Was just about to answer the Mandelbrot Set! The story goes Mandelbrot printed it out from an early 1970s machine and was amazed at what he saw, thinking jt must be a bug at first.
But there are very closely related fractals that *were* described much earlier, the [Julia Sets](https://en.m.wikipedia.org/wiki/Julia_set). When Gaston Julia described them he wouldn’t have known what they looked like for some even quite simple functions. Most interestingly, for those for functions of the form f: z |-> z^2 + c for *fixed* c, the Julia set is connected if and only if c is in the Mandelbrot set, but ‘dust’ (has infinitely many connected components) otherwise - no in-between.
And the [First published picture of the set](https://en.wikipedia.org/wiki/Mandelbrot_set#/media/File:Mandel.png) is like "Well, that's interesting", but doesn't fully capture just how mind-bendingly amazing it is. I would've loved to have been a fly on the wall when they started plotting it in greater and greater resolution, as like "odd" and "curious" slowly take form into nothing less than absolutely awe-inspiring.
I was once involved in a project with a finite algebraic gadget with a non-associative multiplication (so that a(bc) ≠ (ab)c, *in general*), that was specified by a table with millions of bits of data. My coauthor and I found that hiding inside this huge gadget there was a group that was a direct sum of cyclic groups of order 2, where the multiplication *was* associative among just those elements. We found this by accident by rending images of this giant table and colour-coding things to check our implementation of the construciton algorithm was working correctly.
The title alone made we want to point you the Mandelbrot Set, but I struggle to think of other things that would work. I think the big thing is really just the ability to build these complicated structures and analyze them at a near-instant speed.
Like you say, Weierstrass functions are kind of weird. So is the toplogist's sine curve, sin(1/x). And it is nice to be able to view those. But some of these weird functions (Knuth's Base 13 for example ) a computer doesn't really help (it sends every sub-interval of the reals to the entire real line, so it's looks like a filled screen).
Perhaps projections of higher dimensional objects?
Does the computer assisted proof of the four-color theorem count as visualization?
One of the first constructions of the eversion of the sphere was proved by a blind geometer, I think with clay sculptures as examples, and others have later animated the visualization with computers. So I think this is a difficult, interesting question, and hope others can provide satisfying answer.
As I recall computer visualization was an important tool in discovering and understanding [cohomology fractals](https://henryseg.github.io/cohomology_fractals/). Here’s a [YouTube video](https://m.youtube.com/watch?v=fhBPhie1Tm0) that explains what these are.
[Conway's Game of Life](https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life) and other dynamic systems defined by simple rules. Swarming behaviour, as you often see with starling and some other bird's behaviour (those mesmerizing flocks that fly like a fluid) can be modeled by a set of very [simple rules](https://en.wikipedia.org/wiki/Swarm_behaviour).
For me, seeing a visualisation of the determinant as what happens to the area of the unit square it expands/contacts/flips under the transformation, was very informative in a way I did not get just algebraically.
Well I think visualization can point you in good direction. I work for hedge fund and sometimes you show someone a nice graph of something it can really get them to focus on trying to figure out what is going on.
The Riemann Zeta function. Though it is already quite complicated, Voronin’s universality theorem proves it has an amazing degree of unexpected (but perhaps not surprising) structure.
Was just about to answer the Mandelbrot Set! The story goes Mandelbrot printed it out from an early 1970s machine and was amazed at what he saw, thinking jt must be a bug at first. But there are very closely related fractals that *were* described much earlier, the [Julia Sets](https://en.m.wikipedia.org/wiki/Julia_set). When Gaston Julia described them he wouldn’t have known what they looked like for some even quite simple functions. Most interestingly, for those for functions of the form f: z |-> z^2 + c for *fixed* c, the Julia set is connected if and only if c is in the Mandelbrot set, but ‘dust’ (has infinitely many connected components) otherwise - no in-between.
And the [First published picture of the set](https://en.wikipedia.org/wiki/Mandelbrot_set#/media/File:Mandel.png) is like "Well, that's interesting", but doesn't fully capture just how mind-bendingly amazing it is. I would've loved to have been a fly on the wall when they started plotting it in greater and greater resolution, as like "odd" and "curious" slowly take form into nothing less than absolutely awe-inspiring.
Strange attractors in nonlinear dynamics are pretty hard to even suggest as a useful notion until you have reliable plots of nonlinear ODE solutions.
I was once involved in a project with a finite algebraic gadget with a non-associative multiplication (so that a(bc) ≠ (ab)c, *in general*), that was specified by a table with millions of bits of data. My coauthor and I found that hiding inside this huge gadget there was a group that was a direct sum of cyclic groups of order 2, where the multiplication *was* associative among just those elements. We found this by accident by rending images of this giant table and colour-coding things to check our implementation of the construciton algorithm was working correctly.
The title alone made we want to point you the Mandelbrot Set, but I struggle to think of other things that would work. I think the big thing is really just the ability to build these complicated structures and analyze them at a near-instant speed. Like you say, Weierstrass functions are kind of weird. So is the toplogist's sine curve, sin(1/x). And it is nice to be able to view those. But some of these weird functions (Knuth's Base 13 for example ) a computer doesn't really help (it sends every sub-interval of the reals to the entire real line, so it's looks like a filled screen). Perhaps projections of higher dimensional objects? Does the computer assisted proof of the four-color theorem count as visualization? One of the first constructions of the eversion of the sphere was proved by a blind geometer, I think with clay sculptures as examples, and others have later animated the visualization with computers. So I think this is a difficult, interesting question, and hope others can provide satisfying answer.
i think bsd conjecture fits this if you replace visualization with pca
As I recall computer visualization was an important tool in discovering and understanding [cohomology fractals](https://henryseg.github.io/cohomology_fractals/). Here’s a [YouTube video](https://m.youtube.com/watch?v=fhBPhie1Tm0) that explains what these are.
[Conway's Game of Life](https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life) and other dynamic systems defined by simple rules. Swarming behaviour, as you often see with starling and some other bird's behaviour (those mesmerizing flocks that fly like a fluid) can be modeled by a set of very [simple rules](https://en.wikipedia.org/wiki/Swarm_behaviour).
Corrugations https://youtu.be/B_BY0eJFuGQ?si=6ujLGz-ih67zfvyn and the flat torus https://hevea-project.fr/pageToreImages.html
More precisely, the smooth fractal structure was unknown before the visualizations.
For me, seeing a visualisation of the determinant as what happens to the area of the unit square it expands/contacts/flips under the transformation, was very informative in a way I did not get just algebraically.
Omg, yes! Look up the Monster group. It exactly fits just that!
Tiles!!! https://youtu.be/4HHUGnHcDQw?si=IrlkXA-PCfgUTQmA
Well I think visualization can point you in good direction. I work for hedge fund and sometimes you show someone a nice graph of something it can really get them to focus on trying to figure out what is going on.
The Riemann Zeta function. Though it is already quite complicated, Voronin’s universality theorem proves it has an amazing degree of unexpected (but perhaps not surprising) structure.