chaosrack

An analog computer in the browser: a rack of instruments for dynamical systems and audio signals. The core is Go compiled to WebAssembly; the same program runs on the host as a terminal panel, a native rack and a renderer. An homage to the analog computers at glensstuff.com.

Live demo · TinyGo build · Model pages · Manual

• 79 models. Flows (Lorenz, Rössler, Chua, Sprott A–S, 4-D hyper-Rössler, …), maps (Hénon, Ikeda, Clifford, de Jong, Gumowski–Mira, Tinkerbell, Chirikov), parametric figures, an integer-sequence turtle with rigid-body physics, geometry and STL solids, terminals and a desktop drawn into the scene, delay embeddings of live audio, audio analyzers, and a ripple tank.
• A rack control surface. Programmable encoder banks, selector knobs with LED rings, seven-segment and dot-matrix displays, a pin-matrix Mixer and Mod matrix, four oscilloscopes and four signal generators.
• Audio in and out. Microphone, built-in generators (0.1 Hz – 28 kHz), or this machine's audio streamed by the server (--audio). Any parameter can be driven by the signal; Model Out plays the model's own equations.
• Measurement. Largest Lyapunov exponent, bifurcation, recurrence; THD+N, RTA, transfer function, LUFS, wow & flutter, waterfall.
• Custom systems typed as derivatives or maps; permalinks and presets; WebM/GIF recording.

• Contents
• Run
  • The live demo with this machine's audio
• Features
• Models
  • Attractors
  • Maps
  • Solids
  • Geometry
  • Sequences
  • Scope
  • Embeddings
  • Audio
• Controls and measurement
• Remote access
• The rack
  • Visual
  • Visual · Model
  • View
  • Grid
  • Display
  • Layers · Colors
  • Console
  • Record
  • Style
  • Presets
  • Counter
  • Distortion
  • Loudness
  • Wow & Flutter
  • Features
  • Timing
  • Scope 1
  • Gen 1
  • Scope 2
  • Gen 2
  • Scope 3
  • Gen 3
  • Scope 4
  • Gen 4
  • Synth
  • String
  • FM
  • Wave · Noise
  • Filter
  • Amp
  • Keys
  • Matrix
  • Mixer
  • Mod
• Physical dimensions
• Solid models (STL)
• Recording
• Gallery
• Audio
• FVF — Harmonic Wobbulator
• Testing & tooling
• Build
  • Building the wasm into the host binary
  • Layout
• Related / prior art
• Dependency Graph
• Lines of Code

The WebAssembly is embedded in the binary; there is no build step.

go install github.com/0magnet/chaosrack@master
chaosrack serve            # serve the rack at http://127.0.0.1:8080/
chaosrack serve --audio    # also stream this machine's audio to the page

Or without installing: go run github.com/0magnet/chaosrack@master serve --audio.

Open 127.0.0.1:8080/ (Go build) or 127.0.0.1:8080/tinygo/ (TinyGo build).

--audio records the default PulseAudio/PipeWire monitor — whatever this machine is playing — and the page connects to it by itself. Without it the page uses the microphone or the built-in generators.

The page at chaosrack.magnetosphere.net can take its audio and rack link from a server on your own machine, so you don't need to open a page that server serves itself:

1. Run chaosrack serve --audio --pair. It prints a link to the live demo: https://chaosrack.magnetosphere.net/#…&pair=…&wt=127.0.0.1:8080.
2. Open that link once, in the browser you'll use.
3. Allow the browser's prompt to let the site reach "other apps and services on this device" (Local Network Access). Until it's allowed, the page waits and its console says why.

That browser is now paired: plain visits to the demo reconnect by themselves, even after the server restarts. Each run prints a new link, which only a browser that hasn't paired yet needs. The paired browsers are listed in ~/.config/chaosrack/paired-tabs; delete a line to unpair one.

For a browser on another machine, run chaosrack serve --bind 0.0.0.0 --audio and open the link it prints there. WebTransport doesn't go through a proxy, so if that browser uses one, add this machine's address to its bypass list. Details, and the threat model: docs/remote-access.md.

Other commands, which also run in the page's own terminal:

The page also runs with no server at all: see SERVERLESS.md.

• Flows are integrated in the page every frame; every built-in flow's defaults are tested in CI for a positive largest Lyapunov exponent under the app's own integrator.
• Maps have no dt and are drawn as points. Hénon's determinant (−b) and Chirikov's (1) are asserted in tests. The standard map is drawn as an ensemble of orbits; its invariant curves break up as K passes 0.9716.
• Custom equations: type dx/dt, dy/dt, dz/dt (optional 4th dimension); every named parameter gets a knob. iterate reads the same expressions as a map.
• Turtle Path walks an integer sequence mod m (period: the Pisano period) in 2-D or 3-D, classifies the result (closed, drifting, helical), and can be given mass and dropped into a box. Built on 0magnet/pisano.
• Analysis: largest Lyapunov exponent of whatever is on screen, per unit time for flows and per iterate for maps; a bifurcation explorer; a recurrence plot.
• Delay embeddings of live audio: Takens (τ measured by average mutual information, dimension by false nearest neighbors), stereo and polar.
• Trails: scan (the whole curve re-integrated each frame) or Ring (only the head advances); Persist accumulates frames; Points breaks the line into N points.
• Color: the gradient follows x, y, z, trail position or the audio spectrum, through the two-color, three-color, hue or six colormaps shared with the spectrogram (heat, blue, gray, turbo, viridis, magma). CRT phosphor presets with afterglow.
• Generators: four, sine to noise plus test signals (pink noise, log sweep, polarity pulse, …), 0.107 Hz – 28 160 Hz on an octave-a-turn encoder. Below ~20 Hz they are modulators.
• Model Out runs the model's equations at audio rate; its x, y, z are Mixer columns, playable and measurable.
• Mixer: every source (capture, generators, Model Out, keys, drums, tone matrix) is a column of one pin matrix; rows are the speakers, the rack's signal (read by the scopes, meters and models) and two modulation sends.
• Mod matrix: any model or view parameter driven by the rack's signal, a send or the model's own position, with depth, EQ band weighting and a shape (push, swing, flip, or wave — the signal itself).
• Terminals and desk: websh (Bash compiled to wasm) on a quad in the scene; a desk window manager over or inside the scene, with four reproduced 3-D desktops. See docs/desk.md.
• State: permalinks carry the full state; named presets use the same serialization; the rack layout persists.
• Recording: WebM via MediaRecorder or GIF encoded in Go, full canvas or a region.
• Audio transport: float32 samples over WebSocket, or unreliable WebTransport datagrams for lossy links, with automatic fallback.

Also published as one page per model at chaosrack.magnetosphere.net/models/, each with its equations, figure and a link that opens it running. Both this section and those pages are generated from attractor.Catalog(), the selector's own registry (make site, make site-check).

Every position of the MODEL selector, in knob order; a model in two categories is listed in both. Per model: its bank as programmed for it, a still, an animated loop, its description (the Info overlay's text) and its equations. Images are captured by uitool portraits.

Rossler · Lorenz · Chua · Aizawa · Sprott · Thomas · Halvorsen · Chen · Dadras · Rabinovich-Fabrikant · Burke-Shaw · Lü · Newton-Leipnik · Arneodo · Chen-Lee · Dequan Li · Three-Scroll · Wang-Sun · Lorenz Mod 2 · Simone · Rucklidge · Genesio-Tesi · Shimizu-Morioka · Rikitake Dynamo · Moore-Spiegel · Bouali · Lorenz-84 · Finance · Hindmarsh-Rose · Wei (no equilibrium) · Sprott Jerk · Hyper-Rössler (4D) · Sprott Morph · Sprott A · Sprott B · Sprott C · Sprott D · Sprott E · Sprott F · Sprott G · Sprott H · Sprott I · Sprott J · Sprott K · Sprott L · Sprott M · Sprott N · Sprott O · Sprott P · Sprott Q · Sprott R · Sprott S · Custom equation

Rossler	animated

Rössler Attractor — Proposed by Otto Rössler in 1976 as a simpler system that produces chaotic behavior. Unlike the Lorenz system's two-lobed shape, the Rössler attractor has a single folded-band structure with an outward spiral that occasionally makes a large excursion in the z-direction.

dx/dt = −(y + z)
dy/dt = x + ay
dz/dt = b + z(x − c)

#rossler · 3-D flow

Lorenz	animated

Lorenz Attractor — Discovered by Edward Lorenz in 1963 while modeling atmospheric convection. The butterfly-shaped trajectory arises from a simplified system of three coupled differential equations. It was one of the first systems shown to exhibit deterministic chaos, where tiny differences in initial conditions lead to vastly different outcomes.

dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz

#lorenz · 3-D flow

Chua	animated

Chua's Circuit (Double Scroll Attractor) — Invented by Leon Chua in 1983, this is the first electronic circuit proven to exhibit chaos. The system features a piecewise-linear nonlinearity (the Chua diode) that creates the characteristic double-scroll pattern. It is also the basis for multi-scroll attractor generalizations.

dx/dt = α(y − x − h(x))
dy/dt = x − y + z
dz/dt = −βy
h(x) = m₁x + ½(m₀ − m₁)(|x+1| − |x−1|)

#chua · 3-D flow

Aizawa	animated

Aizawa Attractor — A chaotic system that produces a toroidal structure with a tendril extending from the center. The attractor has a visually striking shape that resembles a sphere with a tail, exhibiting both rotational symmetry and chaotic wandering.

dx/dt = (z − b)x − dy
dy/dt = dx + (z − b)y
dz/dt = c + az − z³/3 − (x² + y²)(1 + ez) + fzx³

#aizawa · 3-D flow

Sprott	animated

Sprott Attractor — One of many simple chaotic systems cataloged by Julien Clinton Sprott. These systems were discovered through systematic computer searches for chaotic flows with minimal terms, demonstrating that chaos can arise from remarkably simple equations.

dx/dt = y + Axy + xz
dy/dt = 1 − Bx² + yz
dz/dt = x − x² − y²

#sprott · 3-D flow

Thomas	animated

Thomas' Cyclically Symmetric Attractor — Introduced by René Thomas, this system has the elegant property of cyclic symmetry: each variable is damped and driven by the sine of the next variable in the cycle. The parameter b controls dissipation; as b decreases the system transitions from stable points through limit cycles to chaos.

dx/dt = −bx + sin(y)
dy/dt = −by + sin(z)
dz/dt = −bz + sin(x)

#thomas · 3-D flow

Halvorsen	animated

Halvorsen Attractor — A chaotic system with three-fold rotational symmetry, producing a distinctive pinwheel-like shape. The attractor consists of three intertwined lobes that spiral around each other, creating a visually complex but structurally symmetric trajectory.

dx/dt = −ax − 4y − 4z − y²
dy/dt = −ay − 4z − 4x − z²
dz/dt = −az − 4x − 4y − x²

#halvorsen · 3-D flow

Chen	animated

Chen Attractor — Discovered by Guanrong Chen in 1999, this system was found as a dual of the Lorenz system in a specific mathematical sense. It exhibits chaotic behavior with a distinctive two-scroll structure that differs from both the Lorenz and Rössler attractors.

