Title: More fractal cellular automata Post by: Tglad on May 22, 2017, 03:22:39 AM I stumbled upon a tweet from someone called R4_Unit, who is making some nice fractal cellular automata:
https://twitter.com/r4_unit/status/856042458714648578 (https://twitter.com/r4_unit/status/856042458714648578) (https://pbs.twimg.com/media/C-FFn3lXsAA2Bmv.jpg) It is similar to some previous ones I made https://sites.google.com/site/tomloweprojects/scale-symmetry/automataFinder (https://sites.google.com/site/tomloweprojects/scale-symmetry/automataFinder) but he's using a 3D version of the diamond square algorithm, which is nice... and he's added a little colour to the rules. He will be releasing some code soon. Title: Re: More fractal cellular automata Post by: Sabine on May 22, 2017, 07:49:08 PM Thank you for sharing! Bringing 3d to cellular automata is such a great idea O0
Title: Re: More fractal cellular automata Post by: Softology on May 24, 2017, 02:45:14 AM Interesting.
I had a go at converting the code so I can render them using solid cubes. Red for state 1 and blue for state 2. I must be doing something wrong, because the majority of mine create a more filled in box with patterns, ie (https://c1.staticflickr.com/5/4245/34691253882_03d9742854_o.png) But then a few did get some hollowed out correct looking results (maybe) (https://c1.staticflickr.com/5/4195/34854421295_07bc0bedf9_o.png) (https://c1.staticflickr.com/5/4226/34011466074_325bb153f7_o.png) Same as the last one, but more divisions (https://c1.staticflickr.com/5/4269/34691252802_c9413ea89d_o.png) I need to get a big integer routine so I can replicate his rule strings. Right now I just fill the 3 rule arrays with random numbers between 0 and 2. Jason. Title: Re: More fractal cellular automata Post by: Softology on May 24, 2017, 05:03:44 AM My mistake. Down to a type when assigning random values to the rules array. Now I get the right "carved out" results.
Right-click View Image to see these at their full 4K resolution. (https://c1.staticflickr.com/5/4245/34856101945_b64fc91cb1_o.png) (https://c1.staticflickr.com/5/4276/34856101405_e50301dda5_o.png) (https://c1.staticflickr.com/5/4220/34470075790_bfb9db7a37_o.png) (https://c1.staticflickr.com/5/4251/34693099772_a400105aca_o.png) (https://c1.staticflickr.com/5/4227/34856289215_eda210390c_o.png) Really nice variety in these so far. Jason. Title: Re: More fractal cellular automata Post by: Tglad on May 24, 2017, 07:56:43 AM The green one is my fave. I'm wondering whether it is worth doing the same rule for both the octahedron steps... just to cut down on the search space. I'm not sure if it would add more symmetry or not.
What renderer is that? seems to have nice ambient occlusion. Title: Re: More fractal cellular automata Post by: DarkBeam on May 24, 2017, 09:00:58 AM Should be Visions of Chaos! Lovely images :drink:
Title: Re: More fractal cellular automata Post by: Tglad on May 24, 2017, 01:18:01 PM The 2D case is interesting as the diamond and square can use the same rule, giving a pseudo octagonal symmetry. There are really only 6 configurations of the 4 parents (0 set on, 1, 2 adjacent, 2 opposite, 3 or 4), so, for the single colour case you can show all of them in an 8x8 image:
(https://2.bp.blogspot.com/-XKRD5YgBxm8/WSVidYifc9I/AAAAAAAABRU/pIUWZzNPSR88LNqmsQ3h0OiMUmTvH_ChQCLcB/s640/automata.png) This starts with a 4x4 array designed to have all configurations of 4 parents. For a two-colour case there'd be 3^6 which is 729 automata. Title: Re: More fractal cellular automata Post by: Softology on May 25, 2017, 02:39:34 AM Yes, those images are created with Visions of Chaos ("Voxel Terrain Automata" will be a new mode included with the next release, hopefully uploaded over the next few days). I am using the Mitsuba Renderer which gives the nice ambient occlusion. I also tweak the brightness and contrast a bit to try and bring out more details.
