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.

Per Alexandersson

Posted 2014-08-09T07:01:31.603

Reputation: 6 151

12This should maybe be community wiki... – Per Alexandersson – 2014-08-09T07:01:55.030


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 http://arxiv.org/abs/1208.4839 and http://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 – 2014-08-09T07:27:51.313

@SamHopkins: This should be an answer! I have seen sandpile-models, and Apollonian gaskets, but never expected a connection! – Per Alexandersson – 2014-08-09T07:35:48.943

There are also quite some "famous" examples, e.g. the ulam spiral – PlasmaHH – 2014-08-10T18:26:04.210

1Would <experimental-mathematics> or <visualization> be relevant tags for this question? – J W – 2014-08-11T10:39:09.573

@JW perfect suggestions! I added them now. – Per Alexandersson – 2014-08-11T11:37:53.917


also see eg Phase Plots of Complex Functions: a Journey in Illustration / Wegert, used to visualize the Riemann zeta fn & related ones

– vzn – 2014-08-14T15:29:03.950



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:

Original image

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:

least squares reconstruction

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:

TV reconstruction

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.

Terry Tao

Posted 2014-08-09T07:01:31.603

Reputation: 53 985


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

– Alex R. – 2014-08-10T05:08:30.057


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)

enter image description here


Posted 2014-08-09T07:01:31.603

Reputation: 101

6This is really interesting! – Per Alexandersson – 2014-08-10T21:05:38.483

54$N(n)$ is the number of times an integer $n$ occurs in the database. (This wasn't clear to me from the plot.) – Kirill – 2014-08-11T10:00:25.147

Finding interesting correlations in seemingly trivial concepts. I love it. – AndreasT – 2014-08-12T01:52:36.067

42From 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 – 2014-08-26T11:51:23.387


11630 is in five OEIS sequences, for example, http://oeis.org/A163279, which was posted in 2009. 12067 is in 9 sequences, including http://oeis.org/A163675, from 2009. 12407 is in two sequences, 12887 in seven, 13258 in three.

– Gerry Myerson – 2015-06-22T23:54:11.313

1What is the first number that isn't member of any OEIS sequence at this time (2016) ?! – None – 2016-03-11T19:12:44.080

1@2000 Just went through the database: 14972 is your answer. – paw – 2016-03-11T23:39:35.990

8It is better to link to arxiv abstract pages and not the PDF directly – Mariano Suárez-Álvarez – 2016-03-13T01:50:05.070

@MarianoSuárez-Alvarez, since this fortuitously changes more than 6 characters, it is very easy to fix (so I have done so). – LSpice – 2016-06-09T21:08:26.383


@paw, 14972 is in http://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 – 2016-06-09T22:58:56.510

Note that on the first page of PDFs from the arXiv there is a direct link (in the left hang margin) to the arXiv abstract page. So you can get quickly from the PDF to the abstract, not just vice versa. Still, because loading PDFs is slower than HTML, I agree with Mariano Suárez-Alvarez that linking to the abstract is preferred. – Jess Riedel – 2017-02-26T18:34:30.113


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: http://imgur.com/a/edAhR

– Dan Brumleve – 2017-02-26T21:08:16.267


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.

enter image description here

stan wagon

Posted 2014-08-09T07:01:31.603

Reputation: 101

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 – 2017-02-27T02:27:47.620

7The 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 – 2017-02-27T06:57:24.897


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: http://demonstrations.wolfram.com/TheBanachTarskiParadox/

– stan wagon – 2017-02-28T02:51:10.123


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:

alt text

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.

Stefan Kohl

Posted 2014-08-09T07:01:31.603

Reputation: 11 527


Really a nice picture! Got people interested on MSE, me included.

– MvG – 2014-09-01T14:06:17.850

3It would look even nicer under stereographic projection, since all of the patterns would be circular rather than elliptical. – Adam P. Goucher – 2015-06-22T23:48:28.450


@AdamP.Goucher: You mean like in the second image here?

– Stefan Kohl – 2015-06-23T08:55:36.080

@StefanKohl Yes, precisely! – Adam P. Goucher – 2015-06-24T19:35:57.020

2What would the fourier transform of that look like? – R_Berger – 2016-07-06T21:55:04.593


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.


