Super Motivational Function

I’ve presented a Motivational Function, a while back.

$f(z) = e^{\left(-\frac{1}{z^{2}}\right)}$

It is infinitely flat at the zero point: all its derivatives are zero there. Yet, it manages to lift its head – as it is not analytic at zero! If you think of it as a function of a complex argument, its weirdness becomes more obvious. Turn by ninety degrees, and follow the imaginary axis. Then its value at zero is infinite!

I was thinking about how to do this function’s beauty justice. Last time, I chose a complex plot which encodes absolute values as brightness and phase as hues. But I wanted to see the absolute value’s three-dimensional structure.

Its absolute value is again an exponential function! It has to be: Separating real and imaginary part in the exponent of f, the absolute value can be read off as the exponential function of the real part of the exponent (as the modulus of the exponential of an imaginary value is 1). With z = x + iy …

$|f(x,y)| = e^{\frac{y^2 - x^2}{(x^2 + y^2)^2}}$

This function |f(z)| consists of two enormously elongated towers sitting on a large platform; thus I want to create a logarithmic plot.

$log(|f(x,y)|) = g(x,y) = \frac{y^2 - x^2}{(x^2 + y^2)^2}$

The function g consists of four towers, two of them extending to minus infinity, the other two to plus infinity. Diagonal lines on the base plane separate the four towers from each other. In two of the quadrants between the diagonals, the real part x is greater than the imaginary part y. Thus the exponent is negative and those towers extend to minus infinity – the motivational function becomes zero. In the other quadrants, y is larger than x, and the towers are (plus) infinitely tall. When x is equal to y, g’s value is zero; so the motivational function f is 1. Or better: For reasons of symmetry between the four towers, g’s value can only be zero on the diagonals – even though it looks like this 0/0 at zero should be investigated more carefully.

Looked at all that from above, a cloverleaf-shaped pattern emerges, traced out by the walls of the towers: Loops shaped like the number eight or the symbol for infinity are stacked upon one another.

In the zero point the towers touch each other. Down on the x-y plane, the ∞-shaped loop becomes ∞-ly large and finally turns into the two diagonal lines between x and y axes.

I call it the super motivational function. I see it as composed of colorful symbols for infinity.

_____

I’ve created these plots with SageMath – which is probably the most enjoyable piece of software I have touched since a long time. Grid lines and axis removed on purpose, for artistic effect.

These are not three-dimensional plots of a single function, but a huge number of ∞-shaped parametric curves stacked onto each other. The design is actually a result of my working around the limitation of not being able to cut off a function of two variables at a certain value. These functions rise quickly to values that can hardly be handled by 3D plotting software. Lonely cloverleaves and infinity shapes were the only thing I saw – when attempting to create true three dimensional plots.

A Python script generates all the wire loops, based on the solutions of the equation of my g(x,y) having to have a certain value. As I need to craft all the loops separately, I can or have to tune their colors and their vertical positions. The distance between the loops gradually changes with the height, in order to distribute them somewhat evenly along the walls of the towers, as well as on the x-y plane below.

1. I did not know that Cliff Stoll had a celebrated history of tracking criminal hackers (I did remember him when I saw his workshop–his interest in Klein bottles and playing with old tech). I love the way he describes his expertise: “To a mathematician, I’m a pretty good physicist,” Stoll deadpans. “To a physicist, I’m a fairly good computer maven. To real computer jocks, they know me as somebody who’s a good writer. To people who know how to write … I’m a really good mathematician!” :)

1. elkement says:

2. Beautiful. I love the perspective of the last frame; I feel that I am looking up to the surface while infinitely sinking below. It compliments the first view of the plot; with everything in between, it is a lovely progression. There should be a genre of math art prints like there is of architectural or botanical drawings. I can see it printed out large and hung in a big room.

Thanks for sharing the link for SageMath again. I bookmarked it.

1. elkement says:

Thanks, Michelle!! I’ve also imagined how these curves would look like on a gigantic canvas – or even as a real, physical installation made up of colored, curved wires. I think seeing the surfaces as spanned by wires was perhaps also an inspiration (besides obtaining a de facto limit in the vertical direction) in the sense of: How would I build them, theoretically?

I picked SageMath because it seems this is what all the Mathematica/Maple/MathLab etc. open source alternatives have merged into. Maybe the commercial software packages have more options, but I am super impressed by what SageMath can do, and pricing of the commercial alternatives seemed very prohibitive for a tiny business or single user last time I looked. I like the Jupyter notebooks in particular, and the seamless integration with Python!

1. A sculpture of the curves would be fascinating. If you ever watch any Numberphile videos on YouTube you may have encountered the episodes about some mathematician’s strange obsession with something… almost always involves something cool like Klein bottles. This one includes unusually cut torus sculpture:

Thus, I think there is precedent for turning the Super Motivational Function into sculpture.

1. elkement says:

That’s a fascinating video, thanks Michelle! It’s great to learn that mathematicians have such close connections with artists – and that professional mathematicians also enjoy playing with physical models of these structures!

And I’ve watched a Numberphile video recently, on Klein bottles of all things. It is about a polymath / stereotype “mad scientist” selling them from his home :-) https://www.youtube.com/watch?v=-k3mVnRlQLU
He is also the author of the first “hacker detective memoir”, much revered in cybersecurity circles: https://www.wired.com/story/meet-the-mad-scientist-who-wrote-the-book-on-how-to-hunt-hackers/

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