dx/dt = a(y − x)
dy/dt = (c − a)x − xz + cy
dz/dt = xy − bz

#chen · 3-D flow

Dadras	animated

Dadras Attractor — A three-dimensional autonomous chaotic system with five parameters, introduced by Sara Dadras and Hamid Reza Momeni. The system exhibits rich dynamical behavior including period-doubling routes to chaos.

dx/dt = y − px + qyz
dy/dt = ry − xz + z
dz/dt = sxy − ez

#dadras · 3-D flow

Rabinovich-Fabrikant	animated

Rabinovich-Fabrikant Attractor — Derived by Mikhail Rabinovich and Anatoly Fabrikant from physical equations modeling the stochasticity of three interacting waves. The system is known for its complex topology and extreme sensitivity to parameters, producing intricate folded structures.

dx/dt = y(z − 1 + x²) + γx
dy/dt = x(3z + 1 − x²) + γy
dz/dt = −2z(α + xy)

#rabinovich · 3-D flow

Burke-Shaw	animated

Burke-Shaw Attractor — Introduced by Bill Burke and Robert Shaw, this system exhibits chaotic behavior with a distinctive two-winged structure. It arises from the study of nonlinear dynamics and produces complex trajectories confined to a compact region of phase space.

dx/dt = −S(x + y)
dy/dt = −y − Sxz
dz/dt = Sxy + V

#burkeshaw · 3-D flow

Lü	animated

Lü Attractor — Discovered by Jinhu Lü and Guanrong Chen (2002), the third member of the Lorenz–Chen–Lü family; it forms a bridge between the Lorenz and Chen systems and produces a two-scroll butterfly.

dx/dt = a(y − x)
dy/dt = cy − xz
dz/dt = xy − bz

#lu · 3-D flow

Newton-Leipnik	animated

Newton–Leipnik Attractor — Arises from a rigid-body rotation model with a linear feedback torque; it has two coexisting scroll-shaped attractors.

dx/dt = −ax + y + 10yz
dy/dt = −x − 0.4y + 5xz
dz/dt = bz − 5xy

#newtonleipnik · 3-D flow

Arneodo	animated

Arneodo Attractor — Arneodo, Coullet and Tresser (1981): a third-order jerk equation, x‴ = −ax − bx′ − x″ + dx³, written as three first-order equations. A spiral attractor whose loops fold back on themselves through the cubic term.

dx/dt = y
dy/dt = z
dz/dt = −ax − by − z + dx³

#arneodo · 3-D flow

Chen-Lee	animated

Chen–Lee Attractor — Chen and Lee (2004): the Euler equations of a rigid body in rotation, with linear feedback added to each axis. A two-wing attractor whose wings stand on the z axis.

dx/dt = ax − yz
dy/dt = by + xz
dz/dt = cz + xy/d

#chenlee · 3-D flow

Dequan Li	animated

Dequan Li Attractor — Li (2008): a three-scroll attractor, the scrolls joined through the z axis. Stiff: its default step is small, and larger steps of dt make the trajectory leave the attractor.

dx/dt = a(y − x) + cxz
dy/dt = ex + fy − xz
dz/dt = bz + xy − dx²

#dequanli · 3-D flow

Three-Scroll	animated

Three-Scroll Attractor — the three-scroll unified chaotic system (TSUCS): the Dequan Li system without the x term in dy/dt, which opens its three scrolls wider.

dx/dt = a(y − x) + cxz
dy/dt = ey − xz
dz/dt = bz + xy − dx²

#threescroll · 3-D flow

Wang-Sun	animated

Wang–Sun Attractor — Wang and Sun (2010): a four-wing attractor, the trajectory passing between four lobes arranged around the z axis rather than the usual two.

dx/dt = ax + cyz
dy/dt = bx + dy − xz
dz/dt = ez + fxy

#wangsun · 3-D flow

Lorenz Mod 2	animated

Lorenz Mod 2 Attractor — the second modified Lorenz system: the quadratic terms of the Lorenz equations rearranged so that the two wings fold into a single rotating body.

dx/dt = −ax + y² − z² + ac
dy/dt = x(y − bz) + d
dz/dt = −z + x(by + z)

#lorenzmod2 · 3-D flow

Simone	animated

Simone Attractor — a point drawn toward s·(sin(ay) + cos(bz), sin(az) + cos(bx), sin(ax) + cos(by)): the trigonometric map of the Simone attractor written as a flow, each coordinate driven by the other two. From Shashank Tomar's strange attractors. Chaotic for some (a, b) and periodic for others; b = 2.49 is chaotic.

dx/dt = s(sin(ay) + cos(bz)) − x
dy/dt = s(sin(az) + cos(bx)) − y
dz/dt = s(sin(ax) + cos(by)) − z

#simone · 3-D flow

Rucklidge Attractor — A. M. Rucklidge's 1992 model of double convection: a layer of fluid heated from below while a magnetic field or the frame's rotation pushes back. Two scrolls like Lorenz's, wound more tightly.

dx/dt = −κx + λy − yz
dy/dt = x
dz/dt = −z + y²

#rucklidge · 3-D flow

Genesio–Tesi Attractor — from Genesio and Tesi's 1992 study in control theory of when a feedback system stops oscillating regularly and turns chaotic. A jerk system: the third derivative of x set by x and its first two derivatives, with one square.

dx/dt = y
dy/dt = z
dz/dt = −cx − by − az + x²

#genesiotesi · 3-D flow

Shimizu–Morioka Attractor — Shimizu and Morioka's 1980 system, a relative of Lorenz's in which the butterfly's wings meet in a lip, a fold where the orbit hesitates before choosing a side.

dx/dt = y
dy/dt = x − λy − xz
dz/dt = −αz + x²

#shimizumorioka · 3-D flow

Rikitake Dynamo — T. Rikitake's 1958 model of the Earth's magnetic field: two disk dynamos, each feeding the other's coil. x and y are their currents, and they reverse sign at irregular times, as the geomagnetic field has reversed through geological time.

dx/dt = −μx + zy
dy/dt = −μy + (z − a)x
dz/dt = 1 − xy

#rikitake · 3-D flow

Moore–Spiegel Attractor — Moore and Spiegel's 1966 model of a parcel of gas in a star's atmosphere, rising and sinking through convection while it radiates heat away; T and R set the thermal and the buoyant forces.

dx/dt = y
dy/dt = z
dz/dt = −z − (T − R + Rx²)y − Tx

#moorespiegel · 3-D flow

Bouali Attractor — Safieddine Bouali's 2012 strange attractor with a stretched loop: a scroll with one long excursion out and back.

dx/dt = x(a − y) + αz
dy/dt = −y(b − x²)
dz/dt = −x(c − sz) − βz

#bouali · 3-D flow

Lorenz-84 — Edward Lorenz's 1984 model of the atmosphere's general circulation: x is the westerly jet, y and z the eddies that carry heat poleward, F the pole-to-equator heating and G its seasonal part. The same Lorenz, twenty years on, with weather as a whole rather than one convecting cell.

dx/dt = −y² − z² − ax + aF
dy/dt = xy − bxz − y + G
dz/dt = bxy + xz − z

#lorenz84 · 3-D flow

Finance System — Ma and Chen's 2001 model of an economy: x the interest rate, y the demand for investment, z the price index, a the saving, b the cost of investment and c the elasticity of demand. An economy can be chaotic without any shock from outside.

dx/dt = z + (y − a)x
dy/dt = 1 − by − x²
dz/dt = −x − cz

#finance · 3-D flow

Hindmarsh–Rose Neuron — Hindmarsh and Rose's 1984 model of a nerve cell: x the membrane voltage, y a fast recovery current, z a slow adaptation current, I the current injected. It fires in bursts of spikes, and at I = 3.25 how many spikes a burst has is chaotic. The slow current is slow: give it a long trail.

dx/dt = y − x³ + 3x² − z + I
dy/dt = 1 − 5x² − y
dz/dt = r(s(x + 1.6) − z)

#hindmarshrose · 3-D flow

Wei (no equilibrium) — Zhouchao Wei's 2011 system, chaotic though it has no equilibrium at all: no fixed point for the orbit to circle or be thrown from, which was long assumed to be what chaos needs. An attractor like this is hidden: it cannot be found by starting near a fixed point, because there is none.

dx/dt = −y
dy/dt = x + z
dz/dt = 2y² + xz − a

#wei · 3-D flow

Sprott Jerk — J. C. Sprott's 1997 simplest dissipative chaotic flow: x‴ = −A x″ + x′² − x, one parameter and one quadratic term. Chaotic in a narrow window of A around 2.017, from a small basin: a start outside it escapes, and the rack starts it again.

dx/dt = y
dy/dt = z
dz/dt = −Az + y² − x

#sprottjerk · 3-D flow

Hyper-Rössler (4D)	animated

Rössler Hyperchaos (1979) — Otto Rössler's 4-dimensional extension of his attractor, with two positive Lyapunov exponents (hyperchaos). A hidden fourth coordinate w feeds back into y; the plot is the (x,y,z) projection.

dx/dt = −y − z
dy/dt = x + ay + w
dz/dt = b + xz
dw/dt = −cz + dw

#hyperrossler · 4-D flow

Sprott Morph	animated

Sprott Morph — the glensstuff.com Self-Programming Analog Computer, which stepped itself through the Sprott catalog by re-patching. Every Sprott A–S system is a quadratic 3-D flow, a point in the 30-dimensional coefficient space of Sprott's 1994 search. With the rate knob up, the machine re-patches itself from one system to the next, in systems per minute, while ONE trajectory keeps integrating; the wired readout under the monitor shows the current wiring. B–S are chaotic attractors, stepped with RK4 as their own models are; A is conservative, a chaotic sea rather than an attractor. With the rate at zero the sys knob parks the patch anywhere in the cycle, including between two systems, where the coefficients are blended linearly. Most blends are not attractors: measured at quarter steps, the large majority diverge or settle onto a cycle or a fixed point, and a guard reseeds the trajectory when a blend flings it away. Coefficients are extracted numerically from the catalog's own equations.