Posted 2014-08-09T07:01:31.603

Reputation: 2 005

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 – 2017-03-12T04:16:22.887


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.)

Joseph O'Rourke

Posted 2014-08-09T07:01:31.603

Reputation: 79 882

That is really a nice example! – Per Alexandersson – 2014-08-09T12:27:57.003

Joseph, link to the image appears broken to me. – joro – 2015-01-12T11:44:15.270

@joro: Looks like a server is down. Several of my images have gone missing. – Joseph O'Rourke – 2015-01-12T12:28:52.443

2OK, 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 – 2015-01-12T12:30:10.133


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:

enter image description here

  • 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.

Dr. Goulu

Posted 2014-08-09T07:01:31.603

Reputation: 223

11About 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 – 2016-03-11T15:12:03.043

What do the axes mean? And what are the blue dots? – darij grinberg – 2016-03-12T23:33:54.487

4the 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 – 2016-03-14T06:22:21.993


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: http://imgur.com/a/edAhR

– Dan Brumleve – 2017-02-26T21:05:28.200


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.

enter image description here

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].

Alex R.

Posted 2014-08-09T07:01:31.603

Reputation: 2 481


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

Barnsleys fern

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.): Flame fractal

The most common applications of the latter algorithm seems to be producing abstract book covers for books about the universe: The Grand design enter image description here enter image description here

Per Alexandersson

Posted 2014-08-09T07:01:31.603

Reputation: 6 151


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 $\mathbb{Z}^2$:

enter image description here

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?).

Sam Hopkins

Posted 2014-08-09T07:01:31.603

Reputation: 2 858


enter image description here

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.

Ilias Amrani

Posted 2014-08-09T07:01:31.603

Reputation: 1 373

Looking at this again I can't help but think of Ramsey Theory. – Jp McCarthy – 2015-08-21T12:49:57.230


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".

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.

greater modular chaos

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.


Posted 2014-08-09T07:01:31.603

Reputation: 51


The Hofstadter butterfly enter image description here

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

André Henriques

Posted 2014-08-09T07:01:31.603

Reputation: 25 849


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.


Per Alexandersson

Posted 2014-08-09T07:01:31.603

Reputation: 6 151

2ok 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 – 2016-03-11T18:47:32.663

1@user1952009 These patterns are not random at all. These are the result of edge detecting filters and other features useful for image recognition. – Houshalter – 2017-02-26T17:55:34.350


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} $

enter image description here


Posted 2014-08-09T07:01:31.603

Reputation: 288

5Why 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 – 2016-11-13T00:11:39.603


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 http://qr.ae/TklUCI.

– Anders Kaseorg – 2017-02-26T22:36:01.577


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

$$x^2+y^2, n=503$$ enter image description here

$$4x^2+y^2, n=503$$ enter image description here

$$x^3+x-y^2, n=503$$ enter image description here

$$xy, n=1961=37\cdot 53$$ enter image description here


Posted 2014-08-09T07:01:31.603

Reputation: 13 851

What is your $n$ ? – darij grinberg – 2014-08-09T10:59:35.677

@darijgrinberg for the first 3, n=503. For the last n=1961 = 37*53 (to show relation to factoring). – joro – 2014-08-09T11:05:09.013

1@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 – 2014-08-09T11:35:45.230

@PerAlexandersson I don't claim novelty. Saw the first as avatar on a forum loooong ago. – joro – 2014-08-09T11:38:35.737

These types of images I found very rare. Thanks for uploading it. – None – 2014-08-09T11:02:29.017

Nice! In your first picture there appear horizontal and some vertical stripes at a constant distance of ~20. Please reassure me that those are not part of the picture, but due to resolution/pixel problems. – Wolfgang – 2014-08-09T12:54:06.180

@Wolfgang I am not sure the lines are real. PNG compresses, so they might be artifacts of compression. Will try with something lossless. – joro – 2014-08-09T13:07:58.447


@Wolfgang The lines appear artifacts of scaling. On a bigger plot they disappear: http://s12.postimg.org/fpi9upxgt/x_2_y_2_2.png

