# PhysSandbox

> PhysSandbox (https://physandbox.com) offers 603+ free interactive physics, chemistry, math, and engineering simulators that run in the browser — no installation or sign-up. Each simulator page explains the underlying physical model, its key equations and assumptions, includes an FAQ, and lets the user vary parameters and watch real-time graphs. Content is available in English (no URL prefix) and Russian (/ru prefix).

Every simulator page is self-contained: model overview, governing equations in plain text, intended audience, key terms, and frequently asked questions. Pages are server-rendered — all educational text is present in the HTML without JavaScript.

## Simulator categories

- [Classical Mechanics](https://physandbox.com/mechanics): Projectile motion, forces, energy, collisions, and rotational dynamics (85 simulators)
- [Waves & Sound](https://physandbox.com/waves): Wave propagation, interference, Doppler effect, and sound visualization (37 simulators)
- [Electricity & Magnetism](https://physandbox.com/electricity): Electric fields, circuits, Ohm's law, and electromagnetic induction (65 simulators)
- [Optics & Light](https://physandbox.com/optics): Reflection, refraction, lenses, prisms, and diffraction patterns (50 simulators)
- [Gravity & Orbits](https://physandbox.com/gravity): Orbital mechanics, solar system, tidal forces, and N-body simulations (27 simulators)
- [Astronomy & The Sky](https://physandbox.com/astronomy): Moon phases, eclipses, sky geometry, stars, spectra, cosmology sketches, and comets (51 simulators)
- [Thermodynamics](https://physandbox.com/thermodynamics): Gas laws, heat transfer, entropy, and engine cycles (46 simulators)
- [Biophysics, Fluids & Geoscience](https://physandbox.com/earth-life-fluids): Neural excitation, porous flow, atmospheric and ocean dynamics, and solid-Earth wave cartoons (44 simulators)
- [Math Visualization](https://physandbox.com/math): Vectors, trigonometry, graphing, Fourier series, and fractals (85 simulators)
- [Engineering](https://physandbox.com/engineering): Logic gates, bridge building, gears, and structural analysis (48 simulators)
- [Chemistry](https://physandbox.com/chemistry): Molecules, periodic table, reactions, pH scale, and titration (65 simulators)

## Virtual labs

Guided lab workflows (goal → procedure → measurements → error analysis → conclusion) built on top of the simulators: [Virtual Labs index](https://physandbox.com/lab).

- [Determining g with a Pendulum](https://physandbox.com/lab/pendulum-gravity): Measure the period of a simple pendulum at several lengths, fit T² vs L, and recover the local gravitational acceleration.
- [Hooke's Law: Spring Constant](https://physandbox.com/lab/spring-hookes-law): Apply known forces to a horizontal spring model; measure extension x, linear fit F vs x, read stiffness k.
- [Coefficient of Static Friction (Critical Angle)](https://physandbox.com/lab/inclined-plane-static-friction): Tilt a model ramp until the block slips; record the critical angle θ and estimate μ_s ≈ tan θ from several trials, then compare to the reference value.
- [Coefficient of Kinetic Friction (From Acceleration)](https://physandbox.com/lab/inclined-plane-kinetic-friction): Slide a block down a rough ramp at a known angle; measure the along-ramp acceleration a and recover μ_k = tan θ − a/(g cos θ), averaged over several runs.
- [Archimedes' Principle: Density of a Solid (Floating Sphere)](https://physandbox.com/lab/archimedes-density): A mystery sphere floats in water of known density. Read the submerged volume fraction at equilibrium and estimate the sphere’s density from ρ = f·ρ_w, averaged over noisy readings.
- [Ohm's Law: Finding an Unknown Resistance (V–I)](https://physandbox.com/lab/ohms-law-resistor): Sweep the supply voltage across a fixed mystery resistor; record (I, V) pairs with small instrument noise and recover R as the slope of the linear fit V versus I.
- [Focal Length of a Converging Thin Lens](https://physandbox.com/lab/thin-lens-focal-length): Place a real object at several distances d_o along the optical axis; read noisy image distances d_i and recover the unknown focal length f from the Gaussian lens law, averaged over trials.
- [Kirchhoff's Current Law at a Series Junction (I₁ = I₂)](https://physandbox.com/lab/kirchhoff-kcl-series): A fixed two-resistor series branch carries one unknown loop current. Record noisy ammeter readings I₁ and I₂ for several supply voltages; a linear fit of I₂ versus I₁ should recover unit slope — KCL at the intermediate node.
- [Speed of Sound from an Open–Open Resonance Tube](https://physandbox.com/lab/resonance-tube-speed-of-sound): A fixed air column open at both ends; the model hides the true speed of sound. Tune to harmonic n, read noisy resonance frequencies fₙ, recover v from v = 2 fₙ L / n and average.
- [Spring–Mass Period versus Mass (T² vs m)](https://physandbox.com/lab/spring-mass-period-vs-m): A fixed horizontal spring with unknown stiffness k; vary the attached mass m, record the small-amplitude period T with timing noise, and recover k from a linear fit of T² versus m.