#sprottmorph · 3-D flow

Sprott A	animated

Sprott A (Nosé–Hoover oscillator) — the first of J. C. Sprott's 1994 systems and the only conservative one: rather than settling onto a strange attractor it fills a chaotic sea.

dx/dt = y
dy/dt = −x + yz
dz/dt = 1 − y²

#sprotta · 3-D flow

Sprott B	animated

Sprott B — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Five terms, two of them nonlinear (yz and xy). ∇·F = −1 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every unit of time. The equations are unchanged by (x, y, z) → (−x, −y, z), so the attractor is symmetric under a half turn about the z axis, or has a twin that is its half-turned copy.

dx/dt = yz
dy/dt = x − y
dz/dt = 1 − xy

#sprottb · 3-D flow

Sprott C	animated

Sprott C — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Five terms, two of them nonlinear (yz and x²). ∇·F = −1 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every unit of time. The equations are unchanged by (x, y, z) → (−x, −y, z), so the attractor is symmetric under a half turn about the z axis, or has a twin that is its half-turned copy.

dx/dt = yz
dy/dt = x − y
dz/dt = 1 − x²

#sprottc · 3-D flow

Sprott D	animated

Sprott D — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Five terms, two of them nonlinear (y² and xz). ∇·F = x, which changes sign at x = 0: volume grows where x > 0 and shrinks where x < 0, so the flow is dissipative only on average along the attractor.

dx/dt = −y
dy/dt = x + z
dz/dt = xz + 3y²

#sprottd · 3-D flow

Sprott E	animated

Sprott E — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Five terms, two of them nonlinear (yz and x²). ∇·F = −1 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every unit of time.

dx/dt = yz
dy/dt = x² − y
dz/dt = 1 − 4x

#sprotte · 3-D flow

Sprott F	animated

Sprott F — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (x²). ∇·F = −0.5 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every 2 time units.

dx/dt = y + z
dy/dt = −x + 0.5y
dz/dt = x² − z

#sprottf · 3-D flow

Sprott G	animated

Sprott G — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (xz). ∇·F = −0.6 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every 1.7 time units.

dx/dt = 0.4x + z
dy/dt = xz − y
dz/dt = −x + y

#sprottg · 3-D flow

Sprott H	animated

Sprott H — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (z²). ∇·F = −0.5 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every 2 time units.

dx/dt = −y + z²
dy/dt = x + 0.5y
dz/dt = x − z

#sprotth · 3-D flow

Sprott I	animated

Sprott I — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (y²). ∇·F = −1 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every unit of time.

dx/dt = −0.2y
dy/dt = x + z
dz/dt = x + y² − z

#sprotti · 3-D flow

Sprott J	animated

Sprott J — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (y²). ∇·F = −2 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every 0.5 time units.

dx/dt = 2z
dy/dt = −2y + z
dz/dt = −x + y + y²

#sprottj · 3-D flow

Sprott K	animated

Sprott K — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (xy). ∇·F = y − 0.7, which changes sign at y = 0.7: volume grows where y > 0.7 and shrinks where y < 0.7, so the flow is dissipative only on average along the attractor.

dx/dt = xy − z
dy/dt = x − y
dz/dt = x + 0.3z

#sprottk · 3-D flow

Sprott L	animated

Sprott L — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (x²). ∇·F = −1 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every unit of time.

dx/dt = y + 3.9z
dy/dt = 0.9x² − y
dz/dt = 1 − x

#sprottl · 3-D flow

Sprott M	animated

Sprott M — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (x²). ∇·F = −1 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every unit of time.

dx/dt = −z
dy/dt = −x² − y
dz/dt = 1.7 + 1.7x + y

#sprottm · 3-D flow

Sprott N	animated

Sprott N — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (z²). ∇·F = −2 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every 0.5 time units.

dx/dt = −2y
dy/dt = x + z²
dz/dt = 1 + y − 2z

#sprottn · 3-D flow

Sprott O	animated

Sprott O — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (xz). ∇·F = x, which changes sign at x = 0: volume grows where x > 0 and shrinks where x < 0, so the flow is dissipative only on average along the attractor.

dx/dt = y
dy/dt = x − z
dz/dt = x + xz + 2.7y

#sprotto · 3-D flow

Sprott P	animated

Sprott P — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (y²). ∇·F = 2y, which changes sign at y = 0: volume grows where y > 0 and shrinks where y < 0, so the flow is dissipative only on average along the attractor.

dx/dt = 2.7y + z
dy/dt = −x + y²
dz/dt = x + y

#sprottp · 3-D flow

Sprott Q	animated

Sprott Q — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (y²). ∇·F = −0.5 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every 2 time units.

dx/dt = −z
dy/dt = x − y
dz/dt = 3.1x + y² + 0.5z

#sprottq · 3-D flow

Sprott R	animated

Sprott R — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (xy). ∇·F = −1 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every unit of time.

dx/dt = 0.9 − y
dy/dt = 0.4 + z
dz/dt = xy − z

#sprottr · 3-D flow

Sprott S	animated

Sprott S — one of J. C. Sprott's simple chaotic flows (1994), realized as an analog circuit at glensstuff.com. Found by systematic search for the algebraically simplest systems that still produce chaos.

Six terms, one of them nonlinear (z²). ∇·F = −1 everywhere, so a cloud of starting points loses volume at one steady rate, by a factor of e every unit of time.

dx/dt = −x − 4y
dy/dt = x + z²
dz/dt = 1 + x

#sprotts · 3-D flow

Custom equation	animated

Custom Equations — type your own system. The Equation module takes dx/dt, dy/dt and dz/dt (plus dw/dt for a 4-D system) as text and integrates them with forward Euler, the same way the classic attractors are stepped. The iter switch under the monitor reads the same expressions as a discrete MAP instead — x = f(x,y,z) rather than x += dt·f(x,y,z) — which has no dt and no path between iterates, so it draws as a cloud of points the way Henon and Ikeda do.

There is no eval: expressions are tokenized, converted to RPN and interpreted over a small value stack, so nothing but arithmetic can run. Grammar is + − × ÷ ^ with grouping and implicit multiplication (2x, 3(x+1)), the functions sin cos tan asin acos atan exp log ln sqrt abs sign sinh cosh tanh floor, the constants pi e tau, and the variables x y z w t. Any other name becomes a free parameter and is auto-exposed as its own knob.

#custom · 4-D flow

Henon · Ikeda · Clifford · Peter de Jong · Gumowski-Mira · Tinkerbell · Chirikov Standard Map · Bifurcation · Poincaré Section

Henon	animated

Hénon map (M. Hénon, "A two-dimensional mapping with a strange attractor", 1976) — the example that showed a strange attractor needs neither a flow nor three dimensions, only a stretch and a fold. Zoom in anywhere on a filament and it resolves into more filaments: locally a line crossed with a Cantor set. Its Jacobian determinant is −b, so area contracts by exactly b every iterate no matter where you are.

x' = 1 − a·x² + y
y' = b·x

#henon · discrete map

Ikeda	animated

Ikeda map (K. Ikeda, 1979) — light circulating in a nonlinear optical cavity, where each pass rotates the complex field by an amount that depends on its own intensity. That intensity-dependent rotation is the whole mechanism: the more energetic part of the beam turns further, and the attractor's hook is the result.

t = 0.4 − 6/(1 + x² + y²)
x' = 1 + u·(x·cos t − y·sin t)
y' = u·(x·sin t + y·cos t)

#ikeda · discrete map

Clifford	animated

Clifford attractor (Clifford Pickover) — a trigonometric map with no physical derivation, kept because of what it draws. Every parameter change reshapes it entirely, which makes the four knobs worth turning slowly.

x' = sin(a·y) + c·cos(a·x)
y' = sin(b·x) + d·cos(b·y)

#clifford · discrete map

Peter de Jong	animated

Peter de Jong attractor — the same idea as Clifford's with the terms paired differently, and a different family of figures for it. Four knobs, an enormous space of shapes, and no way to predict which you will get.

x' = sin(a·y) − cos(b·x)
y' = sin(c·x) − cos(d·y)

#dejong · discrete map

Gumowski-Mira	animated

Gumowski-Mira map (I. Gumowski and C. Mira, CERN, 1980) — written to study how a particle beam drifts turn after turn around an accelerator ring, and it produces organic, almost biological figures that look nothing like their origin. The shared nonlinearity g is applied twice per iterate, to x and then to the NEW x. Unlike the others here it is QUASI-PERIODIC at these settings rather than chaotic: the curves it draws are invariant curves the orbit winds around forever, not a fractal dust. Two orbits started a hair apart drift apart slowly instead of separating exponentially, which is why the figure looks woven rather than sprayed.

g(v) = μ·v + 2(1−μ)·v²/(1+v²)
x' = y + a(1 − b·y²)·y + g(x)
y' = −x + g(x')

#mira · discrete map

Tinkerbell	animated

Tinkerbell map — a quadratic map of the plane whose orbit sweeps a curved, wing-like region. Bounded only for a narrow band of parameters; push the knobs and it escapes to infinity, which the renderer catches and reseeds rather than drawing NaNs.

x' = x² − y² + a·x + b·y
y' = 2·x·y + c·x + d·y

#tinkerbell · discrete map

Chirikov Standard Map	animated

Chirikov standard map (B. Chirikov, 1979) — the canonical AREA-PRESERVING map, and the one system here with no attractor at all. A kicked rotor: each iterate kicks the momentum by K·sin θ and lets the angle advance. Because area is preserved nothing contracts onto anything, so it is drawn as an ensemble of orbits rather than one — a single orbit would trace its own invariant curve and tell you nothing about the rest. Turn K up and watch the intact curves break into a chaotic sea with islands of stability stranded in it; near K ≈ 0.9716 the last curve spanning the phase space goes, and orbits can wander in momentum without bound.

p' = p + K·sin θ
θ' = θ + p'   (both mod 2π)

#standardmap · discrete map

Bifurcation	animated

Bifurcation Explorer — the fig-tree diagram, computed live. One parameter of the most recent flow mode sweeps its whole knob range across the x axis; each column integrates the system fresh at that value and plots the local maxima of z. Thin branches are periodic orbits, fan-outs are period-doubling cascades, filled bands are chaos — the route between them is the route to chaos. Pick the swept parameter on the sweep selector in the model bank, with drive beside it; visit an attractor and tune it to change the source system.

#bifurcation · parametric

Poincaré Section — the continuous flow read as a discrete point set. The most recent flow mode is integrated privately and sampled only where it pierces a plane, going one way through it; what is left is the cross-section of the attractor, and the sheets that are invisible in the tangle are the whole picture here.

AXIS and POS place the plane (POS is a fraction of the attractor's own reach along that axis, so 0 is through the middle whatever the system's size); DIR chooses which way through it counts. One way is the default, and it is not a preference: a bounded flow that goes up through a plane has to come back down through it, so keeping both superimposes two different sections and the return map stops being a function.

VIEW picks the picture — PLANE draws the crossings where they physically are, FLAT lays the section out face on in the plane's own coordinates, and MAP is the FIRST-RETURN MAP: each crossing plotted against the next one. That last is where the route to chaos is legible — a periodic orbit is a handful of dots, a period-doubling is that set doubling, and a chaotic attractor is a single-humped curve, which is the logistic map's parabola surfacing inside a differential equation. The dotted 45° line is y = x, where the map's fixed points are.

The crossing point is INTERPOLATED between the two samples that straddle the plane rather than snapped to the nearer of them — snapping smears the section by up to half a step of arc, which on these attractors is the same size as the gap between the sheets it is supposed to show.

The same section is available as an overlay on any flow: the S button on LINE (Display), or ON on AXIS in the model bank.

#poincare · parametric

STL File · Terminal · Terminal Animation · Host Shell · Desk · Lattice

STL File	animated

STL File — Load a stereolithograph (.stl, binary or ASCII) from disk with the load switch under the monitor and it renders as a rotating wireframe. Very large files are decimated to fit the 16-bit index pipeline.

