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

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

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

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

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

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

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

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    This should maybe be community wiki... – Per Alexandersson Aug 9 '14 at 7:01
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    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
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    Would <experimental-mathematics> or <visualization> be relevant tags for this question? – J W Aug 11 '14 at 10:39
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    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

37 Answers 37

Picture based on research by Christopher Hoffman, Alexander Holroyd and Yuval Peres

enter image description here

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    This is neat, but probably needs a couple of qualifications: (1) The apparent randomness in the picture is due to the centers of the circles being chosen at random (there is no deterministic chaos here), and (2) the concentric circles are (as far as I can tell) just an artistic way to make the various regions easier to distinguish from each other. – darij grinberg Aug 20 '15 at 13:26
  • en.wikipedia.org/wiki/Voronoi_diagram – reuns Mar 11 '16 at 18:49

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

  • What an accident! It reminds my favourite joke on ships in bottles. – Fedor Petrov Jun 10 '16 at 22:00
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    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
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    @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

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.

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

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

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.

Just stumbled on this. I am currently working with cyclic elements in simple Lie algebras - these are elements of the form $e+F$ where $e$ is a generic element of degree 2 and $F$ a generic element of lowest possible degree, with respect to the grading defined by a semisimple element.

I took one such in E$_8$, took the matrix of its $\operatorname{ad}$, and made from it a graph, with 248 vertices, and edges connecting those $i$, $j$ with $(i,j)$th entry of that matrix nonzero. Voilà.

enter image description here

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).

protected by Lucia Mar 11 '16 at 13:32

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