- [RC Time Constant (Discharge τ = RC)](https://physandbox.com/lab/rc-circuit-tau): Discharge a capacitor through a known resistor. Record (t, V) and fit ln V versus time to get τ = RC.
- [RL Time Constant (τ = L/R on Rise)](https://physandbox.com/lab/rl-circuit-tau): Step the voltage on a series RL. Record i(t) and fit ln(1−i/i∞) vs t; slope = −1/τ, τ = L/R.
- [Snell's Law: Index of Refraction n₂](https://physandbox.com/lab/snell-refractive-index): Measure θ₁ in air and θ₂ in glass; n₁ sin θ₁ = n₂ sin θ₂, with n₁ = 1, fit sin θ₂ vs sin θ₁ with slope 1/n₂.
- [Malus's Law: I = I₀ cos²θ](https://physandbox.com/lab/malus-cos2-law): Rotate a polarizer; record I(θ) and fit I versus cos²θ. The slope is I₀ in arbitrary units.
- [Diffraction Grating: Wavelength from sin θ = mλ/d](https://physandbox.com/lab/grating-wavelength): For a known line density 1/d, record diffraction orders m and sin θ, fit sin θ vs m; λ = d·(slope).
- [Calorimetry: Specific Heat of a Metal](https://physandbox.com/lab/calorimetry-specific-heat): Known masses: hot metal (m_m) in cool water (m_w). Read equilibrium T_f with thermometer noise; c_m is computed from the energy balance.
- [Boyle's Law: PV at Constant T](https://physandbox.com/lab/boyles-law-pv): Isothermal: PV = nRT. Record P and V; fit or mean the product PV.
- [Charles / Gay-Lussac: V ∝ T at Constant p](https://physandbox.com/lab/charles-law-vt): At fixed p, V/T = nR/p = const. Record V, T, compute V/T, take mean or fit.
- [Range vs Angle: g from R = (v₀²/g) sin 2θ](https://physandbox.com/lab/projectile-range-g): With fixed v₀, measure horizontal range R(θ) on flat ground, fit y = R vs x = sin2θ, slope = v₀²/g.
- [1D Collisions: Coefficient of Restitution e](https://physandbox.com/lab/collision-coefficient-restitution): One mass at rest, measured v₁′ and v₂′, e = −(v₂′−v₁′)/v₁.
- [Foucault Precession: Ω = Ω_E sin|λ|](https://physandbox.com/lab/foucault-angular-precession): Set latitude; read the effective precession rate; compare to Ω_E sin|λ|.
- [Planck's Constant h from the Photoelectric Effect](https://physandbox.com/lab/planck-constant-photoelectric): V_stopping(f): slope dV/df = h/e. Fit V vs f; h = e·slope.
- [Decay Constant from ln N(t)](https://physandbox.com/lab/radioactive-decay-constant): N(t)=N₀e^(−λt), lnN = lnN₀ − λt, T½ = ln2/λ.
- [Wave Speed on a String: f_n = nv/(2L)](https://physandbox.com/lab/standing-waves-string-velocity): Record harmonics f_n, L fixed; v = 2f_n L/n, mean v.
- [μ₀ from Long Straight Wire: B = μ₀I/(2πr)](https://physandbox.com/lab/biot-savart-mu0): At fixed r, measure B for several I. Slope of B vs I is μ₀/(2πr).
- [Wheatstone Bridge: Unknown R_x at Balance](https://physandbox.com/lab/wheatstone-balance-rx): At null galvanometer, R_x = R2·R3/R1. Rebalance with different R1; mean R_x.
- [Solar Cell: η = P_mpp / P_in](https://physandbox.com/lab/solar-cell-efficiency-lab): From I–V sweeps, read P_mpp, divide by incident P_in; average η over light levels.

## All simulators

### Classical Mechanics

- [Vortex Ring (Smoke Ring)](https://physandbox.com/mechanics/vortex-ring): Self-propelling toroidal vortex shown as a 2D cross-section: counter-rotating Lamb–Oseen pair drifts at V_self ≈ Γ/(4πa).
- [Rayleigh–Taylor Instability](https://physandbox.com/mechanics/rayleigh-taylor): Heavy fluid above light: Fourier modes grow as exp(σt) with σ = √(A g k); mushrooming fingers reach down.
- [Faraday Waves](https://physandbox.com/mechanics/faraday-waves): Vertical parametric drive of a thin liquid layer: Mathieu sub-harmonic instability locks stripes, squares or hexagons.
- [Airfoil Streamlines (Joukowski)](https://physandbox.com/mechanics/airfoil-streamlines): Potential flow over a Joukowski airfoil: stream-function bands with the Kutta condition fixing circulation.
- [Projectile Motion](https://physandbox.com/mechanics/projectile-motion): Launch projectiles with adjustable angle, velocity, and gravity. Trace parabolic paths with real-time graphs.
- [Trebuchet](https://physandbox.com/mechanics/trebuchet): Counterweight-driven beam: torque launches a projectile at a chosen release angle. Explore range vs masses and arm lengths.