#stlfile · geometry

Terminal — a live terminal, drawn as a model. It is the same texture-on-a-plane path the spectrogram and the recurrence plot use, so it rotates, zooms and takes a gradient like any other model; what is on the texture is xterm-go (https://github.com/0magnet/xterm-go), a Go port of xterm.js, rendering a real terminal grid with WebGL2. That last part is what makes this possible rather than merely desirable: a terminal drawn as DOM could not be sampled into a texture at all, and rasterizing DOM every frame is not a thing worth doing. xterm-go renders into a canvas, and a canvas is a texture source. Behind it runs websh (https://github.com/0magnet/websh), a Bash interpreter compiled to wasm over an in-memory filesystem, so this is a session you can work in and not a picture of one: pipes, globs, redirection, for loops, all of it rotating with the model. Double-click the canvas to type into it and Esc to give the keyboard back — a double click because a single one is how you rotate the model, and the two must not be the same gesture. Focusing it silences the app's own key bindings automatically, since Pong, the Keys module and the hovered-knob arrows all already stand aside for a focused textarea, which is what a terminal captures keys on. It can also be the BACKDROP behind another model, the way the spectrogram can.

#terminal · geometry

Terminal Animation — the same terminal-on-a-plane, with a drawing program on it instead of a shell. The catalog is tuiwasm (https://github.com/0magnet/tuiwasm): animations — fire, plasma, a matrix rain, an aquarium, a bonsai growing branch by branch, Langton's ant, falling sand — plus charts, tables and styled text. Pick one from the prog selector on this model's panel. The animations draw at half-block resolution: every cell is an upper or lower block glyph carrying its own foreground and background, which is two independently colored pixels per cell and roughly square ones, since a terminal cell is about twice as tall as it is wide. They run at the frame rate — the cells are written straight into xterm-go's buffer rather than encoded as escape sequences and parsed back, which is the difference between sixty frames a second and a wedged tab. Nothing is wired to the keyboard here, unlike the Terminal model: these draw, they do not read, and the animations quit on a keystroke. Like the terminal it can also be the BACKDROP behind another model.

#termanim · geometry

Host Shell — a real shell on the machine serving this page, drawn as a model. The Terminal model beside it is websh (https://github.com/0magnet/websh), a Bash interpreter compiled into the wasm over a filesystem that exists only in the browser; this one is a pty on the host, reached through the same agent the desk's host pane uses. It needs the server to have been started with --shell, which also forces a loopback bind — the server refuses that flag on a listener the network can reach rather than warning about it. Without the flag the terminal still opens and says so, because an empty rectangle with no explanation is the worse answer. Double-click the canvas to type into it and Esc to give the keyboard back, exactly as for the Terminal model. It reached the screen before this as a window inside the Desk; as a model it is the shell without the window manager around it.

#hostterm · geometry

Desk — a window manager, drawn as a model. The same texture-on-a-plane path the spectrogram, the recurrence plot and the Terminal use, with a whole desk (https://github.com/0magnet/desk) on it: winbox windows, a websh (https://github.com/0magnet/websh) shell in each, a file manager, all of them running while you rotate them.

It needed something from desk to be possible at all. A window is more than its pane — its title, buttons and border are DOM, and DOM cannot be sampled into a texture — so texturing the panes alone would give a desk of frameless rectangles. desk's WebGL compositor can now REDRAW the frames instead: each title bar is rasterized with Canvas2D, text and buttons and all, into the same canvas it draws the panes into, and that canvas is a complete picture of the desk.

Nothing types into it while it is a model, and that is the arrangement rather than an omission for the MOUSE: a click on a rotated quad would have to be cast through it to a texture coordinate and synthesized back into a DOM event at a place nothing is. What you get instead are two gestures. Ctrl-drag reaches the desk, so ctrl-dragging a title bar moves a window while an ordinary drag still turns the model; the pass switch under the monitor (Pass-through) swaps the two if you would rather drag windows directly and hold ctrl to turn. And the KEYBOARD does reach it: double-click the canvas to type into the focused window and Esc to give the keyboard back, the same pair the Terminal and Host Shell models use.

Aiming is the honest limitation — you are pointing at a projection, so a title bar is not where the pointer says it is. The Desk switch (Console → Rack) puts the same windows on the page as ordinary DOM, which is the one to use for arranging them; rearrange there, then come back here to look at it. Flatten to work, rotate to admire. It can also be the BACKDROP behind another model, the way the spectrogram and the terminal can.

#desk · geometry

Lattice — a solid drawn in depth by terminals. lattice (https://github.com/0magnet/lattice) is an ordinary terminal program that draws one cross-section of a turning solid in characters. Here 3N copies of it run, each in a terminal of its own, and the terminals are stacked as transparent sheets: N across each axis, three families each perpendicular to the other two. Whichever way the stack is turned, one family faces you and the other two are edge on, so the solid has depth from every side, and each family is brightened as it comes face on and dimmed as it goes edge on, so the solid stays as bright as it turns.

The terminals are real: each is xterm-go's emulator, and each program writes into it the same escape sequences it writes into any terminal. Only the drawing is the rack's, every character on a clear sheet, because a cell at the default background is clear, as it is in a translucent terminal. Seen from behind, a sheet reads mirrored, as glass does.

The solid is turned by the time of day, not by the rack: lattice -axis z -slice 6 in a real terminal of 24 columns by 12 rows beside the page draws the same slice at the same moment, because a terminal's size is the resolution — zoom it out and it draws a finer one.

Its cells in bay 1 are how the sheets are drawn: their rows, style, look, opacity, which families there are and what the view turns. SRC, the first of them, chooses what is drawn: one of the lattice's own solids, or any other model, drawn the same way from its trail, mesh or dots while it runs as itself, its constants in the bank beside the lattice's. The Turtle is the exception that fits: it walks the same integer grid, so without weight it is drawn from its walk rather than its trail, every step a whole number of voxels (or every voxel a whole number of steps) along an axis, and every corner the end of a straight run.

#lattice · geometry

Polyhedron · Nested Cube · Globe · Torus · Magnetosphere

Polyhedron	animated

Polyhedron — The regular solids and tilings, by Schläfli symbol {p,q}: faces of p sides, q of them at every corner. {3,3} is the tetrahedron, {4,3} the cube, {3,4} the octahedron, {5,3} the dodecahedron, {3,5} the icosahedron; where the corners make a full turn the faces tile the plane, and past it they tile the hyperbolic plane, drawn in Poincaré's disk. On a solid, morph cuts the corners through the rectified solid to the dual, op applies a Conway operator and kis raises a pyramid on every face.

#polyhedron · geometry

Nested Cube	animated

Nested Cube — A cube within a cube, two separate wireframes, illustrating the relationship between inner and outer geometric structures.

#nestedcube · geometry

Globe	animated

Globe — A wireframe sphere showing lines of latitude and longitude, similar to the graticule on a geographic globe. Latitude lines are horizontal circles parallel to the equator, longitude lines are great circles passing through the poles.

#globe · geometry

Torus	animated

Torus — A doughnut-shaped surface of revolution generated by revolving a circle (radius r) around an axis at distance R from the center of the circle.

#torus · geometry

Magnetosphere	animated

Magnetosphere — A visualization of magnetic field lines surrounding a dipole, similar to Earth's magnetosphere that shields the planet from solar wind.

#magnetosphere · geometry

Turtle Path

Turtle Path	animated	in 2-D (mod 30, closed)

Turtle Path — Reduce an integer sequence modulo m and the remainders repeat; the length of the repeat is the Pisano period. Read each term as an instruction — odd turns left and steps forward, even turns right and steps forward, zero does neither — and the walk draws a figure.

In three dimensions the parity of the NEXT term decides whether the turn is a yaw or a pitch, which is not arbitrary: the pair (F_n, F_n+1) mod m is the state of the recurrence, and the Pisano period is the period of that pair. One pass decides the rest, without walking further: the figure either closes, drifts in a straight line, or screws away along an axis.

The walk never finishes and is never restarted — the turtle is held mid-stride and extended at the Speed knob's rate, forever, and TRAIL is how much stays behind it (whole, long, short, comet) as the oldest scrolls off the far end.

CAM is where it is watched from. Only follow cares where the head is; every other setting places the figure by its DRIFT, the one direction it is going overall, known exactly from one pass of the period rather than guessed at from the last few frames — and for a screw it centers on the axis the classification locates, so the figure turns on the spot instead of swinging around it. fit then only decides how big it is drawn, lock holds that size too, and auto fits a closed figure and locks one that drifts.

TINT is what a color means — step, pass, visits, heading, turn, term, age — and set SRC on Layers · Colors to trail to see it, or leave it on X/Y/Z to read position instead (a flat DIM 2 figure has no Z to follow).

MUL multiplies the Fibonacci sequence, CAP limits the terms read for sequences that may not repeat, CYCLE steps to the next modulus every so many seconds, and MOD 0 draws the sequence unreduced.

Physics gives the figure weight, with ON on GRAV, the first of its four positions in the model bank after the Turtle's own: it becomes a rigid body in the plane of the screen — three degrees of freedom, the figure's own shape as its mass — inside a solid box the size of the frame, where GRAV (either way up), FRIC, BOUNCE and SPIN say how it behaves. Press on the figure and you can pick it up and throw it; press beside it and you are turning the view as usual. The walk keeps extruding while it lies there, new points arriving at the head and old ones dropping off the tail, so the figure slides through its own body and marches. Switched off, CAM places the figure again.

#turtle · parametric

Lissajous · Graphic Artist · Scope Pong · Fourier Text · Scope Clock · Bouncing Ball · XY Scope

Lissajous	animated

Lissajous Curve — Named after Jules Antoine Lissajous (1822–1880), these are parametric curves formed by combining sinusoidal motions along each axis. Not a chaotic system — the curves are periodic and their shape depends on the frequency ratios and phase relationships between the three oscillations.

x(t) = sin(at)
y(t) = sin(bt)
z(t) = sin(ct)

#lissajou · parametric

Graphic Artist	animated

Graphic Artist — a digital re-creation of Mitchell Waite's "Oscilloscope Graphic Artist" (Popular Electronics, November 1975), which drove a scope's X/Y inputs with harmonically related signals to draw 3-D-looking wireframes. Four relaxation oscillators A/B/C/D, each square or triangle: A is the fixed master and B/C/D lock to it at integer harmonics — the sync the original circuit was meant to enforce, and which a wiring error on real builds defeated, leaving them drawing an unstable blur. Carrier C is split ±45° and the B envelope modulates each phase; the two perpendicular components are what create the volume illusion on a flat scope, and here that component is lifted onto a real Z so the figure genuinely has depth.

VERT = levelA·A + levelB·(B · C₊₄₅)
HORIZ = levelD·D + levelB·(B · C₋₄₅)

#graphicartist · parametric

Scope Pong	animated

Scope Pong — An homage to the glensstuff.com analog Oscilloscope Pong: the whole game is one multiplexed beam tour driving the X/Y inputs, so the court, net, score ticks, paddles and ball share a single stroke — the faint diagonal beams are the retrace an unblanked scope really shows.

W / S move the left paddle, ↑ / ↓ the right; a side left alone for ~10 s returns to the machine player (it boots as a self-playing demo). The match starts over when a side passes 9, or at once with the rset switch under the monitor. Knobs: speed (the ball's), paddle (its size), skill (the machine player's), and left and right, the two paddle pots.

