Title: Does anybody know this fractal? Post by: no_gravity on October 16, 2012, 04:19:45 PM Did some experiments on the divisibility of real numbers. When I transposed them to complex numbers, I got this. I wonder what it reminds me of. Is it something well known?
(http://www.gibney.de/images/size00/-1_-1_to_1_1.jpg) More renderings can be found here: http://www.gibney.de/does_anybody_know_this_fractal Im not even sure if its a fractal. Although it seems to have self-similarity and infinite detail. Title: Re: Does anybody know this fractal? Post by: s31415 on October 16, 2012, 09:21:51 PM Hi,
You can simplify your picture by replacing g/c in your algorithm by g*c, ie. performing an inversion. The characteristic flower-shaped pattern in the center comes from the inversion of a pattern consisting of parallel and orthogonal lines. It would be interesting to see what you get without this inversion. Sam Title: Re: Does anybody know this fractal? Post by: no_gravity on October 16, 2012, 09:54:55 PM Here it is with multiplication instead of division:
(http://www.gibney.de/images/size00/inversion.jpg) (http://www.gibney.de/does_anybody_know_this_fractal_-_part_2) Very similar, but without the nice circles. Everything is straight now. So your assumption seems to be right! Title: Re: Does anybody know this fractal? Post by: hobold on October 17, 2012, 12:43:38 AM The straightened version reminds me of a perspective projection of a regular (cartesian) grid. Conceptually, you place a little sphere in the center of each cubical cell, and then look from any point into any direction at an infinity of spheres that seem to be all sizes (due to perspective shortening). The resulting images have various interlocked "stars" of straight rays emanating from shared center points. I am not sure, though, if an exact analogue of your picture could be found this way.
Title: Re: Does anybody know this fractal? Post by: no_gravity on October 23, 2012, 05:51:47 PM Got a lot of great feedback on this thing. I summarized it in this update: http://www.gibney.de/does_anybody_know_this_fractal_-_part_2
Title: Re: Does anybody know this fractal? Post by: hobold on October 24, 2012, 04:06:40 AM And here is an example image of the projected grid I was reminded of:
(http://www.vectorizer.org/cartespheres.png) Title: Re: Does anybody know this fractal? Post by: no_gravity on October 25, 2012, 08:26:48 PM Pretty interesting! Thats just space evenly filled with spheres?
Title: Re: Does anybody know this fractal? Post by: hobold on October 25, 2012, 09:34:49 PM Thats just space evenly filled with spheres? Yes, just an ordinary cartesian grid, as seen from some oblique angle with the usual perspective projection. Only a limited part of space was actually filled for this example.(Nitpicking: to be a real fractal point cloud, one would have to render infinitesimal points rather than spheres, so that no background is hidden behind foreground objects.) Title: Re: Does anybody know this fractal? Post by: jehovajah on October 27, 2012, 04:46:52 PM The original version is the well known Sacred Geometrical pattern. The inversion is the quadrature of this pattern and is of course less interesting.
The Sacred Geometry is a fractal pattern of long, long provenance, but I do not think I have seen it generated by division algorithms before. That is interesting. The underlying reason is related to the circular relation in the generation of the numbers you use. This circular relation derives from Spherical trigonometry and spherical geometry, the ultimate sources of our numeral coefficients, both Rational and Real. Also the basis of our complex vector algebras. Title: Re: Does anybody know this fractal? Post by: DarkBeam on November 28, 2012, 05:22:15 PM Gaussian Integer? :D |