Mathematically interesting screensavers A screensaver is a computer program that fills a computer screen with a moving pattern that eluminates each pixel for approximately the same proportion of time. Originally designed to prevent burn-in of computer screens based on cathode-ray tubes, screensavers today are primarily works of art.
I would love to have a screensaver that animates the screen in a mathematically meaningful and interesting way. There are many dynamical processes that could be used to design beautiful screensavers, and many mathematical objects can be beautifully animated. I have found various examples of e.g. mathematically beautiful screensavers and suchlike, but the mathematics behind these does not seem to be of research interest in any meaningful sense (perhaps only fractals).
While it is not so difficult to write a screensaver oneself (although it would require some artistic taste to write a good one), I wonder whether there is anything out there that is beautiful and of research level that I could download.
NSF has a screensaver for download that contains scientifically interesting simulations, but not for mathematics, HERE.

Could you please suggest screensavers that animate the screen in a research-level mathematically interesting, meaningful way? 

 A: As promised, here is my “screensaver” that I just made from  algebraic structures that arise from the very large cardinals . The screensaver works by outputting heat maps of endomorphic Laver tables where the $i$-th image at coordinate $(j,k)$ has temperature $t^{\sharp}(\mathfrak{l}_{1},\mathfrak{l}_{2},\mathfrak{l}_{3})(1^{i}0^{j}1^{k})$ and where $t^{\sharp}$ is the functional endomorphic Laver table operation and $\mathfrak{l}_{1},\mathfrak{l}_{2},\mathfrak{l}_{3}$ are simple-to-describe objects in the functional endomorphic Laver table. In other words, the animation is simply a small slice from the functional endomorphic Laver tables, but this small slice still exhibits a great deal of complexity. The animation is slow since so far computing functional endomorphic Laver tables takes up quite a bit of processing power. Of course, one can make the animation go fast if one pre-computes the animation instead of computing it in real-time. 
A: Carsten Steger has contributed a number of screenshots based on 4d surfaces and polytopes to Jamie Zawinski's xscreensaver. There are also screensavers in the package based on tilings, fractals, and cellular automata. Not exactly research level, but a sounder claim on being mathematical than the list on mathsavers.com.
A: Since forever there's a x11 &Linux screensaver called xscreensaver. It consists of main screensaver and about 50 modules, 80% of which shows graphic based on various simulations of interesting phenomena, from swarm behaviour modeling, trough percolation to various nonlinear dynamics and fractals. Every module comes with information about simulated process, numerical methods and is if course open source.
Here: https://www.jwz.org/xscreensaver/screenshots/ you may find a couple of screenshots and download software for Linux, Mac and android.
A: This particular screensaver did not just nicely illustrate math, it actually motivated research:
A Tisket, a Tasket, an Apollonian Gasket, Dana Mackenzie 

In the spring of 2007 I had the good fortune to spend a semester at
  the Mathematical Sciences Research Institute in Berkeley. Someone had
  installed a screen-saver program on the computer. Of course, it had to
  be mathematical. The program drew an endless assortment of fractals of
  varying shapes and ingenuity. Every couple minutes the screen would go
  blank and refresh itself with a completely different fractal. I have
  to confess that I spent a few idle minutes watching the fractals
  instead of writing.
One day, a new design popped up on the screen (see below). It was
  different from all the other fractals. It was made up of simple
  shapes—circles, in fact, and unlike all the other screen-savers, it
  had numbers! My attention was immediately drawn to the sequence of
  numbers running along the bottom edge: 1, 4, 9, 16 ... They were the
  perfect squares! Seeing those numbers awakened the math geek in me.
  What did they mean? And what did they have to do with the fractal on
  the screen? Quickly, before the screen-saver image vanished into the
  ether, I sketched it on my notepad, making a resolution to find out
  someday.


Here you can watch the screensaver in action, illustrating Descartes' theorem.
A: The free Cellular Automata program Capow contains a great screensaver as part of the download. http://www.rudyrucker.com/capow/download.html
A: BOINC is a software for distributed community computing that runs on your extra cpu (or gpu) time usually when a normal screensaver would be running. The project does not focus on creating visualizations, but rather, on performing computational work for research projects. However, the software is distributed with a screensaver, and some projects include visualizations that correspond to the computation being performed. 
BOINC Screensaver

The BOINC screensaver has 3 modes:
The overview screensaver displays general information, such as the
  BOINC status, a list of projects, etc.
The project graphics display the graphics for one of the currently
  running tasks, e.g., SETI@home. However, many projects do not have
  screensaver graphics.
The screensaver coordinator controls the screensaver, selecting either
  the default screensaver or project graphics. It appears when neither
  the overview screensaver nor project graphics are available, and
  displays a moving BOINC logo with messages such as "Connecting to
  BOINC application" or "BOINC screensaver loading."

A list of mathematical projects is available including several different projects in number theory and computation.
Work done on the SAT@home project (boolean satisfiablity problem) has lead to 3 publications, so while the visualizations might not be so interesting, this is truly a research-level screensaver!  
A: Electric Sheep is one of the original screensavers to make use of distributed computing.  It displays beautiful flame fractals that have evolved (via genetic algorithms) in response to user feedback.
A: Game of Life
Game of Life is a cellular automaton. The interesting part is that complex patterns show up even though the rules are simple. One interesting mathematical property is that it is turing complete.
There are various people who have turned this algorithm into a screensaver.
A: The XlockMore screensaver has some nice mathematical "modes"; perhaps the fanciest one is "invert" which everts the sphere.  The possibility of sphere eversion was a significant result when Smale first proved it back in the 1950s.
A: I've made a few "screensavers" in JavaScript for web browsers:


*

*Periodic Complex Functions (code)

*Circles of Appollonius (code)


But I'm not sure how to make these into screensavers for an operating system.