#pong · parametric

Fourier Text	animated

Fourier Text — An homage to the glensstuff.com Fourier Synthesis Character Generator, which built alphanumerics on a scope from summed harmonics. The banner's whole beam tour (strokes and retrace jumps alike) is one complex periodic signal x(t)+i·y(t); what's drawn is its reconstruction from only the first N harmonics — real harmonic synthesis, not a blur. One harmonic is an ellipse, so at low N the letters melt into loops; raise the harm knob and overtones sharpen them into legibility. Type the banner in the text field on the line under the monitor (Visual bay); Model Out plays the actual harmonic stack.

#scopetext · parametric

Scope Clock — the time, drawn the way a scope draws anything: as ONE beam tour. The face, the twelve ticks and the three hands are a single path walked in order, because an X/Y display has one beam and no way to lift it — so the jumps between the end of one stroke and the start of the next are drawn too, and show as the faint diagonals an unblanked scope really shows. Scope Pong and the Fourier banner are built the same way; this is the same instrument told the time.

The beam is spaced evenly along the whole path rather than given a fixed number of points per stroke: the face is a full circle and a tick is a few hundredths of one, so per-stroke spacing would put most of the beam on the ticks and leave the circle dotted. Even spacing is also what a real scope does, since the beam moves at a rate rather than at a number of samples.

The hands are fractional throughout — the hour hand creeps with the minutes, the minute hand with the seconds — because a clock whose hands jump looks stopped in between. TRAIL and Persist apply as they do to any trace, so a long trail smears the second hand into a comet.

#scopeclock · parametric

Bouncing Ball	animated

Bouncing Ball — The classic analog-computer demo (Telefunken shipped it to sell integrators): two integrator chains under constant gravity, a comparator that flips the vertical velocity at the floor with a restitution loss, and wall reflections for the drift. The scope draws the familiar train of shrinking parabolic arcs; when the bounces decay away the machine re-kicks the ball, as the unattended trade-show loop did. Floor hits blip at a pitch set by impact speed (once audio is unlocked by any interaction). Knobs: grav, bounce (the restitution), drift and height; the drop switch under the monitor lets the ball go again from that height. Try Persist to paint the full arc family.

y'' = −g, bounce: v ← −e·v at the floor

#bounceball · parametric

XY Scope	animated

X/Y Scope — the classic two-channel oscilloscope figure, drawing the live audio's (left, right) sample pairs as a line strip. This is the goniometer, or stereo vectorscope, that sits on a mastering desk: correlated channels lie on a diagonal, anti-correlated on the other, and a phase difference opens the diagonal into an ellipse — which is how the display doubles as a stereo phase meter and a mono compatibility check, since what lies along the anti-correlated diagonal is exactly what disappears when the mix is summed to mono. The deflection is squared against the shorter side of the window, so the mono diagonal reads at 45° and a quarter-cycle phase difference draws a circle — not an ellipse the window's shape put there. An angle that cannot be trusted is the one thing a phase display may not have. A mono source is plotted against a lagged copy of itself, since a raw mono signal would otherwise be a featureless diagonal; the Stereo Embedding next door is the same figure with a third axis, and it does not fake a channel that is not there.

AXES turns the display 45° into mid/side — M=(L+R)/2 and S=(L−R)/2, the orientation broadcast goniometers ship in, where center content lies along one axis and difference content along the other, so width is an extent rather than the eccentricity of a tilted ellipse.

GAIN is the deflection; WIN the time base in milliseconds, short for the instantaneous phase relationship and long for the width of a whole mix; GLOW the afterglow, which is the control a hardware vectorscope is actually used through — at zero the frame is cleared as it always was, above it the trace decays instead, so a transient leaves something to read. LAG is how far a mono source is delayed against itself and SMTH how hard the beam is slew-limited between samples.

CORR is the correlation meter: +1.00 means the channels are identical and the figure is the diagonal line, 0 means unrelated and a round cloud, −1.00 means one is the other's polarity inverted — which is exactly the content that disappears when the mix is summed to mono, and the reason this display is a mono-compatibility check. It reads L against R whatever AXES is set to, because it is a property of the channels rather than of the way they are drawn.

#xy · audio

Takens Embedding · Stereo Embedding · Polar Embedding

Takens Embedding	animated

Takens Delay Embedding — attractor reconstruction from a single signal (F. Takens, "Detecting strange attractors in turbulence", 1981). Each trail point is the delay vector (s(t), s(t−τ), s(t−2τ)) of the live audio: a pure tone draws a closed loop, music and speech trace the geometry of whatever produced them.

τ is the embedding delay, counted in samples at a fixed 48 kHz reference so that one knob position is one DURATION — the same delay whether the sound is arriving from a 48 kHz microphone or a 24 kHz server feed, which is what it was not when the number was read as raw samples of whatever happened to be playing. The default of 72 is 1.5 ms; the MEAS readout says what the current setting is in milliseconds.

MEAS measures τ rather than guessing it: the first minimum of the signal's average mutual information (Fraser & Swinney 1986), reported beside the false-nearest-neighbor embedding dimension m (Kennel et al. 1992). It also runs BY ITSELF, once, as soon as the mode has enough audio to measure — and again if the source is swapped, because τ is a property of what is playing. Once is the whole of it: nothing here re-tunes itself per frame, because a knob that moves with the music makes the figure move with it. Press the button to measure again, or turn the knob and it stays where you put it. An m above 3 means the trail you are looking at is a projection of a higher-dimensional reconstruction.

SRC picks which signal of the live pair is embedded. Takens' theorem takes ONE observable and this is the choice of which: the mix, either channel on its own, mid, or SIDE — what the two channels do not have in common, which on a real mix is the reverb and the room rather than the instruments, and which reconstructs a different manifold from the same recording.

WIN is how much time the figure spans, in milliseconds — short is live and legible, long draws a denser tangle that turns over more slowly; GAIN sets how large a full-scale sample draws. The scale is fixed — nothing auto-ranges, so quiet passages draw small and loud ones large, and the view never moves under you; the camera is fitted once to what full scale can reach, so peaks stay on screen.

SMTH is the beam smoothing between samples, and it is the trade between how much WINDOW is on screen and how smooth the line is, since both come out of one vertex budget: at 1 the figure is raw chords through four times as many samples, which is what to use to see a long window; at 16 it is a glass-smooth beam through a short one. A straight line from one delay vector to the next is a chord, and chords are the hard corners you can see — an artifact of the drawing, not something in the signal. The same knob serves the stereo and polar embeddings.

Audio is the rack's own signal: whatever the Mixer pins to RACK L and R, which at power-up is the capture (the microphone or a server feed), and can be the generators and the instruments.

#takens · parametric

Stereo Embedding — the Takens trail built from the two channels instead of one channel's past. Takens' theorem manufactures the missing axes out of a signal's own history because there is only one signal; a stereo source has already measured two, so this mode plots them against each other and what you see is the real relationship between the channels — phase, polarity, correlation, width — rather than a reconstruction. Looked at head-on it is the goniometer (vectorscope) of a mastering desk, which is the XY Scope's figure; the third axis is what a scope with two deflection plates cannot give you.

AXES picks the assignment. L,R,L(t−τ) is the goniometer with a delay coordinate for depth: a tone that draws one ellipse edge-on unrolls into a helix, and τ still means what it means in the Takens mode. L,R,time sweeps the figure along a ribbon so successive cycles stack instead of overwriting — the only way to see a slow phase drift, which on a flat display just wobbles. The mid/side positions rotate the basis 45°: M=(L+R)/2 and S=(L−R)/2, so center content lies along one axis and difference content along the other and width is an extent rather than a tilt. τ is inert on the two 'g' positions, which have time on the third axis.

CORR is the correlation meter: +1.00 means the channels are identical and the figure is a diagonal line, 0 means they are unrelated and it is a round cloud, −1.00 means one is the other inverted (and the difference vanishes if the mix is summed to mono). A mono source reads "mono" and draws the diagonal, which is the correct picture of a signal with no stereo information in it — nothing here fakes a second channel out of a delayed copy of the first, because that delayed copy is exactly what this mode exists to stop pretending is a channel. The mid/side positions are the ones worth turning to then: S is zero and what is left is an honest two-coordinate delay embedding.

ALGN is an inter-channel DELAY, ±2 ms, and it is the fault a goniometer gets reached for: a spaced pair of microphones, a mis-clocked converter, a plugin reporting its latency wrong. An offset draws as a figure that opens into an ellipse and rotates as the frequency moves; dial the knob until the figure collapses back onto the diagonal and it reads how far apart the channels were. WIDE scales side against mid — the width control of a mid/side processor, applied to the drawing rather than to the audio, so it answers what widening would do and what would collapse if the result were summed to mono. Neither moves CORR, which goes on reading the source as it actually is.

WIN is how much time the figure spans, in milliseconds — a phase display is read over a few tens of them, past a couple of hundred it is a filled blob; GAIN sets how large a full-scale sample draws, and as in the Takens mode the scale is fixed and nothing auto-ranges.

Audio is the rack's own signal: whatever the Mixer pins to RACK L and R, which at power-up is the capture (the microphone or a server feed), and can be the generators and the instruments. The feed asks the capture for the source as it was recorded, over the WebSocket and over WebTransport alike, so a stereo sink gives real stereo either way; a server or a device with only one channel to give reads "mono".

#stereo · parametric

Polar Embedding — the Takens delay vector drawn in a sphere instead of a cube. The Takens mode plots (s(t), s(t−τ), s(t−2τ)) coordinate by coordinate, and each coordinate is one sample bounded to ±1 and multiplied by GAIN, so the reachable set is a cube: loud passages pile up against its faces and hardest of all against its corners, where all three coordinates peak together. That box is an artifact of writing the vector down in coordinates; nothing in the sound knows about the axes.

Here the vector is split into a direction and a length, the direction is kept exactly as it is, and only the length is passed through a curve that cannot exceed 1 — so the figure is bounded by a sphere, which looks the same from every angle, and no rotation finds an edge that is really the arithmetic showing through. The angles between successive delay vectors are untouched, which matters: those angles are the reconstructed geometry Takens' theorem is about, and a map that bent them would be drawing a different manifold.

MAP picks the curve. tanh is the soft clipper: near-linear when it is quiet, asymptotic to the surface when it is loud. Algebraic (soft on the ring) reaches the same surface more slowly, so the mid-range keeps more of its dynamics. dB makes the radius a level in decibels, full scale at the surface and DRV setting how many decibels down the center is (2 is 40 dB, about a level meter's useful range), so quiet program material is drawn at a size it can be read at. Direction only (unit) throws the loudness away completely and puts every point on the surface — what is left is the angular motion the amplitude was hiding, so a figure that merely swelled and shrank before now moves. DRIVE is how hard the signal is pushed into the curve, and it does nothing on the direction-only position, where there is no length left to compress.

SRC picks which signal of the live pair is embedded, as in the Takens mode. τ and WIN mean exactly what they mean in the Takens mode, and the scale is fixed there too — nothing auto-ranges, so quiet draws small and loud draws large. The camera is fitted once to the sphere, which is the whole of what a bounded radius can reach, so peaks stay on screen without the room the cube's corner needed.

Audio is the rack's own signal: whatever the Mixer pins to RACK L and R, which at power-up is the capture (the microphone or a server feed), and can be the generators and the instruments.