- [Chain Sliding Off Table](https://physandbox.com/mechanics/chain-sliding-off-table): Uniform chain on a smooth edge: hanging weight pulls, friction on the tabletop resists. Watch when it slips and how acceleration grows with s.
- [Block Stack & Friction](https://physandbox.com/mechanics/block-stack-friction): Pull the bottom block in a vertical stack: spring-like links with friction caps show who slips — compare with rigid-stack estimates.
- [Bicycle Stability (2D)](https://physandbox.com/mechanics/bicycle-stability): Side view: roll dynamics with fork trail and gyroscopic wheel torques — see speed, trail, and ω = v/R vs lean.
- [Spring Pendulum](https://physandbox.com/mechanics/spring-pendulum): 2D elastic pendulum: swing and spring stretch together — energy swaps between modes; try chaotic-looking presets.
- [Bridge Resonance (1-D mode)](https://physandbox.com/mechanics/bridge-resonance): Damped modal oscillator with harmonic drive: sweep ω near √(k/m) — presets for cadence-like and low-ζ peaks.
- [Rising Bubble (Archimedes)](https://physandbox.com/mechanics/bubble-rise-archimedes): Ideal-gas bubble expands as it rises (p = p_atm + ρgy); buoyancy vs weight + Stokes drag — depth and velocity graphs.
- [Belt Drive & Slip](https://physandbox.com/mechanics/belt-drive-slip): Two pulleys: tension difference limited by μ and wrap θ (e^{μθ}); load torque vs τ_max and simple slip on ω₂.
- [Free Fall](https://physandbox.com/mechanics/free-fall): Drop objects of different masses with optional air resistance. Prove Galileo right.
- [Uniform vs Accelerated Motion](https://physandbox.com/mechanics/uniform-vs-accelerated): Compare constant velocity and accelerating objects side by side.
- [Relative Motion](https://physandbox.com/mechanics/relative-motion): Boat crossing a river, plane in wind. Vector addition visualization.
- [Circular Motion](https://physandbox.com/mechanics/circular-motion): Object on a string with centripetal acceleration and force vectors.
- [Force Diagram Builder](https://physandbox.com/mechanics/force-diagram): Place objects, add force vectors, see net force and resulting acceleration.
- [Newton's Cradle](https://physandbox.com/mechanics/newtons-cradle): Conservation of momentum and energy visualized with swinging balls.
- [Inclined Plane](https://physandbox.com/mechanics/inclined-plane): Adjust angle and friction. See force components and resulting motion.
- [Inclined Plane: Work & η](https://physandbox.com/mechanics/inclined-plane-work): W = F·s, friction and gravity work, ΔU, efficiency η = ΔU/W_F; slide-down preset.
- [Friction Simulator](https://physandbox.com/mechanics/friction): Static vs kinetic friction with adjustable coefficient.
- [Slide to Stop](https://physandbox.com/mechanics/slide-to-stop): Initial velocity on a rough table: constant μ_k g decel, time and distance to rest.
- [Atwood Machine](https://physandbox.com/mechanics/atwood-machine): Two masses on a pulley. Adjust masses to see acceleration and tension.
- [Elevator Physics](https://physandbox.com/mechanics/elevator-physics): Person on a scale in an elevator. See apparent weight change.
- [Energy Conservation](https://physandbox.com/mechanics/energy-conservation): Ball on a roller coaster track with real-time KE, PE, and total energy bars.
- [1D Force Field](https://physandbox.com/mechanics/force-field-1d): U(x) presets, F = −U′, bead on potential, phase (x,v) and E(t).
- [1D Collisions](https://physandbox.com/mechanics/1d-collisions): Elastic and inelastic collisions with momentum and energy tracking.
- [2D Collisions](https://physandbox.com/mechanics/2d-collisions): Billiard-ball style collisions with adjustable angles.
- [Torque & Balance](https://physandbox.com/mechanics/torque-balance): Beam on a fulcrum. Place weights to balance or tip.
- [Lever Classes](https://physandbox.com/mechanics/lever-classes): 1st / 2nd / 3rd class: arms, τ about fulcrum, MA vs Torque & Balance.
- [Blocks & Tackle](https://physandbox.com/mechanics/pulley-blocks): n supporting strands, same rope tension T, ideal MA = n, F = T.
- [Simple Pendulum](https://physandbox.com/mechanics/simple-pendulum): Adjust length, mass, and gravity. Observe period and damping effects.
- [Conical Pendulum](https://physandbox.com/mechanics/conical-pendulum): Steady cone: ω(θ,L), T and mg vectors, T_rev vs simple-pendulum T₀.
- [Physical Pendulum (Rod)](https://physandbox.com/mechanics/physical-pendulum): Thin uniform rod: pivot along L, I and T(δ), equivalent length L_eq.
- [Pendulum Collision](https://physandbox.com/mechanics/pendulum-collision): Two bobs: hit along normal, e elastic; θ¨ between hits vs 1D collisions.
- [Yo-Yo Dynamics](https://physandbox.com/mechanics/yoyo): Unwinding string: a = g/(1+I/mr²), T, α, optional friction torque τ.