I looped through a bunch of these overnight and kept a series of the more interesting results. So far that is 5,270 rules I checked out of the 4,294,967,295 possible rules. Right-click View to see the original 4K resolution images. (https://c1.staticflickr.com/5/4247/34711273072_9218e1a56f_o.png) (https://c1.staticflickr.com/5/4197/34835006746_238334251b_o.png) (https://c1.staticflickr.com/5/4274/34031361434_464b316894_o.png) (https://c1.staticflickr.com/5/4201/34874713035_b4a16b45e8_o.png) (https://c1.staticflickr.com/5/4204/34031358634_9c15cace7d_o.png) If you want symmetric structures you can fill the initial cube side with all state 1 or state 2 cells rather than random cells as the processing source code shows. (https://c1.staticflickr.com/5/4226/34031360184_4d50b2f18f_o.png) More samples in my flickr gallery here https://www.flickr.com/photos/39445835@N05/albums/72157681216191183. Thanks for pointing these out Tom. I would not have seen them otherwise. Jason. Title: Re: More fractal cellular automata Post by: Tglad on May 25, 2017, 01:07:36 PM I like the green one again.
I was wrong about the number of 3-colour 2D fractal automata, there should be 3^24 or 282,429,536,481 of them (for square symmetry). One way to slightly reduce the search space is what I will call 'positivity', meaning that if there are 4 parents white, then the child will be white, and visa versa. (this allows empty areas to remain). A much more extreme way is what I call 'bit symmetry', meaning that the same rules apply if you swap black with white. In the 3 colour case, the equivalent is that you can swap the three colours and the same rules apply. An interesting fact is that this 'bit symmetry' only works here with 3 colours. Not with fewer or with more. And for 3 colours the number of possible fractal automata drops to just three: (https://2.bp.blogspot.com/-TnSUMmuSgOc/WSa35dYR5KI/AAAAAAAABR8/hG6VegcSpDQMrGhdkon_xuBecgpqPCVhACLcB/s1600/automata3colour.png) Of these I quite like the one on the right, here's a close up of the complex bit: (https://3.bp.blogspot.com/-xp7_2VBx6dg/WSa3_tOCMEI/AAAAAAAABSA/k8dR-AArQ88sWBVl4CukUC6u8gcftz66ACLcB/s1600/automata3colourbig2.png) Title: Re: More fractal cellular automata Post by: Softology on May 27, 2017, 05:01:00 AM Voxel Automata Terrain are now included in the latest Visions of Chaos I just uploaded. http://softology.com.au/voc.htm
I ran another automated search last night over thousands of rules. Using a "base/floor" of all state 1 cells on the initial cube leads to these symmetric results. (https://c1.staticflickr.com/5/4196/34915981535_9f6b545343_o.png) (https://c1.staticflickr.com/5/4202/34072544044_0f4740f19c_o.png) (https://c1.staticflickr.com/5/4203/34104994413_60009452a2_o.png) More samples here https://www.flickr.com/photos/39445835@N05/albums/72157681216191183 Jason. Title: Re: More fractal cellular automata Post by: DarkBeam on May 27, 2017, 10:11:02 AM Those all show fractal detail! It would be amazing to see them at higher voxel resolution :o
Title: Re: More fractal cellular automata Post by: Sabine on May 27, 2017, 05:44:18 PM Luca!!! :love: I already thought: Hey, mixpinski-esque up there, gosh must be me, the fractal noob :whistle2: :}
Title: Re: More fractal cellular automata Post by: Softology on May 28, 2017, 06:44:06 AM I have been pushing the detail in the latest batch. These use 8 levels of division/iteration.
