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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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    $\begingroup$ 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. $\endgroup$ Commented Aug 9, 2014 at 7:27
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    $\begingroup$ @SamHopkins: This should be an answer! I have seen sandpile-models, and Apollonian gaskets, but never expected a connection! $\endgroup$ Commented Aug 9, 2014 at 7:35
  • $\begingroup$ There are also quite some "famous" examples, e.g. the ulam spiral $\endgroup$
    – PlasmaHH
    Commented Aug 10, 2014 at 18:26
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    $\begingroup$ also see eg Phase Plots of Complex Functions: a Journey in Illustration / Wegert, used to visualize the Riemann zeta fn & related ones $\endgroup$
    – vzn
    Commented Aug 14, 2014 at 15:29
  • $\begingroup$ Sadly, I don't have enough reputation, but here's one thing that in my opinion is really fascinating: how dragon curves appear when roots of certain polynomials are plotted. $\endgroup$
    – IS4
    Commented Aug 14, 2019 at 17:24

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

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

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

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

                             configuration lattice of the SPM with n=40

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

partitions of 80 into powers of 2

They are also mentioned in this discussion.

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    $\begingroup$ This is beautiful! $\endgroup$ Commented Jan 22, 2021 at 19:14
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    $\begingroup$ Candy floss.... $\endgroup$ Commented Jan 23, 2021 at 4:19
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    $\begingroup$ Thanks! The first one is often called bee nest but I like candy floss better ;) $\endgroup$ Commented Jan 23, 2021 at 9:00
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The discovery of the special nature of Costa's minimal surface has been made on a visualization.

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

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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

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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

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

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  • $\begingroup$ Do you know any instances where one of these pictures have lead to an insight, or unexpected development? $\endgroup$ Commented Aug 9, 2014 at 8:15
  • $\begingroup$ @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. $\endgroup$
    – Wolfgang
    Commented Aug 9, 2014 at 8:44
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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 $A_n=(\{1,\ldots,2^n\},*_n)$ where

  1. $x*_n(y*_n z)=(x*_n y)*_n(x*_n z)$, and

  2. $x*_n 1=x+1\bmod 2^n$.

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 $L_n:\{0,\ldots,2^n-1\}\rightarrow\{0,\ldots,2^n-1\}$ 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 $\{0,\ldots,2^n-1\}$ 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 $512\times512$ 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.

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

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In the Jones monoid, a monoid associated to the Temperley-Lieb algebra, the idempotents were discovered (see Organic Semigroup Theory for an overview) to exhibit fern-like qualities when the Green's $\mathcal{D}$-classes are drawn out. The below picture, taken from the earlier link and generated by James East in GAP, illustrates this quite beautifully, by having a black pixel for a group $\mathcal{H}$-class, indicating the presence of an idempotent, and otherwise a white pixel:

enter image description here

Why such beautiful patterns appear still seems to be a very exciting mystery!

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Consider a root system of type D. The Hasse diagram is built up by writing, as bottom row, a node for each simple root, and then on each row above, connect up two roots if their sum is a root.

Hasse diagram of root system D8

The Hasse diagram has an obvious symmetry in the last two roots (7 and 8, in the picture), from the symmetry of the Dynkin diagram. We can picture that symmetry as flipping the box kite in the picture inside out. But the picture is also symmetric under the obvious diagonal reflection through the lower right corner of the picture, exchanging the bottom row with the right hand side. The bottom row is the simple roots. The right hand side is various sums of simple roots. As far as I can see, no other root system has a symmetry in its Hasse diagram not induced by a symmetry of the Dynkin diagram. This has very complicated consequences about the decomposition of the tangent bundle of any flag variety over D, for example. I saw this symmetry when I drew this picture, and I don't know if it was already known.

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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

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    $\begingroup$ Looks cool - is it 'unexpected'? $\endgroup$ Commented Mar 28, 2018 at 20:49
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    $\begingroup$ @PerAlexandersson Well I was not expecting that much symmetry - the element was chosen by randomly picking coefficients in the root vector basis $\endgroup$ Commented Mar 28, 2018 at 21:59
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This may be late to the party, but hidden symmetries lurk in higher-dimensional lattices.