- [Satellite Yo-Yo Despin](https://physandbox.com/mechanics/satellite-yoyo-despin): L = const: ω_f = ω_0(I+2mr_i²)/(I+2mL²) as tethers pay out from r_i to L.
- [Rubber Sheet & Ball](https://physandbox.com/mechanics/rubber-sheet-gravity): Sheet height ∝ −Σm/r; ball rolls along −∇h — embedding metaphor, not GR.
- [Slinky Drop (Springs)](https://physandbox.com/mechanics/slinky-drop): 1D chain: release top anchor; bottom lags until stress wave — toy masses & k.
- [Huygens Pendulum Sync](https://physandbox.com/mechanics/huygens-pendulum-sync): Two pendula + κ(θ₁−θ₂) on shared beam; phases drift toward lock.
- [Ballistic Pendulum](https://physandbox.com/mechanics/ballistic-pendulum): Bullet hits block: embedded vs e; ω₀ = v/L, θ_max, energy graph.
- [Center of Mass System](https://physandbox.com/mechanics/center-of-mass-system): 2–4 bodies or rod: R_cm, V_cm; explosion with ΣΔp = 0; |P| and |V_cm| graphs.
- [Car on a Curve](https://physandbox.com/mechanics/car-on-curve): Flat: v_max = √(μgR), F_c vs μmg. Banked ideal: tan θ = v²/(gR). Top view + wedge.
- [Quarter-Car Suspension](https://physandbox.com/mechanics/quarter-car-suspension): ¼-vehicle vertical model: sprung vs unsprung masses, Kₛ, Cₛ, tire Kₜ, sinusoidal road — RK4 time histories.
- [Capstan (Rope on Cylinder)](https://physandbox.com/mechanics/capstan-rope): T₂ = T₁ e^{μφ}: μ, wrap angle φ, top view + T₂/T₁ vs φ graph.
- [Spring Work (F–x)](https://physandbox.com/mechanics/spring-work-graph): F = kx graph, shaded W = area = ½kx²; U(x) curve vs Spring-Mass dynamics.
- [Double Pendulum](https://physandbox.com/mechanics/double-pendulum): Mesmerizing chaotic motion with path tracing.
- [Spring-Mass System](https://physandbox.com/mechanics/spring-mass): Simple harmonic motion with adjustable spring constant and damping.
- [Coupled Oscillators](https://physandbox.com/mechanics/coupled-oscillators): Two masses, three springs: normal modes ωₛ, ωₐ and beats.
- [Forced Oscillator](https://physandbox.com/mechanics/forced-oscillator): Driven damped harmonic oscillator: transients, resonance curve A(ω).
- [Buoyancy Simulator](https://physandbox.com/mechanics/buoyancy): Drop objects in water. Float or sink based on density.
- [Bernoulli Flow](https://physandbox.com/mechanics/bernoulli): Pipe narrows: speed up, pressure down. Continuity + Bernoulli.
- [Hydraulic Press](https://physandbox.com/mechanics/hydraulic-press): Pascal: same pressure, large piston → large force. F₂ = F₁·A₂/A₁.
- [Rocket Propulsion](https://physandbox.com/mechanics/rocket-propulsion): Variable mass: thrust ṁu, Tsiolkovsky Δv, vertical launch with gravity.
- [Angular Momentum](https://physandbox.com/mechanics/angular-momentum): Two masses on a rod: I = 2mr², change r or m and watch ω adjust to keep L constant.
- [Rolling & Sliding Disk](https://physandbox.com/mechanics/rolling-disk): No-slip v = ωR vs sliding: translational vs rotational KE, disk vs hoop inertia.
- [Gyroscope Precession](https://physandbox.com/mechanics/gyroscope-precession): Gravity torque τ = mgd, spin L = Iω, steady precession Ω ≈ τ/L — schematic 3D view.
- [Coriolis Effect](https://physandbox.com/mechanics/coriolis-effect): Puck on a rotating platform: curved path in the rotating frame vs straight line inertial.
- [Wall Bounce](https://physandbox.com/mechanics/wall-bounce): 2D box: frictionless walls, restitution e, optional gravity. Trails and kinetic energy graph.
- [Oblique Wall Impact](https://physandbox.com/mechanics/oblique-wall-bounce): One tilted segment: e, angles to normal n, open boundaries vs closed Wall Bounce.
- [Rutherford Scattering](https://physandbox.com/mechanics/rutherford-scattering): Repulsive 1/r² orbit sketch vs impact parameter and energy.
- [Ideal Line Vortex](https://physandbox.com/mechanics/ideal-vortex): v_θ = Γ/(2πr) arrow field; circulation and potential flow cartoon.
- [Forearm Lever (class 3)](https://physandbox.com/mechanics/forearm-lever): Elbow fulcrum, load at hand, muscle moment arm — torque estimate.
- [Communicating Vessels & Manometer](https://physandbox.com/mechanics/communicating-vessels): Hydrostatic balance, U-tube ΔP = ρgΔh, and inclined-tube vertical head.
- [Simple Machines](https://physandbox.com/mechanics/simple-machines): Wedge, wheel & axle, and screw: ideal mechanical advantage vs geometry.