#polar · parametric

Spectrogram · FVF Wobbulator · Recurrence Plot · RTA — Octave Bands · Transfer Function · Waterfall — Spectral Decay

Spectrogram	animated

Spectrogram — a scrolling short-time Fourier transform of the live audio: frequency up the plane, time scrolling right to left, magnitude as color. It is a display inspired by Vanya Sergeev's audioprism, written in Go (github.com/0magnet/audioprism-go) and drawn here as a texture on a plane in the same 3-D pipeline as every other model, so it rotates and zooms like one — and the same texture can be painted onto other geometry with the skin switch. Its controls in the model bank expose the whole chain: transform size, overlap, window function, channel, magnitude scale and limits.

#spectrogram · audio

FVF Wobbulator	animated

FVF — Harmonic Wobbulator. A software analog of the Frequency→Voltage→Frequency converter with balanced modulator designed at bunkerofdoom.com (hardware built 1984). The live audio's pitch is tracked, scaled/offset into a new carrier frequency, and ring- or AM-modulated back by the original signal — the metallic, glitchy 'very strange' timbre. Shown here as the processed spectrogram.

#fvf · audio

Recurrence Plot	animated

Recurrence Plot — the picture of when a signal returns to where it has already been (J.-P. Eckmann, S. O. Kamphorst and D. Ruelle, "Recurrence Plots of Dynamical Systems", 1987). Time runs along both axes, and the cell (i, j) is lit when the two samples are within ε of each other, so the main diagonal is always lit and everything else says how the signal repeats. A steady tone draws unbroken diagonals spaced by its period; a chaotic or noisy source breaks them into short segments; a held sound is a solid block, and the instant the source changes is an edge running across the square.

SRC picks what is being plotted: audio is the raw samples, embed is the Takens delay vector (s, s−τ, … s−(m−1)τ) of that same audio — the reconstructed phase space a recurrence plot is properly defined on, which removes the anti-diagonals a scalar signal shows because sin t equals sin(T/2 − t) — and traj is the most recent attractor's own trajectory, the way the Bifurcation explorer picks its system. τ is the Takens mode's own τ knob, so its MEAS button measures the delay for this plot too, and m is the embedding dimension.

WIN is how much history the square covers: milliseconds for the audio sources, tenths of that in the system's own time units for a trajectory. ε is the threshold as a fraction of the source's scale — full scale for audio, the attractor's own diameter for a trajectory, which is what makes one default readable across systems that differ in width by fifty times. It is never taken from the current level, deliberately: a threshold that follows the music makes the plot's density pulse with it instead of describing it.

RQA reads the picture back as three numbers — RR, the percentage lit and the number to turn ε by; DET, the share of that lying on diagonals, which is what separates a system from noise; and LAM, the share on verticals, which is states it sat in rather than passed through. Drawn as a texture on a square plane in the same 3-D pipeline as every other model.

#recurrence · audio

RTA — Octave Bands. The real-time analyzer a room is measured with: the spectrum split into fractional-octave bands and shown as a bar per band. A spectrogram shows every bin, which is right for watching sound move and wrong for asking what a room is doing to it — linearly spaced bins put nine-tenths of the picture above 2 kHz and squeeze the bass into a few pixels. Bands of equal RATIO drawn at equal widths is a logarithmic frequency axis, which is how hearing is organized and how every acoustics standard reports.

BAND picks the width: 1/1 is a hi-fi graphic equalizer's ten, 1/3 is what room measurement and ISO use, and the two finer settings find a single narrow resonance — a 1/12-octave band is about 6% wide, roughly the ear's own resolution in the midrange. The centers are the standard ones (ISO 266 / ANSI S1.11, the base-ten series), so a reading here is comparable with anybody else's.

FEED IT PINK NOISE from a generator's pink wave: fractional-octave bands get wider in hertz as they go up, so equal power per octave is what reads FLAT — a flat display on pink noise is the definition of a flat system, and it is the convention room measurement is done in. White noise rises 3 dB per octave on the same display, which is the difference between the two and the reason pink is the one used.

TOP and RNGE place the scale in dBFS; AVG is the meter's averaging, quick to rise and slow to fall as every level meter is; HOLD is the peak-hold decay in dB per second, which is what makes the display readable on music rather than only on noise — music excites part of the band at a time and the held peaks are the envelope that accumulates into the answer. SRC picks the channel.

#rta · audio

Transfer Function — what the thing between two channels did to the sound. Every other measurement here asks about ONE signal; this one asks about the RELATIONSHIP between two, which is the question a system is actually tuned by. Send a signal into a loudspeaker, a room, a filter or a cable, capture what comes back, and read what happened in between. That is what a system-tuning rig does, and chaosrack already carries two channels end to end.

MAGNITUDE is the frequency response in dB — flat is a system that changed nothing. PHASE is how far the output lags the input; a pure DELAY is a phase that falls linearly with frequency, and the slope IS the delay, which is what the DLY readout fits and what gets dialed into a delay line to line a speaker up. COHERENCE is the number that says whether to believe the other two: 1 means the output is fully explained by the input, and noise, a second source, a nonlinearity or a system that moved during the measurement all drive it down. A dip in the magnitude with the coherence still high is the system; the same dip with the coherence collapsed is the measurement giving up, and the curves are drawn together so the two can be told apart.

AVERAGING IS NOT OPTIONAL. From a single window coherence is exactly 1 at every frequency — for any two signals whatever, including two unrelated noises — so a display built on one window is a row of perfect scores that means nothing. AVG sets how many windows are folded in before the result is shown, and the curves are drawn only where the coherence clears COH and the stimulus actually reached the band.

REF says which channel is the reference, BAND how finely the bands are smoothed, RNGE the magnitude scale, SHOW which curves are drawn. Feed it pink noise or the log sweep from a generator (its pink and swp waves).

#xfer · audio

Waterfall — Cumulative Spectral Decay. The measurement a loudspeaker is characterized by, and the one display here that could only exist in this app: every other analyzer draws its own flat panel and this is a genuine 3-D surface, so it rides the same pipeline the attractors do and drags, rotates, zooms and takes the gradient like any other model. Frequency runs left to right, logarithmically, because hearing is organized in ratios; level runs up; and TIME runs into the screen. Each line is the spectrum of what is left of the impulse response from a moment onwards, so a flat loudspeaker's surface falls away evenly and a RESONANCE is a ridge running back into the screen at one frequency. That is what this is for: a frequency response cannot tell a resonance from a broad lift, because they are identical in magnitude and nothing alike in time.

Feed it a generator's log sweep (its swp wave), with the sweep in one channel as the reference and what came back in the other. The impulse response is recovered by deconvolution and the surface rebuilt each time a sweep pass completes — so unlike every other audio mode the picture HOLDS STILL between passes, which is what makes it something to rotate and look at rather than something to freeze first.

LINE is how many slices the surface has and STEP how far apart they are in milliseconds, so the two are how deep it reaches in TIME: sixteen at 5 ms is the 80 ms a loudspeaker's resonances live in, thirty-two at 40 ms the 1.3 seconds a bar of music takes. FFT is the transform each slice is taken through, and it is a genuine trade — 1k resolves time best and the bass worst, 8k the other way, and a window longer than STEP means consecutive slices see the same audio. TOP and RNGE place the level scale in dBFS as they do on the RTA. DPTH is how far back the surface reaches on screen, which is geometry rather than time. CHAN is the channel the live surface analyses; REF is which channel the decay surface calls the reference, because that one needs both. LINE, STEP and FFT follow the SRC switch to what each surface wants, and stop following once you turn one.

Turn SRC on Layers · Colors to Y: it colors by LEVEL across exactly the decibels TOP and RNGE show, which is how every published CSD plot is colored. Z and trail color by age, X repeats the frequency axis, and AUDIO is one flat tint here.

A sweep takes four seconds to cross the band, so the measurement needs a whole pass: a third of one is 20 Hz to 200 Hz and recovers an impulse five milliseconds wide.

SRC PICKS WHICH SURFACE. DCAY is the decay above, which is a measurement and needs the sweep. LIVE is the other thing the word waterfall means: successive spectra of whatever is playing, stacked into the screen as they age, so the depth axis stops being time-since-the-impulse and becomes time-ago. Live is the one for music — a note is a ridge that rises at the front and travels back as it decays — and decay is the one for a measurement rig.

RT60 is the room's reverberation time from the same impulse, by Schroeder backward integration as T20 between -5 and -25 dB: the one number the surface is far too shallow to show. Blank on the live surface and blank through a wire, neither of which has a decay to measure.

#waterfall · parametric

Every control is always on the panel, including on models where it does nothing. docs/controls.md lists which global controls are inert on which models, the audio models' own controls, and the pairs of controls that change the same thing by different means.

The analyzers, with their methods and verification, are in docs/measurements.md:

The stimuli are positions of each generator's waveform knob.

Off by default. The server binds 127.0.0.1; nothing that reaches the machine is served unless asked for.

chaosrack serve --shell                    # a real pty in a desk window
chaosrack serve --fs --fs-root ~/projects  # host files for the shell and file manager, confined
chaosrack serve --shell --auth             # on a shared machine: token printed, not embedded
chaosrack serve -b 0.0.0.0 --audio         # serve the LAN; other browsers pair once
chaosrack serve --audio --pair             # chaosrack.magnetosphere.net gets this machine's audio

• --shell and --fs refuse a non-loopback bind.
• Requests are checked by Origin and Sec-Fetch-Site, and carry a per-run token from crypto/rand.
• The audio capture and the rack link accept only pages served as localhost, 127.0.0.0/8, ::1 or a --host name (DNS-rebinding safe).
• --fs-root resolves symlinks, so a link inside the root cannot lead out.
• --reconnect lets a named host shell (host NAME in the desk) outlive its window.
• LAN pairing: the printed link carries the server's WebTransport address, certificate hash and a pair secret in its fragment; the page signs each session with an ed25519 key. Paired keys are in ~/.config/chaosrack/paired-tabs.
• --pair does the same on loopback, so the public page in a browser on this machine gets the server's audio and rack link (after the browser's one-time "access this device" prompt).

Threat model and details: docs/remote-access.md.

The panel is a rack of 3U modules in bays. A module's header is a drag handle; the Console's Modules switches hide or show each one. Tiling is rack-go and docking/floating is winbox-go. The window-manager modes (Desk, Contain) and the rack frame are described in docs/desk.md.

Every module, captured from the running rack by uitool modules. Each description is the module header's tooltip; addresses are bay.column.row. Expand Controls for every cell's tooltip.

1.1 · Visual — every model, in one bay: attractors, maps, solids, geometry and sequences drawn in three dimensions, and the scope pictures, embeddings and audio displays drawn from a signal. The outer ring of the MODEL knob picks which kind, the inner ring which model, and the bank beside it is reprogrammed to that model's constants The head of the bay: the monitor, the switches under it, and the readouts of the model playing. The MODEL knob and the model's step head the bank beside it, which the MODEL knob reprograms.

1.3 · Visual · Model — a bank of 30 programmable controls: the MODEL knob, its step, then every constant, pot and selector of the model it is set to, and the groups the model brings into play: the section's plane, the Turtle's weight, the water lens and the spectrogram's analysis, dark until switched on. Each display names what the control under it does for that model; turning the MODEL knob reprograms them, and each model keeps its own settings while another is showing. With Back on (Layers · Colors) they are the backdrop's.

