Clouds from equations

• flows
• maps
• searched
• surfaces
• curves
• fractals
• grown
• patterns
• alive
• the rest
• sources

flows

Differential equations, integrated from one point on the attractor and recorded in time order. Consecutive points are consecutive moments, which is what makes the cloud crawl along itself rather than shimmer.

maps

Iterated rather than integrated: a rule applied over and over to a point in a plane, lifted into depth by delay embedding — Takens' theorem says a delayed coordinate unfolds the dynamics rather than merely decorating them.

searched

Nobody chose these. Thirty coefficients are drawn at random and the result is kept only if it is chaotic, which about one draw in a few hundred is. The seed is the whole parameter, so the same seed is the same attractor on every machine.

surfaces

Swept in scan order over two parameters, so the cloud reads as a surface rather than a fog.

curves

Traced along a curve and thickened into a tube, in order, so the crawl runs along the curve.

fractals

Self-similar at every scale, found by the chaos game, by escape time, or by marching rays inward until they stop escaping.

grown

Built step by step by a rule, rather than evaluated: rewriting an alphabet, or letting particles wander in and stick.

patterns

Sampled where a pattern is, rather than swept: a point is kept if the field says so.

22 that are alive

These are not made once. Each runs on a thread of its own at the frame rate and streams into the live slot, driven by whatever the audio input is doing: the four bands, the loudness, and where the bar is. That is the whole interface they get — deliberately, because the point is a cloud that happens to be alive rather than a second engine with its own controls.

The plates below are each one caught a couple of seconds in, with no audio playing. The ones that are about growing were given longer.

curves

Drawn as one path, in order, so the crawl runs along it and the lines stroke draws it as wire.

grown

Built step by step by a rule, rather than evaluated: rewriting an alphabet, or letting particles wander in and stick.

fields

A grid of numbers stepped forward sixty times a second. The cloud is what rides it.

many bodies

Many things, each with a position of its own, and rules about how they see each other.

The rest of the picture

• shapes
• layers
• colour
• light
• randomness
• audio
• frame

shapes

A filled slot is not the only way vizz gets a form on screen. These are what /shape/mode sweeps through, and most of them never touch memory at all: the particle's index is hashed into four numbers and the shape is a function of those four, evaluated in the vertex stage. That is why the sweep can morph one into the next — a particle keeps its hashes, so the same point travels between forms rather than the field being scattered again.

layers

The print-look counterpart to the particle field, and the other half of what vizz puts on a screen.

colour

What comes out of the shader, and what stops it turning to white.

light

A cloud of dots has no surface, which makes lighting one a question rather than a setting.

randomness

Where every particle that is not read out of a slot comes from.

audio

The four bands, the loudness and the bar — the whole of what the simulations above are driven by, and most of what the rest of the app listens to.

frame

Everything that happens to the picture after the clouds are in it.

Sources

Every entry above, in one list, with whoever found it. Where the original paper is behind a paywall the link goes to an encyclopaedia article instead — the citation is the thing to keep.