(https://c1.staticflickr.com/5/4225/34094426554_998e68a528_o.png) (https://c1.staticflickr.com/5/4276/34094353674_02cef3fcec_o.png) (https://c1.staticflickr.com/5/4220/34549947230_fe31a800f2_o.png) Many do have sierpinski like patterns in them. (https://c1.staticflickr.com/5/4273/34773716862_3d3ea6af5a_o.png) Jason. Title: Re: More fractal cellular automata Post by: Sabine on May 28, 2017, 10:54:03 AM Marvellous, thanks for sharing! :)
Title: Re: More fractal cellular automata Post by: DarkBeam on May 28, 2017, 03:55:19 PM :thumbsup1: woow thanks
Title: Re: More fractal cellular automata Post by: Tglad on June 01, 2017, 04:28:43 AM Hi all, I've had a deeper look at the 2D case, written up here: http://tglad.blogspot.com.au/2017/05/diamond-square-fractals.html (http://tglad.blogspot.com.au/2017/05/diamond-square-fractals.html)
There is also a bit-symmetric automaton possible with 4 colours using a triangle lattice (so a triandle-triangle algorithm as opposed to diamond-square or cube-squashedOctahedron-octahedron in 3D): (https://4.bp.blogspot.com/-RO06KS5PROk/WS6muM-6ubI/AAAAAAAABSc/XbXSr9cJOTE8u4AWiC8cR9ZrwJm4ukHVgCLcB/s1600/automataTriangle.bmp) (https://1.bp.blogspot.com/-uq0PW_DXofQ/WS6nGZKYC2I/AAAAAAAABSk/hR-dA4Ok_XM2zpGPRJ1RS135WFM8f14qwCLcB/s1600/automataTrianglebig.bmp) Title: Re: More fractal cellular automata Post by: Softology on June 04, 2017, 04:50:32 AM Here are some level 9 recursion examples. Rendered this time using Pixar's RenderMan renderer. Right-Click View to see 4K resolution.
(https://c1.staticflickr.com/5/4220/34694934520_a9c72c5276_o.png) This next one is over 33 million tiny cubes and took just over 2 hours (!!) for RenderMan to render. (https://c1.staticflickr.com/5/4210/34917932512_096f1606f6_o.png) The same one, but at 8 levels of recursion (https://c1.staticflickr.com/5/4273/34837741762_3726f361c8_o.png) Jason. Title: Re: More fractal cellular automata Post by: Softology on June 04, 2017, 11:19:49 PM And finally, here is a movie showing a few of these being rotated.
https://youtu.be/iT8aZz3GPMI Jason. Title: Re: More fractal cellular automata Post by: Sabine on June 04, 2017, 11:33:07 PM :thumbsup1:
Title: Re: More fractal cellular automata Post by: Alef on August 31, 2017, 11:51:05 AM There are research about natural cellular automata: https://www.nature.com/nature/journal/v544/n7649/abs/nature22031.html (https://www.nature.com/nature/journal/v544/n7649/abs/nature22031.html) A living mesoscopic cellular automaton made of skin scales Quote In vertebrates, skin colour patterns emerge from nonlinear dynamical microscopic systems of cell interactions. Here we show that in ocellated lizards a quasi-hexagonal lattice of skin scales, rather than individual chromatophore cells, establishes a green and black labyrinthine pattern of skin colour. We analysed time series of lizard scale colour dynamics over four years of their development and demonstrate that this pattern is produced by a cellular automaton (a grid of elements whose states are iterated according to a set of rules based on the states of neighbouring elements) that dynamically computes the colour states of individual mesoscopic skin scales to produce the corresponding macroscopic colour pattern. Using numerical simulations and mathematical derivation, we identify how a discrete von Neumann cellular automaton emerges from a continuous Turing reaction–diffusion system. Skin thickness variation generated by three-dimensional morphogenesis of skin scales causes the underlying reaction–diffusion dynamics to separate into microscopic and mesoscopic spatial scales, the latter generating a cellular automaton. Our study indicates that cellular automata are not merely abstract computational systems, but can directly correspond to processes generated by biological evolution. (http://elementy.ru/images/news/cellular_automaton_made_of_skin_scales_1_1037.jpg) (http://elementy.ru/images/news/cellular_automaton_made_of_skin_scales_6_703.jpg) There were more popular article about this but in russian. In generaly what seems as cellular automata is generated by reaction difussion systems operating in larger "cells". Were skin scale is like cell. In the article code is included. They did both, research and then computer simulation of process. |