n-1}a_{k}2^{k})=\sum_{k=0}^{n-1}a_{k}2^{n-1-k}$.

Define an operation $\#_{n}$ on $\{0,...,2^{n}-1\}$ by $x\#_{n}y=L_{n}(((L_{n}(x)+1)*_{n}(L_{n}(y)+1))-1)$.

In the following image, each pixel of the form $(x,x\#_{n}y)$ (we use matrix coordinates here) is colored white while all of the other coordinates are colored black ( here $n=9$ so the image is a 512x512 image). As $n\rightarrow\infty$, the resulting image will give one finer and finer detail about the classical Laver tables.

At this link, you may zoom into the above image of the classical Laver tables.

All of the information about $A_{9}$ is contained in the above image.

The white points actually form a subset of a Sierpinski-like triangle. However, the white points are so sparse that the resulting image hardly resembles the Sierpinski triangle. However, while the white points do not quite make the Sierpinski triangle, if there exists a rank-into-rank cardinal, then every white point in this image has fractal structure if you zoom extremely far into the image and let $n\rightarrow\infty$.

This image is not the only image you may obtain from the classical Laver tables since on this answer, I have posted other images obtainable from the classical Laver tables and generalized Laver tables. You may also generate your own images obtainable from the generalized Laver tables here.