The clouds

1. aggregate — Thomas Witten & Leonard Sander, “Diffusion-limited aggregation, a kinetic critical phenomenon”, Phys. Rev. Lett. 47 (1981)
2. Aizawa — Yoji Aizawa & Tatsuo Uezu (1982), as catalogued by Chaoscope
3. Arneodo — Alain Arnéodo, Pierre Coullet & Charles Tresser (1981)
4. Boy's surface — Werner Boy (1901); this parametrisation is François Apéry’s (1986)
5. Buddhabrot — Melinda Green, the Buddhabrot (1993); Lori Gardi’s Nebulabrot
6. Burke–Shaw — Bill Burke & Robert Shaw (1981)
7. Chen — Guanrong Chen & Tetsushi Ueta, “Yet another chaotic attractor”, Int. J. Bifurcation and Chaos 9 (1999)
8. Chladni — Ernst Chladni, Entdeckungen über die Theorie des Klanges (1787)
9. Chua — Leon Chua (1983); Matsumoto, Chua & Komuro, IEEE Trans. Circuits and Systems 32 (1985)
10. Clifford — Clifford Pickover, Computers, Pattern, Chaos and Beauty (1990)
11. cloth — Xavier Provot, “Deformation constraints in a mass-spring model”, Graphics Interface 1995; Thomas Jakobsen, “Advanced Character Physics”, GDC 2001
12. colonised tree — Adam Runions, Brendan Lane & Przemysław Prusinkiewicz, “Modeling Trees with a Space Colonization Algorithm” (2007)
13. coral — The same rewriting, branching three ways a third of a turn apart
14. Coullet — Pierre Coullet, Charles Tresser & Alain Arnéodo (1979) — a jerk system
15. crystal — Clifford A. Reiter, “A local cellular model for snow crystal growth”, Chaos, Solitons & Fractals 23 (2005); the morphologies are Ukichiro Nakaya’s (1954)
16. cyclic — Robert Fisch, Janko Gravner & David Griffeath, “Cyclic cellular automata in two dimensions” (1991); David Griffeath on the spirals’ four snapshots (1994)
17. Dadras — Sara Dadras & Hamid Reza Momeni, Physics Letters A 373 (2009)
18. de Jong — Peter de Jong, in A. K. Dewdney’s Computer Recreations, Scientific American (1987)
19. Dini — Ulisse Dini (1870); a pseudosphere dragged along a helix
20. Duffing — Georg Duffing (1918); the chaotic forced case is Ueda’s (1979)
21. Enneper — Alfred Enneper (1864)
22. fern — The same rewriting, in the fern’s own rule
23. figure-eight knot — The 4₁ knot, the simplest amphichiral one
24. finance — Junhai Ma & Yushu Chen, Applied Mathematics and Mechanics 22 (2001)
25. flame — Scott Draves & Erik Reckase, “The Fractal Flame Algorithm” (2003)
26. flock — Craig Reynolds, “Flocks, Herds and Schools: A Distributed Behavioral Model”, SIGGRAPH 1987
27. fluid — Jos Stam, “Stable Fluids”, SIGGRAPH 1999, and “Real-Time Fluid Dynamics for Games”, GDC 2003; vorticity confinement from Fedkiw, Stam & Jensen, SIGGRAPH 2001
28. foam — Georgy Voronoy (1908); the walls are the set of points equidistant from the two nearest centres
29. Four-wing — Wang, Sun & van Wyk, the four-wing system (2009)
30. Genesio–Tesi — Roberto Genesio & Alberto Tesi, Automatica 28 (1992)
31. growth — Anders Hoff, “Differential Line” (inconvergent, 2016); Andy Lomas, “Cellular Forms” (2014); Nervous System’s Floraform (2014)
32. Gumowski–Mira — Igor Gumowski & Christian Mira, CERN (1980)
33. gyroid — Alan Schoen, Infinite Periodic Minimal Surfaces Without Self-Intersections, NASA TN D-5541 (1970); Schwarz’ P and D are from 1865