2.1 · View — orientation: per-axis angle knobs and continuous spin rates

2.3 · Grid — how many views of the model are drawn, what differs between them, and which of them the knobs are pointed at. It sits beside View, Display and Layers · Colors because it is a question about the picture, not about the signal: Display is how one trajectory is drawn, View is where the camera is, and this is how many cameras there are.

2.5 · Display — trace rendering: line width, point decimation, trail length and mode, and the overlays drawn on the trace (Poincaré section, graticule). Three P-units; the switches are on their buttons.

2.6 · Layers · Colors — how the picture is made up and colored. BEHIND draws a second picture filling the canvas behind the model and SKIN paints one onto its surface; BG is the color behind it all. SRC is what the color follows, MAP how a value becomes a color, PERIOD and SHIFT the window on the map, and START, MID and END the gradient's own colors. The buttons: F (fill) and B (back) on BEHIND, I (invert) on MAP, H (held) on SRC.

2.10 · Console — Reset All and Normalize, MIDI, and the switches for the rack itself and for motion. The model is chosen with the MODEL knob in bay 1.

3.1 · Record — a monitor showing exactly what will be captured, and the transport that captures it. The picture is the recorded area, not the whole canvas, so what you see here is what lands in the file.

3.3 · Style — knob face + LED color, and CRT phosphor

3.4 · Presets — the current view, saved under a name. A preset holds everything the permalink holds: the model, every knob and color, the effect switches, the parameters and the pose. Recalling one resets to defaults and re-applies it, exactly as the numbered patch memories on this module do; the difference is that these have names, and the address bar does not have to carry them.

3.5 · Counter — a frequency counter for the rack, in the spirit of the glensstuff.com NAND-gate counter: it counts trigger crossings of the live audio source over the gate window and shows cycles per second, the way the discrete-logic original did. It counts the rack's own signal: whatever the Mixer pins to RACK L and R.

3.6 · Distortion — THD, THD+N, SINAD and ENOB of the live audio, measured against whichever tone is in it. Feed it a generator's sine at 1 kHz (type 1000 into its Hz readout) through whatever you want measured and read how much of what comes back is not that tone. THD is the harmonics alone, which is the number a specification quotes; THD+N is everything that is not the fundamental, which is the honest one and always the larger; SINAD is the same ratio in decibels, the way a converter is specified; ENOB runs the ideal-converter relation backwards and says how many bits would sound this clean. The gap between THD and THD+N is how much of the rubbish is hiss rather than distortion. The instrument's own floor is about 0.0015% THD+N (96.5 dB SINAD, 17.3 effective bits), measured on a synthesized pure tone — that is the Blackman-Harris window's leakage past the notch, and nothing quieter than it can be read.

3.8 · Loudness — LUFS to ITU-R BS.1770 and EBU R 128, the scale everything is delivered against. M is the momentary loudness over 400 ms, S the short-term over 3 s, I the INTEGRATED reading over everything since the last reset — gated, so silence and quiet passages do not drag a programme's number down, which is what makes it the number a delivery spec means. LRA is the loudness range, how far the loud parts sit above the quiet ones: one number for how dynamic the material is, and the thing a loudness target alone says nothing about. TP is the TRUE peak in dBTP, the peak of the reconstructed signal rather than of the samples — a full-scale tone at a quarter of the sample rate can have every sample at −3 dBFS and still clip the converter, which is why a delivery ceiling is a true-peak ceiling. K-weighting is derived from the analog prototype at whatever rate the audio is arriving at, not the 48 kHz coefficients the standard prints, so the weighting is the same filter at every rate.

3.9 · Wow & Flutter — speed stability, measured off a 3150 Hz test tone. Every other analyzer here asks about amplitude; this asks whether the TIME AXIS is steady, which is what a turntable, a tape deck or a cassette is judged by. Play a generator's sine at 3150 Hz through the deck, capture the result, and read it here: SPEED is the mean frequency error as a percentage (a constant error is a pitch shift, a different fault from a wobble), WOW the slow 0.5–6 Hz modulation that once-per-revolution faults produce, FLUTTER the fast 6–100 Hz kind from capstans and idlers, and W&F the DIN-weighted quasi-peak that a specification quotes — the deviation through a filter peaked at 4 Hz, where the ear is most sensitive to pitch movement. 3150 Hz is the figure every test record carries (DIN 45507, IEC 60386). With no deck in the loop at all it reads 0.000% on the generator itself, which is the check that the instrument is not inventing the number.

3.11 · Features — what the Mod matrix's audio routes read, as a meter bridge: six LED bar graphs, one for each feature of the incoming audio's mono mix, the same numbers a route modulates a knob by. Each ladder is twenty segments, green to 70%, amber to 90% and red above. The levels are relative, not absolute: each feature is scaled by its own recent peak (instant rise, slow fall), so a quiet passage still fills the ladders and a loud one does not pin them. Dark while modulation is off — nothing routed and no feed from the server — since then nothing is measured.

3.12 · Timing — the rack metering itself. FRAME is the average interval in milliseconds and FPS its reciprocal, taken over the mean rather than the median so one hesitation a second cannot hide behind fifty-nine good frames. MIN and MAX are the best and worst frame in the window. LATE is the share of frames that missed the display's own interval, which is measured rather than assumed: nothing renders faster than vsync, so the shortest frame in a window IS the interval, and the reading means the same on a 60 Hz panel and a 144 Hz one. The four figures below it are where the frame went — MODEL the passes that draw the instrument, METERS everything the rack does to its own audio, SCOPE the tube, and REST the browser's own style, layout, raster and compositing, which is usually the largest share of the four. A rack that draws in 4 ms and spends 9 ms measuring its own audio is not slow, it is mis-apportioned, and no single frame rate says so.

4.1 · Scope 1 — one of the rack's four oscilloscopes, each beside its generator. Under VOLTS and TIME, the INPUT knobs 1 and 2, each beside a display naming its source, pick what each input is fed from — the rack's signal (RACK L or R, as the Mixer makes it), the capture (CAP L or R, wherever the Mixer sends it), one generator by itself, or a coordinate of the model (Model Out) — named on the face. The tube, and at its right the SOURCE buttons — what the beam draws: IN 1, IN 2, the two summed (SUM), or one against the other (X-Y) — between the small H and V knobs that move the trace sideways and up and down. Under it the beam's knob, INTENSITY, with two mini knobs beneath: I, SCALE ILLUM, which lights the graticule and, turned down past the bottom of its travel, clicks into OFF and switches the scope off; and F, FOCUS. Beside that the TRIGGER, with its SLOPE and MODE buttons, and below them VOLTS and TIME, how much a division is worth up and across.

4.3 · Gen 1 — a generator, for sound or for modulation. The FREQ knob, an encoder with no stops — a click a semitone, a turn an octave, the ring's color which octave — from 0.107 Hz to 28160 Hz, a one-octave piano strip under it lighting the note it is on; the level knob inside its envelope's ATTACK and DECAY rings, with ENV between them to repeat it; and the waveform knob, OFF at the bottom, with SOLO beside it to hear only this one and OUT under it. Where it goes is OUT: OFF, L, C (both sides) or R, on the speakers and the rack's signal together. The Mixer's GEN 1 column has every other pinning — the speakers or the rack alone, a gain, an inverted side, the sends to modulation — and OUT reads MIX while it does.

4.4 · Scope 2 — one of the rack's four oscilloscopes, each beside its generator. Under VOLTS and TIME, the INPUT knobs 1 and 2, each beside a display naming its source, pick what each input is fed from — the rack's signal (RACK L or R, as the Mixer makes it), the capture (CAP L or R, wherever the Mixer sends it), one generator by itself, or a coordinate of the model (Model Out) — named on the face. The tube, and at its right the SOURCE buttons — what the beam draws: IN 1, IN 2, the two summed (SUM), or one against the other (X-Y) — between the small H and V knobs that move the trace sideways and up and down. Under it the beam's knob, INTENSITY, with two mini knobs beneath: I, SCALE ILLUM, which lights the graticule and, turned down past the bottom of its travel, clicks into OFF and switches the scope off; and F, FOCUS. Beside that the TRIGGER, with its SLOPE and MODE buttons, and below them VOLTS and TIME, how much a division is worth up and across.

4.6 · Gen 2 — a generator, for sound or for modulation. The FREQ knob, an encoder with no stops — a click a semitone, a turn an octave, the ring's color which octave — from 0.107 Hz to 28160 Hz, a one-octave piano strip under it lighting the note it is on; the level knob inside its envelope's ATTACK and DECAY rings, with ENV between them to repeat it; and the waveform knob, OFF at the bottom, with SOLO beside it to hear only this one and OUT under it. Where it goes is OUT: OFF, L, C (both sides) or R, on the speakers and the rack's signal together. The Mixer's GEN 2 column has every other pinning — the speakers or the rack alone, a gain, an inverted side, the sends to modulation — and OUT reads MIX while it does.

4.7 · Scope 3 — one of the rack's four oscilloscopes, each beside its generator. Under VOLTS and TIME, the INPUT knobs 1 and 2, each beside a display naming its source, pick what each input is fed from — the rack's signal (RACK L or R, as the Mixer makes it), the capture (CAP L or R, wherever the Mixer sends it), one generator by itself, or a coordinate of the model (Model Out) — named on the face. The tube, and at its right the SOURCE buttons — what the beam draws: IN 1, IN 2, the two summed (SUM), or one against the other (X-Y) — between the small H and V knobs that move the trace sideways and up and down. Under it the beam's knob, INTENSITY, with two mini knobs beneath: I, SCALE ILLUM, which lights the graticule and, turned down past the bottom of its travel, clicks into OFF and switches the scope off; and F, FOCUS. Beside that the TRIGGER, with its SLOPE and MODE buttons, and below them VOLTS and TIME, how much a division is worth up and across.

4.9 · Gen 3 — a generator, for sound or for modulation. The FREQ knob, an encoder with no stops — a click a semitone, a turn an octave, the ring's color which octave — from 0.107 Hz to 28160 Hz, a one-octave piano strip under it lighting the note it is on; the level knob inside its envelope's ATTACK and DECAY rings, with ENV between them to repeat it; and the waveform knob, OFF at the bottom, with SOLO beside it to hear only this one and OUT under it. Where it goes is OUT: OFF, L, C (both sides) or R, on the speakers and the rack's signal together. The Mixer's GEN 3 column has every other pinning — the speakers or the rack alone, a gain, an inverted side, the sends to modulation — and OUT reads MIX while it does.

4.10 · Scope 4 — one of the rack's four oscilloscopes, each beside its generator. Under VOLTS and TIME, the INPUT knobs 1 and 2, each beside a display naming its source, pick what each input is fed from — the rack's signal (RACK L or R, as the Mixer makes it), the capture (CAP L or R, wherever the Mixer sends it), one generator by itself, or a coordinate of the model (Model Out) — named on the face. The tube, and at its right the SOURCE buttons — what the beam draws: IN 1, IN 2, the two summed (SUM), or one against the other (X-Y) — between the small H and V knobs that move the trace sideways and up and down. Under it the beam's knob, INTENSITY, with two mini knobs beneath: I, SCALE ILLUM, which lights the graticule and, turned down past the bottom of its travel, clicks into OFF and switches the scope off; and F, FOCUS. Beside that the TRIGGER, with its SLOPE and MODE buttons, and below them VOLTS and TIME, how much a division is worth up and across.