- [Arch & Wedge Statics](https://physandbox.com/mechanics/arch-statics): Compressive thrust in a stone arch and wedge force resolution (schematic).
- [Boids (Flocking)](https://physandbox.com/mechanics/boids-flocking): Separation, alignment, cohesion on a torus; cursor as predator cue.
- [Particle Life](https://physandbox.com/mechanics/particle-life): Six types, pairwise matrix forces on a torus — clusters, worms, foam presets.
- [Poincaré Section (Double Pendulum)](https://physandbox.com/mechanics/poincare-double-pendulum): (θ₁, ω₁) when sin θ₂ crosses 0 with ω₂>0; RK4, chaotic return map.
- [Catenary Cable](https://physandbox.com/mechanics/catenary-cable): Uniform chain between level anchors: y ∝ cosh(x/a); sag, arc length, tension directions.
- [Magnus Effect (Ball)](https://physandbox.com/mechanics/magnus-effect): Same v₀ and θ with vs without spin: toy a = (kωv_y, −g − kωv_x); range comparison.
- [Fluid Surface (Accel / Spin)](https://physandbox.com/mechanics/fluid-surface-accel): Linear tank: tan α = a/g; rotating bucket: paraboloid sketch vs rpm.
- [Water Hammer (1D)](https://physandbox.com/mechanics/water-hammer): Linearized P,V waves; valve closes; Joukowsky ΔP ≈ ρaV hint.
- [Foucault Pendulum (Sketch)](https://physandbox.com/mechanics/foucault-pendulum): Ω_eff = Ω_E sin|λ| with time scale; top view rotating swing line.
- [Tippe Top (Schematic)](https://physandbox.com/mechanics/tippee-top): Offset COM, friction at rim, spin — qualitative flip when μ is high enough vs spin.
- [Kapitza Pendulum](https://physandbox.com/mechanics/kapitza-pendulum): Pivot shakes vertically: fast driving can stabilize the inverted equilibrium — parametric pumping in θ̈ + (g/L) sin θ ≈ (Aω²/L) cos(ωt) sin θ.
- [Rattleback (Celt)](https://physandbox.com/mechanics/rattleback-celt): Chiral hull couples spin and rocking in a toy ODE: one rotation sense reverses after growing wobble.
- [Maxwell Wheel](https://physandbox.com/mechanics/maxwell-wheel-disk): Disk on a thin axle unwinds string: shared linear and rotational kinetic energy from falling potential, a ≈ g / (1 + I/(mr²)).
- [Brachistochrone Race](https://physandbox.com/mechanics/brachistochrone-race): Beads race down a line, a circular arc, and a cycloid with v = √(2gy); the cycloid wins the descent-time minimum.
- [Coupled Pendulum Chain](https://physandbox.com/mechanics/coupled-pendulum-chain): N pendula with neighbor springs in θ: watch traveling waves and reflections after a center kick.
- [Sliding Ladder](https://physandbox.com/mechanics/sliding-ladder): Frictionless wall and floor: θ̈ = −(3g/2L) cos θ; released from rest, wall contact is lost when sin θ = (2/3) sin θ₀.
- [Angle of Repose](https://physandbox.com/mechanics/angle-of-repose): 1D sandpile lattice: avalanches relax until slopes stay below μ = tan α — compare the green guide line.
- [Cart & Inverted Pendulum](https://physandbox.com/mechanics/cart-inverted-pendulum): Horizontal force on a cart with a hinged rod: coupled (M+m)ẍ and mL²θ̈ integrated for manual balancing.
- [Woodpecker Toy](https://physandbox.com/mechanics/woodpecker-toy): Stick–slip sleeve on a rod: static hold, spring loading, then kinetic slip — a qualitative hop cycle.

### Waves & Sound

- [Acoustic Levitation (Schematic)](https://physandbox.com/waves/acoustic-levitation): Standing wave cos(kx)cos(ωt); pressure nodes marked; bead cartoon near a node.
- [Acoustic Phased Array (Line)](https://physandbox.com/waves/acoustic-phased-array): N sources, spacing d/λ and phase ramp: steer audio/sonar-like beams (array factor sketch).
- [Chladni Figures](https://physandbox.com/waves/chladni-plate): Plate mode sin(mπx)sin(nπy); nodal contrast + grains drift to nodes (model).
- [Cymatics: Circular Membrane](https://physandbox.com/waves/cymatics-membrane): Drum eigenmodes J_m(k_{mn}r) cos(mθ); angular m, radial n, shimmer.
- [Echo & Echosounder](https://physandbox.com/waves/echo-sounder): Round-trip time t = 2d/v; pulse to a wall and back with adjustable sound speed.
- [Wave Speed: String vs Rod](https://physandbox.com/waves/wave-speed-media): v = √(T/μ) for a string vs v ≈ √(E/ρ) for longitudinal bar waves.
- [LC Oscillator (Undamped)](https://physandbox.com/waves/lc-oscillator): Ideal series LC: q(t), I(t), ω₀ = 1/√(LC); U_C + U_L constant; vs RLC AC.
