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Notice, that Nervous System sells some of McCabe’s works as jigsaw puzzles. Softology’s blog entry and W:Blut’s post dissect McCabe’s approach (there is even a reference implementation in Processing). McCabe’s images are created using a more complex multi-scale model. For instance his Bone Music series: or his Turing Flow series: Several artist have used Reaction Diffusion systems in different ways, but the most impressive examples of 2D images I have seen, are the works of Jonathan McCabe. Reaction-Diffusion systems used by artists
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Notice, that this fragment requires a recent source build from the GitHub repository to run. You can even use a picture to modify the concentrations:Ī template implementation can be found as part of the Fragmentarium source at GitHub: ag. It is also possible to enforce some structure by changing the concentrations in certain regions: Here is an example of a typical system created using the above system, though many other patterns are possible: (Robert Munafo has a great page with more information on Gray-Scott systems). Vec2 dV = vec2( Diffusion.x * lv.x - xyy + f*(1.-v.x), Diffusion.y * lv.y + xyy - (f+k)*v.y) Vec4 v = texture2D(backbuffer, position) (see the Fragmentarium source for a nine-point stencil).Ī simple two-component Gray-Scott system may then be modelled simply as: + texture2D( backbuffer, position + P.zy ) + texture2D( backbuffer, position + P.xz ) 4.0 * texture2D( backbuffer, position ) + texture2D( backbuffer, position - P.xz) + texture2D( backbuffer, position - P.zy) The Laplacian may be calculated using a finite differencing scheme, for instance using a five-point stencil: We also use the Fragmentarium host define ‘#buffer RGBA32F’ to ask for four-component 32-bit float buffers, instead of the normal 8-bit integer buffers. On a GPU, we need two buffers to do this: we render the next time step into the front buffer using values from the back buffer, and then swap the buffers.īuffer swapping is a standard technique, and in Fragmentarium the only thing you need to do, is to declare a ‘uniform sampler2D backbuffer ’ and Fragmentarium will take care of creation and swapping of buffers. The time derivative can solved in discrete time steps using forward Euler integration (or something more powerful). To model these, we can represent the concentrations on a discrete grid, which fits nicely on a 2D texture on a GPU. There will be a similar equation for the B field. The \(K\) coefficient determines how quickly the concentration spreads out, and \(P(A,B)\) is a polynomial in the different species concentrations in the system. Where A and B are fields describing the concentration of a chemical species at each point in space. Modelling Reaction-Diffusion on a GPUĪs the name suggests, these systems have two driving components: diffusion, which tends to spread out or smoothen concentrations, and reactions, which describe how chemical species may transform into each other.įor each chemical species, it is possible to describe the evolution using a differential equation on the form: The reaction-diffusion model is a great example of how complex large-scale structure may emerge from simple, local rules.
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Reaction-diffusion systems are interesting, because they display a wide range of self-organizing patterns, and they have been used by several digital artists, both for 2D pattern generation and 3D structure generation.
MANDELBULB 3D ALL FORMULAS ARE GONE SKIN
Here, he suggested, that the pattern formation in animal skin could be explained by a two component reaction-diffusion system.
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An interesting early application was Alan Turing’s theory of Morphogenesis (Turing’s 1951 paper). Reaction-diffusion systems model the spatial dynamics of chemicals.