4.12 · Gen 4 — a generator, for sound or for modulation. The FREQ knob, an encoder with no stops — a click a semitone, a turn an octave, the ring's color which octave — from 0.107 Hz to 28160 Hz, a one-octave piano strip under it lighting the note it is on; the level knob inside its envelope's ATTACK and DECAY rings, with ENV between them to repeat it; and the waveform knob, OFF at the bottom, with SOLO beside it to hear only this one and OUT under it. Where it goes is OUT: OFF, L, C (both sides) or R, on the speakers and the rack's signal together. The Mixer's GEN 4 column has every other pinning — the speakers or the rack alone, a gain, an inverted side, the sends to modulation — and OUT reads MIX while it does.

5.1 · Synth — the sound the keyboard plays. PRST sets every knob on the bay to an instrument, as a place to start from; LVL is how loud. Where it is heard is the Mixer's: its KY column.

5.2 · String — the additive voice that is the piano: partials not quite harmonic, each fading at its own rate. STR its level; PART how many partials; TILT how fast they fall away up the series; INHM the string's stiffness, which sharpens the upper partials (×10⁻⁴); STRK where along the string it is struck, which silences the partials with a node there; UNI the detune between the two strings of each low partial, in cents; DAMP how much faster each higher partial dies.

5.5 · FM — a sine modulated by another: the electric piano. FM its level; RATO the modulator's frequency over the note's; INDX how deep the modulation starts; IDEC how fast it fades, which is how fast the bark becomes a pure tone.

5.7 · Wave · Noise — WAVE is one oscillator, a generator waveform, the organ's drawbars or tuned noise, at a level; NOISE is a burst of filtered noise at the start of each note — a hammer, a quill, a tine's click — at TONE times the note, dying over NDEC.

5.9 · Filter — a lowpass on each note that opens at BRGT times the note and closes towards DARK times it over FDEC: a plucked string brightest at the pluck. RESO is its resonance.

5.11 · Amp — the envelope every layer follows: ATK up, then towards SUS with time constant DEC (at C3), which KTRK shortens up the keyboard and lengthens down it, as a piano's strings ring longer in the bass; REL is how long letting go takes.

6.1 · Keys — a playable polyphonic keyboard: click or drag the keybed (glissando works), or play the computer keyboard on the labeled keys (Z row = lower octave, Q row = upper). The bed is the whole piano, A0 to C8, across the whole bay; what it sounds like, how loud and on which speakers is the Synth bay above it.

7.1 · Matrix — a pentatonic tonematrix step sequencer with a rhythm section under it: paint pads on the grid (click or drag; rows are pitches, columns are sixteenths) and the playhead loops them as pings at the tempo, over as many columns as the steps knob says. The four lanes at the foot are the drums (bass drum, snare, hi-hat, cymbal), on the same beat; they count in sixteenths or, for a shuffle or swing, in triplets, and each beat of them sits under the same beat of the tune. A preset tab loads a pattern into them to play or to edit. The pentatonic rows mean any pattern is consonant.

8.1 · Mixer — every sound on the rack, and where it goes. Each column is a source (the capture, the generators, the model's x, y and z, the instruments); each row is a destination: SPK L and R are the speakers, RACK L and R the rack's own signal, which the meters, the audio models, the scopes and the Mod matrix read, and MOD A and B two sends only the Mod matrix reads, for a source that should turn a knob without being heard or measured. A pin where they cross puts the source there: press it to pin at full level, press again to take it away, and turn the wheel over it, or LVL below, to set its gain; a negative gain inverts it. Several pins on a row are mixed. The capture has no speaker pins: it is the system's own audio, already playing.

8.4 · Mod — audio modulation: the rack's signal turning the rack's knobs. Each column is a source: RACK — the rack's signal, as the Mixer makes it, both sides summed (st) or one (L, R) — SEND, the Mixer's MOD A and B, which only modulation reads — or HEAD, where the model is drawn now (x, y, z, and r, its distance from the center). Each row is a control: the running model's parameters, then the view's. A pin routes the column to the row; another pin in the same row moves the route; a lit pin pressed again takes it away. DEST, DPTH, SHAPE and EQ under it are the selected row's route: which control, how deep, which way it moves, and which frequency bands drive it.

The panel is drawn to scale: 1 mm = 4 CSS px at interface scale 1. The numbers are in pkg/rackspec, and the stylesheet is tested against them.

chaosrack stlgen           # writes docs/stl/*.stl
chaosrack stlgen --list    # what it would write

Modules in one to three slot widths with case and Eurocard, a blank panel, the 19-inch cage, dimensioned variants with their measurements as solid geometry, the Platonic solids, sphere, torus and nested cube, and every flow swept as a tube (parallel-transport frames). The same list is selectable in the app's STL File model. Output is gitignored.

The Record module captures the canvas: WebM through MediaRecorder or GIF encoded in Go with one adaptive palette per take; the full canvas or a dragged region; record, pause, stop and single-frame still. Video bitrate is ~0.10 bits per pixel per frame, at least 8 Mbps.

FVF Wobbulator bank	XY oscilloscope

Per-model animations, contact sheets and the chaos-monkey jam sessions: docs/gallery.md.

Aizawa mandalas. Seen end-on, the Aizawa attractor closes into a petaled mandala, and its trail rainbow can be made to stand still, or to repeat as harmonics: several rings outward, a wheel around, or a spiral. How these were found, and the full table: docs/aizawa-harmonics.md. On a local chaosrack serve, the same links work with http://127.0.0.1:8080/ in place of the site.

• 7 petals, 1 ring
• 7 petals, 2 rings outward
• 7 petals, 1 rainbow around (a color wheel)
• 7 petals, spiral
• 5 petals, 3 rings
• 3 petals, 3 rings
• 8 petals, 2 rings
• Lines still, colors flowing (one color cycle every 5 s)
• Lines moving, colors standing: the turns per loop are set to a golden-ratio value, so the orbit never closes, and the petals keep turning while the color rings hold

The first three found, with f above 0, where the orbit locks without fine tuning: one, two, three.

?audio=mic and ?audio=ws force a source. The wire format and the spectrogram engine are audioprism-go's.

A model of a 1984 frequency→voltage→frequency converter with a ring modulator (bunkerofdoom.com): it tracks the input's pitch, rescales it into a new carrier, and ring- or AM-modulates the input with it. chaosrack serve --audio --wobbulate (or the route switch) routes all system audio through it and plays the result on the speakers without feedback. Signal path, routing and the hardware: docs/fvf.md.

The UI is tested in a real browser over the Chrome DevTools Protocol (0magnet/cdp), by chaosrack uitool:

They attach to a Chromium/Brave tab on --remote-debugging-port 9222, or run their own headless browser with -headless -serve <binary>.

make test-wasm runs the js/wasm tests under Node with a fake DOM. Native tests cover the pure logic and the chaos guard. make lint runs golangci-lint twice, natively and under GOOS=js GOARCH=wasm. CI runs tests, both lint passes, a wasm compile gate and a TinyGo build.

go build .                           # the host binary, with the wasm committed in assets/
GOOS=js GOARCH=wasm go build .       # the page alone
make check                           # lint, vet, test, readme-check
make help                            # every target

The host binary serves the wasm in assets/ with //go:embed. The committed copies are rebuilt by CI after each push, so a binary built from a changed checkout serves the old page until that lands. To serve your own changes, build the wasm first, then the binary:

make wasm          # assets/gowasm/chaosrack.wasm (standard Go)
go install .       # or: go build .
chaosrack serve --audio

make tinywasm (needs TinyGo; slow), make metersworker (the analyzers' Web Worker) and make wasms (all three) build the others. Leave the rebuilt assets uncommitted — CI owns them:

git checkout assets/gowasm/chaosrack.wasm assets/gowasm/built-from.txt

make pages writes the deployed index.html pages, and make onefile the single-file build (SERVERLESS.md).

The repository root is one program: main.go on a host, main_js.go in the page. commands/ assembles the chaosrack command; assets/ holds the embedded wasm and templates; pkg/ holds about 45 packages, chiefly attractor (the app and panel), server, dynamics (the models' math), meters, audiosrc, scope, rackspec, racklayout, racksurface, rackhost (the native rack), preset, equation and stlmodels. manual/ is the help text the tooltips and Info window show.

Browser attractor visualizers and audio tools:

• ibiblio e-notes — Lorenz / Rössler (WebGL, incl. GPU 1M-point)
• Harvard/Knill — animated Lorenz
• daybarr/lorenz-webgl
• Simulations4All — Lorenz with σ/ρ/β sliders
• KinnyTools — audio-reactive Lorenz
• jujiplay — interactive Rössler
• Shashank Tomar — Strange Attractors — fourteen systems as GPU particle systems; chaosrack's Arneodo, Chen–Lee, Dequan Li, Three-Scroll, Wang–Sun, Lorenz Mod 2 and Simone come from its catalog (discussion)
• libpd-wasm — Pure Data in an AudioWorklet
• SuperSonic — SuperCollider's scsynth in WebAssembly
• The Analog Computer Museum's library — manuals of EAI, Telefunken, Dornier and other analog computers
• Reconfigurable Analog Computers — digitally reconfigured analog computers as co-processors
• GitHub strange-attractors topic

Instrument panels and controls in the browser:

• webaudio-controls (demo) — knob, slider, switch and keyboard web components
• input-knobs — <input> elements styled as knobs and switches
• precision-inputs and NexusUI — knob, dial and matrix widgets
• iO-808 — a TR-808 in Web Audio
• Ableton Learning Synths — a guided synth tour
• Audiotool — hardware-style devices patched with cables
• Cardinal (VCV Rack in WebAssembly), Patchcab, DreamRack — Eurorack-style racks
• woscope — WebGL XY oscilloscope

Made with goda:

# GOOS=js: the import edges of a wasm program live in js/wasm-tagged
# files and are invisible to a host-context run
GOOS=js GOARCH=wasm go run github.com/loov/goda@latest graph github.com/0magnet/chaosrack/... | dot -Tsvg -o docs/chaosrack-goda-graph.svg

Made with gocloc (excludes vendor/, node_modules/, .git/):

gocloc --not-match-d='(vendor|node_modules|\.git)' .

-------------------------------------------------------------------------------
Language                     files          blank        comment           code
-------------------------------------------------------------------------------
Go                             614           8691          31925          81751
HTML                            87            684            417          11505
Markdown                        15           2292            706           4724
JSON                             4              0              0           3933
JavaScript                       6            199            287           2103
CSS                              1             98           1039            808
Makefile                         1             34             59            249
YAML                             1              0             19            149
Bourne Shell                     3             19             63             85
XML                              1              0              0             84
BASH                             1              8             25             79
Plain Text                       2              1              0              4
-------------------------------------------------------------------------------
TOTAL                          736          12026          34540         105474
-------------------------------------------------------------------------------