- [Larsen Effect (Feedback Loop)](https://physandbox.com/waves/larsen-feedback): Mic + speaker + delay: loop gain and saturation in a toy discrete feedback model.
- [Duffing Oscillator](https://physandbox.com/waves/duffing-oscillator): m x¨+cx˙+kx+k₃x³=F cos ωt; soft/hard spring; A(ω) scan vs IC.
- [Wave Packet & Dispersion](https://physandbox.com/waves/wave-packet-dispersion): Superpose cos(kx−ωt); ω=ck+αk²; spreading vs Wave on String PDE.
- [Mach Cone (Schematic)](https://physandbox.com/waves/mach-cone): M = v/c > 1: Huygens pulses + cone sin μ = 1/M; not CFD shock.
- [Monochord / Sonometer](https://physandbox.com/waves/monochord-sonometer): f₁ = (1/2L)√(T/μ), harmonics, note label, Pluck; vs Standing Waves shapes.
- [Huygens Principle (Slit)](https://physandbox.com/waves/huygens-principle): Secondary sources on slit; superposition & diffraction sketch; not full optics.
- [Forced Nonlinear Pendulum](https://physandbox.com/waves/forced-nonlinear-pendulum): θ¨+γθ˙+(g/L)sinθ=A cosωt; phase plot; vs Double Pendulum 2-DOF.
- [Seismic P and S (Schematic)](https://physandbox.com/waves/seismic-p-and-s): Longitudinal vs transverse particle motion; v_P, v_S sliders — not layered Earth.
- [Water Wave Dispersion ω(k)](https://physandbox.com/waves/water-wave-dispersion): Shallow k√(gh), deep √(gk), full tanh(kh) — three curves.
- [Tsunami & Shallow Water (1D)](https://physandbox.com/waves/tsunami-shallow-water): Linear η,u on H(x): c = √(gH) drops over shelf; Gaussian uplift impulse.
- [Storm Surge Bathtub + Wind Setup](https://physandbox.com/waves/storm-surge-bathtub): Inverse barometer pressure setup plus wind stress τL/(ρgH) over a shelf; coastal water level and tide timing toy.
- [Tide Constituents & Beats](https://physandbox.com/waves/tide-constituents-beats): M2/S2/K1/O1 harmonic superposition, spring-neap beat cycle, diurnal inequality, and an amphidromic phase cartoon.
- [Kelvin Ship Wake (Cusp Angle)](https://physandbox.com/waves/kelvin-wake): Deep-water wake V: arms at arcsin(1/3) ≈ 19.47° each side of the track (idealized).
- [KdV Solitons (Exact)](https://physandbox.com/waves/kdv-soliton): u_t + 6uu_x + u_xxx = 0; Hirota two-soliton collision or one sech² pulse.
- [Wave on a String](https://physandbox.com/waves/wave-on-string): Shake one end, adjust frequency and amplitude. Standing waves and reflections.
- [Transverse vs Longitudinal](https://physandbox.com/waves/transverse-longitudinal): Side-by-side comparison of wave types with particle motion.
- [Wave Interference](https://physandbox.com/waves/interference): Two sources creating constructive and destructive patterns. 2D ripple tank.
- [Standing Waves](https://physandbox.com/waves/standing-waves): Find harmonics on a string. Nodes and antinodes highlighted.
- [Doppler Effect](https://physandbox.com/waves/doppler-effect): Moving source with pitch change via Web Audio. Wavefront compression visible.
- [Light Doppler & Redshift](https://physandbox.com/waves/light-doppler-redshift): EM vacuum Doppler: f/f₀ and z vs v/c; linear Δλ/λ ≈ v/c vs exact √(1+β)/√(1−β). Ties to spectral lines.
- [Room Reverberation (2D rays)](https://physandbox.com/waves/room-reverb): Specular rays in a shoebox plan; impulse response; RT60 vs Sabine. Absorption per bounce.
- [Active Noise Cancellation (1D)](https://physandbox.com/waves/active-noise-cancellation): Two tones same f: amplitudes and phase; RMS sum vs ideal π phase. Destructive interference demo.
- [Beam Modal Analysis (Euler–Bernoulli)](https://physandbox.com/waves/beam-modal-analysis): First three bending modes: λ from BC; f ∝ (λ/L)²√(EI/μ). Pinned, cantilever, clamped.
- [Group & Phase Velocity](https://physandbox.com/waves/group-phase-velocity): Two-wave beat: ω(k)=ck+αk²; v_g=Δω/Δk vs v_p=ω̄/k̄; envelope vs carrier.
- [Organ Pipe (harmonic series)](https://physandbox.com/waves/organ-pipe): Open–open vs stopped: f_n formulas; pressure shape; table of modes; Web Audio.
- [Sound Wave Visualizer](https://physandbox.com/waves/sound-visualizer): Real-time waveform and frequency spectrum from microphone.
- [Beat Frequency](https://physandbox.com/waves/beat-frequency): Two slightly different frequencies creating audible beats.
- [Resonance Tube](https://physandbox.com/waves/resonance-tube): Open vs closed pipe harmonics. Hear fₙ and see standing pressure wave.