– joro – 2014-08-09T13:15:10.527

@PerAlexandersson Is there explanation of all of my plots? Larger plot of the first remotely resembles the plot in the oldest answer: http://s12.postimg.org/fpi9upxgt/x_2_y_2_2.png

– joro – 2014-08-09T14:58:31.290

@joro: The plots should agree with the corresponding level curves, so my strong suspicion is that it is just the level curve set, but "colored" with a strange function; taking mod is sort of "strange". Thus, the pictures are explained by level curves and interplay with coloring function. – Per Alexandersson – 2014-08-09T15:18:48.653

@PerAlexandersson Actually I am asking why taking mod preserves the corresponding curves. mod is discrete, curves are not. – joro – 2014-08-09T15:43:09.877

3Imagine 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 – 2014-08-09T16:39:15.880


Such pictures were discussed in "Kvant" magazine, see http://kvant.ras.ru/1987/11/pti.htm

– Alexey Ustinov – 2016-08-23T11:33:18.450

that last picture looks a lot like artifacts that appeared on images I generated using fourier transforms of sine waves: https://lh3.googleusercontent.com/-0fNn0QHwwPg/T3FpdUtrs9I/AAAAAAAACtE/rz-ekmKxqXU/w1060-h842-p-rw/inacurrate.png

– jes5199 – 2017-02-27T00:16:20.803


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.


Posted 2014-08-09T07:01:31.603

Reputation: 354


There's a really nice video with some fascinating visualizations of this here, they kind of blow away the traditional visualizations.

– Jason C – 2017-04-30T23:42:23.370


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:

enter image description here

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.

Alexey Ustinov

Posted 2014-08-09T07:01:31.603

Reputation: 6 017


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.

Denis Serre

Posted 2014-08-09T07:01:31.603

Reputation: 26 530


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



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

Gerald Edgar

Posted 2014-08-09T07:01:31.603

Reputation: 25 496

9This is beautiful, but badly lacking context... – darij grinberg – 2015-01-12T16:46:57.583

How would one create these? – Andrei Nemes – 2017-02-28T12:22:24.320


@AndreiNemes ... see here https://en.wikipedia.org/wiki/Rose_(mathematics)

– Gerald Edgar – 2017-02-28T13:57:00.727

@GeraldEdgar thanks a lot! – Andrei Nemes – 2017-02-28T17:04:14.303


Abelian sandpile model:

Abelian sandpile model

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

Alexey Ustinov

Posted 2014-08-09T07:01:31.603

Reputation: 6 017


Abelian sandpile model on a square grid, to be more precise. See other grids at http://www.math.cmu.edu/~wes/sandgallery.html .

– darij grinberg – 2015-08-20T13:30:57.917


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.

Manfred Weis

Posted 2014-08-09T07:01:31.603

Reputation: 3 750


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

unexpected Voronoi diagram

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.

Alexey Ustinov

Posted 2014-08-09T07:01:31.603

Reputation: 6 017


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):

Plane section of the E₈ lattice parallel to a Coxeter 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.


Posted 2014-08-09T07:01:31.603

Reputation: 7 248


Coefficients of cyclotomic polynomials with composite number:

enter image description here

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

Alexey Ustinov

Posted 2014-08-09T07:01:31.603

Reputation: 6 017


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

Steve Huntsman

Posted 2014-08-09T07:01:31.603

Reputation: 10 916


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 $\binom{15}5=\binom{15}{10}=3003$): enter image description here

მამუკა ჯიბლაძე

Posted 2014-08-09T07:01:31.603

Reputation: 7 451


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.


Posted 2014-08-09T07:01:31.603

Reputation: 5 204

Do you know any instances where one of these pictures have lead to an insight, or unexpected development? – Per Alexandersson – 2014-08-09T08:15:59.490

@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 http://oeis.org/A188346/graph with this one http://oeis.org/A187210/graph.

– Wolfgang – 2014-08-09T08:44:21.113


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.

enter image description here


Posted 2014-08-09T07:01:31.603

Reputation: 101


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


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


enter image description here

Payam Seraji

Posted 2014-08-09T07:01:31.603

Reputation: 601

What an accident! It reminds my favourite joke on ships in bottles. – Fedor Petrov – 2016-06-10T22:00:44.790

5Amusing 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 – 2016-06-10T22:31:00.983

@FedorPetrov Do you have a reference? :) – მამუკა ჯიბლაძე – 2017-02-26T06:36:52.843

@მამუკაჯიბლაძე http://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 – 2017-02-26T10:50:11.537

@FedorPetrov Thanks, great! Especially other stuff (yes I understand Russian :)) – მამუკა ჯიბლაძე – 2017-02-26T14:11:20.457


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.

enter image description here

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.