34. Hadley — Edward Lorenz, “Irregularity: a fundamental property of the atmosphere”, Tellus 36A (1984)
35. Halvorsen — Arne Dehli Halvorsen’s cyclically symmetric system, as catalogued by Sprott
36. harmonic — A sphere modulated by a spherical harmonic; the classical functions of Laplace (1782)
37. Hilbert curve — David Hilbert, “Über die stetige Abbildung einer Linie auf ein Flächenstück”, Math. Annalen 38 (1891)
38. Hopf — Heinz Hopf, “Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche”, Math. Annalen 104 (1931)
39. hyperbolic — H. S. M. Coxeter, “Crystal symmetry and its generalizations” (1957), after which Escher drew Circle Limit I; Douglas Dunham, “Hyperbolic symmetry” (1986)
40. Hénon — Michel Hénon, “A two-dimensional mapping with a strange attractor”, Comm. Math. Phys. 50 (1976)
41. Ikeda — Kensuke Ikeda, Optics Communications 30 (1979)
42. Julia — Gaston Julia (1918) and Pierre Fatou
43. Klein bottle — Felix Klein (1882); the figure-eight immersion is the standard one
44. Kleinian — David Mumford, Caroline Series & David Wright, “Indra’s Pearls: The Vision of Felix Klein” (2002), Grandma’s recipe
45. kuramoto — Yoshiki Kuramoto (1975); Steven Strogatz, “From Kuramoto to Crawford”, Physica D 143 (2000)
46. life — Carter Bays, “Candidates for the Game of Life in Three Dimensions”, Complex Systems 1 (1987); the multi-state rules as catalogued by Softology
47. liquid — Miles Macklin & Matthias Müller, “Position Based Fluids”, SIGGRAPH 2013; kernels from Müller, Charypar & Gross, SCA 2003
48. Lissajous — Jules Antoine Lissajous (1857); the knotted case is Bogle, Hearst, Jones & Stoilov (1994)
49. Lorenz-96 — Edward Lorenz, “Predictability: a problem partly solved”, ECMWF Seminar on Predictability (1996)
50. Lyapunov — Mario Markus & Benno Hess, “Lyapunov exponents of the logistic map with periodic forcing”, Computers & Graphics 13 (1989)
51. Mandelbox — Tom Lowe (2010)
52. Mandelbrot — Benoit Mandelbrot (1980); the smooth escape count is Douady and Hubbard’s
53. Mandelbulb — Daniel White and Paul Nylander (2009)
54. Menger — Karl Menger (1926)
55. Newton — Arthur Cayley asked the question in 1879 and could not answer it for the cubic
56. Newton–Leipnik — Tim Newton & Roy Leipnik, “Double strange attractors in rigid body motion”, SIAM J. Appl. Math. 41 (1981)
57. Nosé–Hoover — Shūichi Nosé (1984) and William Hoover (1985)
58. orbital — The real spherical harmonics, as a chemistry textbook draws an atomic orbital
59. orbits — Direct-summation gravity, with Plummer softening (1911)
60. pendulum — The double pendulum, the standard demonstration that four numbers can be unpredictable
61. phyllotaxis — The golden angle; Helmut Vogel’s sunflower model (1979)
62. plant — Aristid Lindenmayer (1968); Przemysław Prusinkiewicz & Lindenmayer, The Algorithmic Beauty of Plants (1990)
63. polytope — H. S. M. Coxeter, “Regular Polytopes” (1948); Thomas Banchoff’s turning hypercube films (1978); the walk is Hierholzer’s Eulerian circuit (1873)
64. quadratic — Julien Clinton Sprott, Strange Attractors: Creating Patterns in Chaos (1993)
65. quadratic flow — The same search over quadratic flows; Sprott, “Some simple chaotic flows”, Phys. Rev. E 50 (1994)
66. quasicrystal — Dan Shechtman et al., Phys. Rev. Lett. 53 (1984); Nobel Prize in Chemistry, 2011
