Is it possible to plot the ECC equation on to the Mandelbrot set to explore more mathematical properties of such amazing equations? If you know of any useful sources please do let me know.
https://en.m.wikipedia.org/wiki/Mandelbrot_setIt is not really clear what you want, but I will try to answer anyways. Can we use an elliptic curve like y^2=x^3+1 to make a fractal? Well, in complex dynamics, we often iterate polynomials or rational functions and the iterations of these functions can be used to make fractals. If you zoom into the Julia sets of these fractals, you will notice that there is no squashing or stretching in these fractals. This is because holomorphic functions are essentially the 2D conformal mappings that preserve angles and do not squash or stretch anything. This also means that it will be difficult to generalize the Mandelbrot set to anything other than iterations of functions of one complex variable. Now, curves like y^2=x^3+1 are not functions since they are multi-valued on the complex plane, and if we want a nice fractal, we do not want to make these curves single valued by cutting out the branch that we don't want. Since elliptic curves have degree greater than 1 and since elliptic curves are the simplest non-rational expressions with degree greater than 1, it seems reasonable that one can get some neat results or at least neat pictures from the dynamics of iterating an elliptic curve. But I have never studied this in depth so I cannot tell you how well this will work out. It seems reasonable enough though unless I am missing something.
The paper THE FATOU AND JULIA SETS OF MULTIVALUED ANALYTIC FUNCTIONS by P. A. Gumenuk (2002) seems to generalize complex dynamics to multivalued functions.
So I did a very basic experiment iterating an elliptic curve to see if I get a fractal, and it looks like I do in fact get a nice looking fractal.
-Joseph Van Name Ph.D.
P.S. These experiments currently have little to do with Bitcoin development. At the moment, it seems like there is nothing special about elliptic curves other than their simplicity (or maybe that is special). The iteration of an elliptic curve to generate fractals does not seem to have much to do with the addition operation on elliptic curves that is used in cryptography. And even if it did, we are working with complex elliptic curves here rather than the elliptic curves over finite fields that are needed in cryptography.