- [Hearing & Loudness (sketch)](https://physandbox.com/waves/hearing-threshold): Qualitative threshold vs frequency; compare to equal-loudness ideas.
- [2D Wave Equation (Membrane)](https://physandbox.com/waves/wave-equation-2d-membrane): Explicit FDM on a clamped rectangle: Gaussian pulse, circular fronts, edge reflections.

### Electricity & Magnetism

- [Cherenkov Radiation Cone](https://physandbox.com/electricity/cherenkov-cone): Charged particle in a medium of refractive index n: spherical wavefronts of phase velocity c/n pile up on a Mach-like cone with half-angle cos θ_c = 1/(βn) once β > 1/n. Animated wavefronts, magenta cone envelope and material presets (water, glass, diamond) explain the blue glow of pool reactors and IceCube/Super-Kamiokande detection.
- [Bar Magnet & Iron Filings](https://physandbox.com/electricity/bar-magnet-filings): Drag-and-drop bar magnet on a card of iron-filing rods; each filing aligns with the local two-pole B field.
- [Biot–Savart Law](https://physandbox.com/electricity/biot-savart): Infinite wire B ∝ 1/r; ring via segment sum; heatmap + probe.
- [Electric Dipole Field (2D)](https://physandbox.com/electricity/electric-dipole-field): ±q on axis: V heatmap, equipotentials, E field lines; formulas.
- [Ideal Op-Amp (feedback)](https://physandbox.com/electricity/ideal-op-amp): Inverting, non-inverting, buffer; sine or DC; optional rail clipping.
- [Van de Graaff Generator](https://physandbox.com/electricity/van-de-graaff): Belt charges a dome; V = Q/C; stylized spark to ground when V_break is exceeded.
- [Kirchhoff's Laws (KCL & KVL)](https://physandbox.com/electricity/kirchhoff-laws): 3-node DC: junction divider + optional R∥V; hints, KCL/KVL, solved currents.
- [Plane EM Wave (vacuum)](https://physandbox.com/electricity/em-plane-wave): E ⊥ B ⊥ k: sin(kz−ωt) fields, Poynting along z; ω = ck (c = 1).
- [DC Motor & Generator](https://physandbox.com/electricity/dc-motor-generator): Coil in B: motor V = IR + kω, generator E = kω into a load — same k, two modes.
- [Gradient-B Drift](https://physandbox.com/electricity/gradient-b-drift): B_z(x,y) with weak gradient; orbit from q(E+v×B) at local B — gyration + drift.
- [Magnetic Mirror (Bottle)](https://physandbox.com/electricity/magnetic-mirror-bottle): Axial B(z) pinch; μ adiabatic invariant, v∥ from −μ ∂B/∂z — Van Allen / mirror sketch.
- [Dipole Radiation Pattern](https://physandbox.com/electricity/dipole-radiation): Time-averaged power ∝ sin² θ in the plane containing the dipole axis (far-field cartoon).
- [Half-Wave Dipole Antenna Pattern](https://physandbox.com/electricity/half-wave-dipole-pattern): Far-field E-plane pattern of a thin λ/2 wire: power ∝ [cos(½π cos θ)/sin θ]², directivity D ≈ 1.64 (~2.15 dBi); optional sin² θ overlay vs a short dipole.
- [Rotating Dipole E-field](https://physandbox.com/electricity/rotating-dipole-field): ±q orbit in the plane: Coulomb superposition arrows — near-field sketch before full radiation.
- [Bode Diagram (RC low-pass)](https://physandbox.com/electricity/bode-diagram): |H| in dB and phase vs log f; f_c = 1/(2πRC) marked — first-order pole intuition.
- [Nyquist Plot (linear systems)](https://physandbox.com/electricity/nyquist-diagram): Open-loop L(jω) in the complex plane vs frequency; critical point −1 and distance teaching aid.
- [DC–DC Buck / Boost](https://physandbox.com/electricity/dcdc-buck-boost): Ideal CCM averages: buck V_out ≈ D·V_in, boost V_out ≈ V_in/(1−D); schematic + ripple cartoon.
- [Buck Converter Ripple](https://physandbox.com/electricity/buck-converter): Step-down DC-DC converter: duty cycle, inductor ripple, CCM/DCM boundary, capacitor ESR ripple, and switching waveform sketch.
- [Boost Converter](https://physandbox.com/electricity/boost-converter): Step-up DC-DC converter: Vout = Vin/(1−D), inductor current ripple, MOSFET/diode states, CCM/DCM hint, and output ripple.
- [Solar Cell I–V & MPP](https://physandbox.com/electricity/solar-panel-iv): Single-diode cell: photocurrent vs irradiance, ideality n, temperature; I(V), power P(V), maximum-power point.
- [Battery Thevenin (SOC)](https://physandbox.com/electricity/battery-thevenin-model): V_oc(SOC), R_int(SOC), Coulomb-counted SOC vs charge/discharge current; terminal voltage trace.
- [Skin Depth & Proximity Effect](https://physandbox.com/electricity/skin-effect): δ = √(2/(ωμσ)), current crowding in round conductors, proximity effect, and estimated AC resistance vs frequency.