67. quaternion Julia — Alan Norton, “Generation and display of geometric fractals in 3-D”, SIGGRAPH 1982
68. Rabinovich–Fabrikant — Mikhail Rabinovich & Anatoly Fabrikant, Sov. Phys. JETP 50 (1979)
69. reaction — The Gray–Scott system in John Pearson’s parameterisation, Science 261 (1993); the mitosis constants are Karl Sims’
70. Rikitake — Tsuneji Rikitake, “Oscillations of a system of disk dynamos”, Math. Proc. Cambridge Phil. Soc. 54 (1958)
71. Rucklidge — Alastair Rucklidge, “Chaos in models of double convection”, J. Fluid Mech. 237 (1992)
72. Rössler — Otto Rössler, “An equation for continuous chaos”, Physics Letters A 57 (1976)
73. Sakarya — The Sakarya system (2010)
74. sandpile — Per Bak, Chao Tang & Kurt Wiesenfeld, “Self-organized criticality”, Phys. Rev. Lett. 59 (1987); Deepak Dhar on the Abelian property (1990)
75. scope — The XY mode of the cathode-ray oscilloscope; Jerobeam Fenderson, “Oscilloscope Music” (2016)
76. seashell — David M. Raup, “Geometric analysis of shell coiling” (1966); Deborah Fowler, Hans Meinhardt & Przemysław Prusinkiewicz, “Modeling seashells” (SIGGRAPH 1992)
77. Shimizu–Morioka — Tsutomu Shimizu & Nobuo Morioka, Physics Letters A 76 (1980)
78. Sierpinski — Wacław Sierpiński (1915), by Michael Barnsley’s chaos game
79. slime — Jeff Jones, “Characteristics of pattern formation and evolution in approximations of Physarum transport networks”, Artificial Life 16 (2010); Tero et al., Science 327 (2010)
80. smoke — The same solver in three dimensions; buoyancy after Fedkiw, Stam & Jensen, “Visual Simulation of Smoke”, SIGGRAPH 2001
81. spirals — Boris Belousov (1951) and Anatol Zhabotinsky (1964); the cellular form is Alasdair Turner’s
82. spirograph — The hypotrochoid, and Denys Fisher’s 1965 toy
83. Sprott B — Julien Clinton Sprott, “Some simple chaotic flows”, Phys. Rev. E 50 (1994) — case B
84. standard map — Boris Chirikov, “A universal instability of many-dimensional oscillator systems”, Physics Reports 52 (1979); the last-torus threshold is John Greene’s (1979)
85. supershape — Johan Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes”, Am. J. Botany 90 (2003)
86. swarmalators — Kevin O’Keeffe, Hyunsuk Hong & Steven Strogatz, “Oscillators that sync and swarm”, Nature Communications 8 (2017)
87. tangle — Thomas Jakobsen, “Advanced Character Physics”, GDC (2001); Miklós Bergou et al., “Discrete Elastic Rods”, SIGGRAPH (2008)
88. Thomas — René Thomas, “Deterministic chaos seen in terms of feedback circuits”, Int. J. Bifurcation and Chaos 9 (1999)
89. three-scroll — The three-scroll unified chaotic system; Dequan Li, Physics Letters A 372 (2008)
90. torus knot — The classical (p,q) torus knots
91. tree — The same rewriting, with a trunk that keeps going
92. twisted gasket — The Sierpiński chaos game with a rotation in the loop — the kaleidoscopic IFS of the fractal-rendering community
93. vortex — Hermann von Helmholtz on vortex motion (1858); Louis Rosenhead’s desingularised kernel (1930); Anthony Leonard, “Vortex methods for flow simulation” (1980)
94. wind — Robert Bridson, Jim Hourihan & Marcus Nordenstam, “Curl-Noise for Procedural Fluid Flow”, SIGGRAPH 2007