Posted 2014-08-09T07:01:31.603

Reputation: 1


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.


Posted 2014-08-09T07:01:31.603

Reputation: 5 204


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.


Posted 2014-08-09T07:01:31.603

Reputation: 1


Gil Kalai

Posted 2014-08-09T07:01:31.603

Reputation: 12 545

12This 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 – 2015-08-20T13:26:15.867

https://en.wikipedia.org/wiki/Voronoi_diagram – reuns – 2016-03-11T18:49:48.603


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

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


Posted 2014-08-09T07:01:31.603

Reputation: 101

1Is this part of your research? – Per Alexandersson – 2014-08-14T20:38:55.353

@Per Alexandersson Yes, I coded the tracer with actionscript and tested functions... – helloflash – 2014-08-14T22:12:31.887

8And so? What mathematical insights did this give rise to? – Todd Trimble – 2014-08-17T14:00:41.697

1@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 – 2014-08-17T19:35:35.583

12Sorry 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 – 2014-08-27T19:36:37.487


See description below


Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges $r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.

Let $G :=\left\langle\tau_{0(2),1(2)}, \tau_{0(5),4(5)}, \tau_{1(4),0(6)}\right\rangle$ be the group from the first remark to this question. The group $G$ has one "exceptional" orbit $0^G = \{0,1,4,5,6,7,8,9,10,11,12,13,14,15,18,19\}$ of length $16$. Also it has four series of orbits of length $2$ (namely $\{2(60), 3(60)\}$, $\{22(60), 23(60)\}$, $\{26(60), 27(60)\}$ and $\{46(60), 47(60)\}$) -- the "trivial" orbits.

There is numerical evidence that the other orbits come all in infinite series, have all length congruent to $8$ modulo $16$, and that all positive integers in the residue class $8(16)$ do occur as orbit lengths -- and that in particular all orbits are finite (but there is no proof of this). Also there is numerical evidence that $16(60) \cup 32(60) \cup 52(60) \cup 56(60)$ is a set of representatives for the non-"trivial" orbits.

The picture above shows the lengths of the orbits whose representative lies in the residue class $16(60)$; each color stands for a particular orbit length -- for example the big light-yellow rectangle on the right stands for the residue class $76(120)$ which is a set of representatives for a series $\{76(120), 77(120), 110(180), 111(180), 114(180), 115(180), 118(180), 119(180)\}$ of orbits of length $8$. The big blue rectangle on the upper left stands for the residue class $16(480)$ (orbit length $24$), the big green rectangle stands for the residue class $1216(1920)$ (orbit length $40$), etc., and the black areas between the colored rectangles stand for larger orbit lengths (in the hundreds and above -- also the orbit of length $47610700792$ discussed in the question linked above belongs here). Presumably the colored rectangles would completely tile the picture if one wouln't have to stop drawing at some point.

Larger versions of the picture can be found here (2.5MB SVG file) and here (17MB SVG file -- may exceed capacities depending on used computer and browser). The longest orbits represented in the largest picture have length $984 = 61 \cdot 16 + 8$, which is still tiny in comparison with the $47610700792$ mentioned above.

The picture below indicates by color in which of the four residue classes $16(60)$, $32(60)$, $52(60)$, $56(60)$ the representative of the orbit of given $n \in 30(60)$ lies -- the trisection alternates between horizontal and vertical, i.e. the left third of the picture corresponds to the residue class $30(180)$, the middle third to $90(180)$, the right third to $150(180)$, the top left $9$-th to $30(540)$, the top right $9$-th to $150(540)$, the bottom left $9$-th to $390(540)$, and the bottom right $9$-th to $510(540)$:

See description above

There is a larger version of the picture available here (1.2MB SVG file).

Stefan Kohl

Posted 2014-08-09T07:01:31.603

Reputation: 11 527