References
[1] Rule 90 (Lyria Shell)
Rule 90 elementary cellular automaton (Wolfram) on the surface of a Lyria shell constructed from equations in Cortie (1989).
[2] Game of Life (Natalina Cafra)
Conway’s Game of Life on the surface of a Natalina Cafra shell constructed from equations in Cortie (1989).
[3] Brian’s Brain (Klein Bottle)
Brian’s Brain cellular automaton (Silverman, 1996) on the surface of a Klein bottle.
[4] Langton’s Ant in Three Dimensions
Based on the definition of Heiko Hamann (2003).
[5] Sierpiński Pillow
Sierpiński Carpet on a pillow shape surface (Bourke, 2000).
[6] Graph Drawing by Stress Majorization
Stress majorization graph drawing algorithm (Gansner et al, 2004) applied to a random graph with 10k nodes and 10k edges.
[7] Vicsek Fractal
The Vicsek fractal in three dimensions (Vicsek, 1989).
[8] Koch Curve in Three Dimensions
The Koch curve in three dimensions, as described by Eric Baird (2014).
[9] Rapidly Exploring Random Tree (Trefoil Knot)
Rapidly exploring random tree (LaValle, 1998) on the surface of a Trefoil knot.
[10] ‘Maze’ Cellular Automaton (Apple Surface)
Maze cellular automaton on an apple surface (Bourke, 2000).
[11] ‘Crystal Growth 1’ Cellular Automaton
Rule 0-6/1,3/2/VN (Jason Rampe). Refer to Softology’s blog for details.
[12] Apollonian Sphere Packing
Apollonian sphere packing constructed from notes by Ronald Peikert via Kean Walmsley’s Blog. Animation ideas are from the Observable implementation by Claudio Esperança.
[13] Particle Lenia
Particle Lenia in three dimensions (Mordvintsev et al.), constructed from implementation notes by Alexander Mordvintsev. The original Lenia system is by Bert Chan.
[14] Accretor Cellular Automata
Accretor cellular automata, a system created by Erwin Driessens and Maria Verstappen. Implemented based on the description and animations by Softology.
[15] Koch Quadratic Fractal
Koch quadratic fractal (Type 1) in three dimensions, constructed from the description by Robert Dickau.
[16] Graph Rewriting
Constructing graphs using simple graph rewriting rules, which I learned about from the Wolfram Physics Project. The graph layouts are updated dynamically using the Fruchterman-Reingold force-directed algorithm.
[17] SmoothLife
SmoothLife continuous cellular automata (Stephan Rafler, 2011) on a sphere.
[18] Parametric Mollusc Shells
Parametric mollusc shells constructed from equations in Cortie (1989).
[19] Particle Life in Three Dimensions
Particle Life in three dimensions based on the description and implementation by Tom Mohr. The original system (Clusters) was created by Jeffrey Ventrella.
[20] Stacked Generations of 2D Cellular Automata
Stacked generations of 2D cellular automata, as studied by Stephen Wolfram and based on the description by Softology. Rules were selected from LifeWiki.
[21] Voronoi Fracturing (Tranguloid Trefoil)
Recursive Voronoi fracturing applied to a Tranguloid Trefoil surface (Roger Bagula).
[22] Self-Organizing Trees
Self-organizing trees generated by the space-colonization algorithm described by Palubicki et al (2009).
[23] Romanesco Broccoli Fractal
Romanesco broccoli fractal, constructed procedurally in Houdini using the Archimedean spiral and scaled cones. I learned of this fractal from the video by Junichiro Horikawa. This is an alternative implementation based around projecting points generated by the Spiral SOP onto cones using the Ray SOP.
[24] Pythagoras Tree Fractal
Pythagoras Tree Fractal (Bosman, 1942), implemented from the description in the Wikipedia article.
[25] Julia Set Fractals
Julia Set fractals projected onto 3D surfaces, implemented from the description and parameters in the Wikipedia article.
[26] Collatz Conjecture AnglePath Sequences
Number sequences related to the Collatz Conjecture based on the AnglePath procedure by Reddit user level1807.
[27] Primordial Particle Systems
Primordial Particle Systems (Schmickl et al., 2016) in three dimensions, implemented from the video describing the algorithm and paper extending the system to 3D.
[28] Multiple Neighborhood Cellular Automata
Multiple Neighborhood Cellular Automata, a system created by Slackermanz. The neighborhood patterns and rules used here are based on the original code by Slackermanz as described in the blog post by Softology.