Everything else

• Solid sphere and hollow shell — the uniform-height trick is Archimedes' hat-box theorem — equal slices of a sphere carry equal area — in the form usually written up as sphere point picking.
• Trefoil knot — classical knot theory; the parametrisation is the standard one, as given on MathWorld.
• Lorenz and Aizawa — Edward N. Lorenz, “Deterministic Nonperiodic Flow”, Journal of the Atmospheric Sciences 20(2), 1963. Aizawa is named for Yoji Aizawa, whose work on chaotic flows it comes out of; unlike Lorenz it has no single canonical paper behind it, and the parameter set vizz uses is the one that circulates in graphics, stated the same way in Analog Paradigm's write-up.
• Torus, grid plane, cloud pair — plain parametric forms with nothing behind them worth citing.
• Wind — the Arnold–Beltrami–Childress flow with A = B = C = 1: a steady solution of the Euler equations, divergence-free by construction, and chaotic in its streamlines — the standard example of a simple field that mixes.
• The stack — 555-5555's visuals for Caribou's live shows, which is the look this was built to reach. Not an algorithm and not a paper — an idiom, borrowed knowingly. The generators themselves are ordinary procedural patterns.
• Moiré — Isaac Amidror, The Theory of the Moiré Phenomenon, Volume I: Periodic Layers, Springer.
• Blend modes — the separable blend modes of W3C Compositing and Blending Level 1. vizz departs from the spec in one deliberate way: it blends on sRGB-encoded values rather than linear ones, because compositing on encoded values is the print-era behaviour this aesthetic grew out of. The single conversion to linear happens on the way out of the shader.
• Analytic antialiasing — the technique Iñigo Quilez describes in “filtering procedural textures”. The per-generator gradient magnitudes are worked out for these generators specifically.
• Cosine palettes — Iñigo Quilez, “palettes”.
• HSV — Alvy Ray Smith, “Color Gamut Transform Pairs”, SIGGRAPH 1978 — the paper HSV comes from.
• The tone shoulder — the c/(1+c) curve from Reinhard, Stark, Shirley and Ferwerda, “Photographic Tone Reproduction for Digital Images”, SIGGRAPH 2002. The 0.15 is tuned for this picture, not from the paper.
• sRGB — IEC 61966-2-1.
• Estimated normals — Hoppe, DeRose, Duchamp, McDonald and Stuetzle, “Surface Reconstruction from Unorganized Points”, SIGGRAPH 1992, where the local plane fit is step one. vizz stops there: the paper goes on to propagate a consistent orientation across the cloud, and vizz instead flips each normal towards the eye, which is a better answer for a scan of a room seen from inside it.
• Lamps and the sun — Lambert's cosine law (Johann Heinrich Lambert, Photometria, 1760) for the directional term. The distance falloff is a smooth r²/(d²+r²) rather than inverse-square, so there is no singularity to divide by on stage.
• PLY — the Stanford polygon format, Greg Turk, Stanford University.
• Depth of field — vizz, on the ordinary optics of the circle of confusion.
• The particle hash — the lowbias32 constants from Chris Wellons' hash prospector, described in “Prospecting for Hash Functions”.
• Window and spectrum — the transform itself is the rustfft crate; the window and its correction factor are from Fredric J. Harris, “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform”, Proceedings of the IEEE 66(1), 1978, and Parseval's theorem for the normalisation.
• Onset detection — the half-wave-rectified spectral flux detector, surveyed and compared in Simon Dixon, “Onset Detection Revisited”, DAFx-06.
• Tempo — autocorrelation of an onset-strength envelope, as set out in Daniel P. W. Ellis, “Beat Tracking by Dynamic Programming”, Journal of New Music Research 36(1), 2007. vizz uses the tempo-estimation half and not the dynamic-programming beat tracker — a VJ needs the rate, and the phase comes from the pads.
• Feedback trails — analogue video feedback, which is where this whole idiom comes from — Steina and Woody Vasulka's work and the Sandin Image Processor (Dan Sandin, 1971–73). The implementation is an ordinary ping-pong of two Rgba16Float textures.
• Kaleidoscope and lens — vizz. Standard screen-space constructions with no particular paper behind them.
• Drawing at twice the size — the line coverage is the box-filter case of Satish Gupta and Robert Sproull, “Filtering Edges for Gray-Scale Displays”, SIGGRAPH 1981. The sprite floor is the screen-space filter of Zehao Yu, Anpei Chen, Binbin Huang, Torsten Sattler and Andreas Geiger, “Mip-Splatting: Alias-free 3D Gaussian Splatting”, CVPR 2024, applied to a disc instead of a Gaussian.