Four dimensions is enough to produce this effect. A four-dimensional lattice of points in $\mathbb{Z}^4$ appears not to have any regular pentagons. And yet the invertible transformation matrix $M$ given by

\begin{pmatrix} 0&0&0&-1\\1&0&0&-1\\0&1&0&-1\\0&0&1&-1\\ \end{pmatrix}

transforms the lattice into itself and satisfies the relation $M^5=I_4$. Thus a fivefold symmetry emerges with the inherited periodicity of the lattice, and may be converted into a quasiperiodic proper fivefold lattice by projecting the four-dimensional lattice into a plane.

Such is a construction of a quasilattice. Since the four-dimensional lattice and its quasilattice projection contain a center of inversion as well, the fivefold axis is further promoted to a tenfold axis.

Below 1 is a diffraction pattern produced by a quasicrystalline alloy possessing long-range tenfold symmetry. The full symmetry of such a quasilattice cannot be rendered in the plane with a discrete collection of points, but tenfold clusters derived from the quasilattice symmetry, and thus the four-dimensional parent lattice, are evident.

enter image description here

Transformation matrices with eight- or twelvefold symmetry can be constructed in the four-dimensional lattice, and higher dimensions open up still more possibilities. Six dimensions, for instance, yield the symmetries of the regular heptagon and enneagon (therefore also the 14-gon and 18-gon), as well as the full symmetry of a regular icosahedron.

Reference

  1. Seki, Takehito & Abe, E.. (2015). "Local cluster symmetry of a highly ordered quasicrystalline Al 58 Cu 26 Ir 16 extracted through multivariate analysis of STEM images". Microscopy (Oxford, England). 64. https://doi.org/10.1093/jmicro/dfv035.
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For better visualizing and understanding fractals like the Mandelbrot set, the idea of color cycling is a great invention.

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

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

This is the first one I made.

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

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

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

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

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

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

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

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

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

enter image description here

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    $\begingroup$ 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. $\endgroup$ Commented Aug 20, 2015 at 13:26
  • $\begingroup$ en.wikipedia.org/wiki/Voronoi_diagram $\endgroup$
    – reuns
    Commented Mar 11, 2016 at 18:49
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See description below

Explanation:

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).

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The Collatz conjecture has already been mentioned here in an answer. I think it is worth however to add explicitly two more images here, both from the OEIS.

The first one simply displays the sum of numbers in the trajectory for each initial value $n$. trajectory And here is the same, but with the horizontal axis also logarithmic (both base $10$). trajectory, log What looks most remarkable to me here are the almost vertical gaps next to the bottom, about five of them between $n$ and $2n$, well visible on both images. Will they continue at that rate for bigger $n$?
And as for the cloud-like concentration of points with values just above $10^5$ (starting with $a(27)=101440)$, its visibility depends heavily on the scale used. But one could wonder whether for different scales/ranges one can see other horizontal point clouds of similar "thicknesses".

The second one displays the number of Collatz steps when starting with primes. The distance of two horizontal "relaying segments" of the slowly decreasing patterns (i.e. certain subsets of primes) seems to be always $5$. (Why?) prime lengths

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

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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    $\begingroup$ Is this part of your research? $\endgroup$ Commented Aug 14, 2014 at 20:38
  • $\begingroup$ @Per Alexandersson Yes, I coded the tracer with actionscript and tested functions... $\endgroup$
    – helloflash
    Commented Aug 14, 2014 at 22:12
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    $\begingroup$ And so? What mathematical insights did this give rise to? $\endgroup$
    – Todd Trimble
    Commented Aug 17, 2014 at 14:00
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    $\begingroup$ @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. $\endgroup$
    – helloflash
    Commented Aug 17, 2014 at 19:35
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    $\begingroup$ 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. $\endgroup$
    – Todd Trimble
    Commented Aug 27, 2014 at 19:36
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