[29] Random Graph Generators
Random graphs constructed using generators described in the igraph documentation (Growing Random Graph, Barabási-Albert, Erdős-Rényi). Graph layouts are updated dynamically using the Fruchterman-Reingold force-directed algorithm.
[30] Random Walk on Boy’s Surface
A random-walk-like algorithm that makes use of the nearpoint VEX function, applied to scattered points on Boy’s Surface.
[31] Pyramid Fractal Surface
Pyramid fractal surface implemented from the description in the paper by Thomas Lowe.
[32] Random Iterated Function Systems
Random iterated function systems (IFS), adapted from the procedure by Paul Bourke (March 1998). Here the equations are modified to include sine and cosine functions of x and y, and sampling is from eight random parameter sets rather than four.
[33] k-Nearest Neighbor Flame Simulations
k-nearest neighbor pyro simulations with POP Advect by Volumes and the nearpoints VEX function in Houdini.
[34] Turbulent Smoke (Möbius Strip)
Turbulent smoke simulation created with the Houdini Pyro Solver, projected onto a Möbius strip.
[35] Mandelbrot Zoom Stacking
3d stacked zoom levels of the Mandelbrot Set. This is similar to the stacking of 2D cellular automata (see [20]) using discrete zoom levels.
[36] K-d Trees
k-d trees (Bentley) constructed from scattered points on the faces of a cube.
[37] Totalistic Cellular Automata (Klein Bottle)
Three-color totalistic cellular automata on the surface of variants of the Klein bottle. The system and rules are from A New Kind of Science pp. 60-70 by Stephen Wolfram.
[38] Space Filling Curves (Borromean Rings)
Hilbert, Gosper and Peano space filling curves on the surface of the Borromean rings. Implemented using L-Systems and equations from articles by Robert Dickau and Paul Bourke.
[39] Diffusion-Limited Aggregation
Diffusion-limited aggregation (Witten & Sander, 1983) in three dimensions implemented with VEX in #houdini based on the neighbor counting technique described by Softology.
[40] Boids (Dupin Cyclide)
Boids (flocking) simulation on the surface of a Dupin cyclide. Implemented from the description on Craig Reynolds’ Boids page.
[41] Reaction-Diffusion
Gray-Scott reaction-diffusion simulations implemented from the descriptions, parameters and animations provided in the article by Karl Sims and the video by Neat AI.
[42] Platonic Solid Fractals
Fractals constructed from Platonic solids (dodecahedron, icosahedron, cube) by recursive scaling and copying onto vertices. Implemented procedurally based on the fractal construction and animation techniques by PurpleCat.
[43] Vector Field Pathlines
Pathlines across vector fields starting from scattered points on parametric surfaces: Saddle Torus (Bagula), Horn (Bourke), Antisymmetric Torus (Rubin). The vector fields were discovered by experimentation.
[44] H Tree Fractal
H tree fractal, implemented from the description in the Wikipedia article.
[45] Rössler Attractor
The Rössler attractor, implemented from equations and parameters in the Wikipedia article.
[46] Continuous Cellular Automata 1
Continuous cellular automata in 3D based on the system by Stephen Wolfram in A New Kind of Science pp. 155-160. System details: Continuous cell states in [0, 1]; updated state = fractional part of (avg neighbor state + a constant); von Neumann neighborhood; cells with states below a threshold are not drawn. After implementing and experimenting with this 3D threshold system I discovered that it has been previously studied by Richard Southwell, who used a threshold of 0.5 to construct an object for 3D printing.
[47] Mobile Automata
Generalized mobile automaton with rule set (f) from Page 77 of NKS by Stephen Wolfram, run for 5000 steps.
[48] 3D Bitwise Functions
3D geometry constructed from bitwise functions applied to x,y,z integer coordinates, based on the x,y systems by Stephen Wolfram in NKS p. 871. Cubes with values below a threshold are not drawn.
[49] Hypergraph Rewriting
Hypergraph rewriting based on systems and rules from the Wolfram Physics Registry of Notable Universe Models. The hypergraph layouts are updated dynamically using the Fruchterman-Reingold force-directed algorithm.
[50] 3D Totalistic Cellular Automata
Two-color 26-neighbor totalistic 3D cellular automata run for 240 steps starting from a single live cell. At each step the set of live cells is scaled to fit inside a 1x1x1 cube. The rule is from NKS p. 183 (top) by Stephen Wolfram.
[51] Hexagonal Prism Cellular Automata
Hexagonal prism totalistic cellular automata with 8 and 20 neighbors run for 240 steps starting from a single live cell, using the space-filling hexagonal grid systems described by Francis A. Bitonti and Softology.
[52] Continuous Cellular Automata 2
See [46].