• The graded finish — the curve is Troy Sobotka's AgX, in the polynomial fit from Benjamin Wrensch's “Minimal AgX Implementation” (2023). The bloom filters are from Jorge Jimenez, “Next Generation Post Processing in Call of Duty: Advanced Warfare”, SIGGRAPH 2014: a thirteen-tap downsample, Karis-averaged on the first step, and a tent on the way up. Metering from a percentile of a log-luminance histogram is how game engines' auto-exposure works, Unreal's among them.
• Gaussian splats — Bernhard Kerbl, Georgios Kopanas, Thomas Leimkühler and George Drettakis, “3D Gaussian Splatting for Real-Time Radiance Field Rendering”, SIGGRAPH 2023, for the format and the method; the projection is the EWA splatting of Matthias Zwicker, Hanspeter Pfister, Jeroen van Baar and Markus Gross, “EWA Volume Splatting”, IEEE Visualization 2001; the sort is Kenneth Batcher's bitonic network, “Sorting networks and their applications”, AFIPS 1968.
• Haze and light beams — Bartłomiej Wroński, “Volumetric Fog and Lighting”, SIGGRAPH 2014 Advances in Real-Time Rendering, for the froxel volume; Sébastien Hillaire, “Physically Based and Unified Volumetric Rendering in Frostbite”, SIGGRAPH 2015, for the integration; single scattering as Nelson Max sets it out in “Optical Models for Direct Volume Rendering”, IEEE TVCG 1995; the phase function of Louis Henyey and Jesse Greenstein, “Diffuse radiation in the galaxy”, Astrophysical Journal 1941.
• Solid surfaces — the discs are surfels, after Hanspeter Pfister, Matthias Zwicker, Jeroen van Baar and Markus Gross, “Surfels: Surface Elements as Rendering Primitives”, SIGGRAPH 2000. The shadow is a shadow map, Lance Williams, “Casting Curved Shadows on Curved Surfaces”, SIGGRAPH 1978, read with the percentage-closer filtering of William Reeves, David Salesin and Robert Cook, “Rendering Antialiased Shadows with Depth Maps”, SIGGRAPH 1987. Normals from the depth buffer are standard deferred-rendering practice; the span that follows the surfel size is vizz's.
• Lines and streaks — lines after Malte Zöckler, Detlev Stalling and Hans-Christian Hege, “Interactive Visualization of 3D-Vector Fields Using Illuminated Stream Lines”, IEEE Visualization 1996. Streaks after William Reeves, “Particle Systems — a Technique for Modeling a Class of Fuzzy Objects”, SIGGRAPH 1983, which drew each particle as a motion-blurred streak.
• Plexus — the motion-graphics “plexus” look, after Rowbyte's Plexus plug-in for After Effects (2010), whose name it has become; the geometry is a k-nearest-neighbour graph cut at a radius.
• Glyphs — Rita Borgo, Johannes Kehrer, David H. S. Chung, Eamonn Maguire, Robert S. Laramee, Helwig Hauser, Matthew Ward and Min Chen, “Glyph-based Visualization: Foundations, Design Guidelines, Techniques and Applications”, Eurographics State of the Art Reports, 2013.
• Fractal solids — sphere tracing is John C. Hart, “Sphere tracing: a geometric method for the antialiased ray tracing of implicit surfaces”, The Visual Computer 12, 1996; the quaternion Julia estimate is Hart, Sandin and Kauffman, “Ray tracing deterministic 3-D fractals”, SIGGRAPH 1989. The Mandelbulb is Daniel White and Paul Nylander's, 2009; the Mandelbox is Tom Lowe's, 2010; the sponge is Karl Menger's, 1926, with the distance function and the tetrahedral normal from Íñigo Quílez.
• Ink — edges from the depth buffer are Takafumi Saito and Tokiichiro Takahashi, “Comprehensible Rendering of 3-D Shapes”, SIGGRAPH 1990; hatching by tone after Emil Praun, Hugues Hoppe, Matthew Webb and Adam Finkelstein, “Real-Time Hatching”, SIGGRAPH 2001, reduced to three screen-space layers; stipple after Adrian Secord, “Weighted Voronoi Stippling”, NPAR 2002, on a jittered grid without the relaxation.
• Liquid — screen-space fluid rendering, after Wladimir J. van der Laan, Simon Green and Miguel Sainz, “Screen Space Fluid Rendering with Curvature Flow”, I3D 2009, with a bilateral blur in place of the curvature flow; the Fresnel term is Christophe Schlick, “An Inexpensive BRDF Model for Physically-based Rendering”, Computer Graphics Forum 13(3), 1994.
• Gravity wells — vizz. A displacement field that reads as gravity, not a gravity model.
• Camera moves — vizz. The shapes are borrowed from camera work rather than from anything written down.