- [Debye Shielding in a Plasma](https://physandbox.com/electricity/debye-shielding): Test charge in an electron plasma: Yukawa potential φ ∝ e^{-r/λ_D}/r, λ_D = √(ε₀k_BT/n_e e²), 2D map and λ_D vs T, n.
- [Langmuir Plasma Oscillations](https://physandbox.com/electricity/langmuir-oscillations): Collective electron oscillations at ω_p = √(n_e e²/ε₀m_e): animated δn wave, Bohm–Gross dispersion ω(k), and ω_p vs density.
- [Alfvén Wave on a Magnetic Field Line](https://physandbox.com/electricity/alfven-wave): Transverse MHD Alfvén wave: v_A = B/√(μ₀ρ), animated field-line displacement, standing modes and pulse reflections at fixed boundaries.
- [Magnetic Reconnection (Sweet–Parker)](https://physandbox.com/electricity/magnetic-reconnection): Toy Sweet–Parker X-point: opposite B-regions, current sheet, inflow v_in ≈ v_A/√S and outflow, Lundquist number S and sheet thickness δ/L.
- [Parker Spiral Solar Wind](https://physandbox.com/electricity/parker-spiral-solar-wind): Rotating Sun plus radial solar wind: interplanetary B spirals with tan ψ = Ωr/v_sw, animated field lines (top view), ψ(r) and |B| falloff.
- [Radiation Belts Drift Shells](https://physandbox.com/electricity/radiation-belts-drift): Van Allen belts: gradient and curvature drift on dipole L-shells, mirror latitude, equatorial loss cone α_lc(L); electrons west, protons east.
- [Eddy Current Tube](https://physandbox.com/electricity/eddy-current-tube): Magnet fall: dv/dt = g in air vs g − k v in copper pipe (toy drag model).
- [Plasma Ball (Stylized)](https://physandbox.com/electricity/plasma-ball-stylized): Glass sphere, center electrode, streamers toward cursor — visual only.
- [PN Junction & Diode](https://physandbox.com/electricity/pn-junction-diode): Band cartoon and Shockley-style I(V); ideality, temperature, I₀ scale.
- [TDR Transmission-Line Pulses](https://physandbox.com/electricity/transmission-line-reflection): Voltage step into a cable: open/short/mismatch reflections, Γ, source re-reflections, time-of-flight, and distance from echo.
- [Phased Array](https://physandbox.com/electricity/phased-array-antenna): Uniform linear array factor vs steering phase and element spacing.
- [Yagi–Uda Antenna](https://physandbox.com/electricity/yagi-uda-antenna): Reflector, driven element, and directors: teaching pattern plus gain vs number of directors (schematic model).
- [Smith Chart & Matching](https://physandbox.com/electricity/smith-chart): Click to set normalized Z; series L/C along constant-r arcs and shunt L/C via Y = 1/Z; Γ, SWR, matching trail.
- [VSWR & Standing Wave (Transmission Line)](https://physandbox.com/electricity/vswr-standing-wave): Complex Z_L → Γ and VSWR; voltage magnitude envelope and animated Re{V} along a lossless line.
- [Rectangular Waveguide TE₁₀](https://physandbox.com/electricity/waveguide-te10): Dominant mode fields (E_y, H_x, H_z), cutoff f_c = c/(2a√ε_r), β and guide wavelength λ_g.
- [Compton Scattering](https://physandbox.com/electricity/compton-scattering): Photon wavelength shift vs angle; λ_C ≈ 2.426 pm.
- [Electric Field Visualizer](https://physandbox.com/electricity/electric-field): Place charges and watch the E arrow grid, streamlines, and equipotentials update in real time.
- [Electric Potential](https://physandbox.com/electricity/electric-potential): 2D heatmap V = Σ kq/r. Drag charges, see equipotential colors.
- [Coulomb's Law](https://physandbox.com/electricity/coulombs-law): Two charges with adjustable magnitude and distance. Force vectors shown.
- [Parallel-Plate Capacitor](https://physandbox.com/electricity/parallel-plate-capacitor): C = ε₀εᵣA/d, E = V/d, Q and energy; plates and C vs d.
- [Three-Phase AC Waveforms](https://physandbox.com/electricity/three-phase): Phase and line voltages, phasors, star/delta relation, line current, power factor, and P/Q/S power triangle.
- [Hall Effect](https://physandbox.com/electricity/hall-effect): U_H = R_H I B / t; R_H sign for e⁻ vs holes; B, I, n, t.
- [Photoelectric Effect](https://physandbox.com/electricity/photoelectric): E = hc/λ vs φ, K_max, V_s; I vs λ and vs intensity (model).
- [Particle in E and B Fields](https://physandbox.com/electricity/particle-em-field): Lorentz force: cyclotron motion, E×B drift, trajectories in sim units.
- [Circuit Builder](https://physandbox.com/electricity/circuit-builder): Drag and drop components: battery, resistor, bulb, switch. See current flow.
- [Ohm's Law](https://physandbox.com/electricity/ohms-law): Adjust voltage and resistance. See current change with interactive V-I graph.
- [Series & Parallel](https://physandbox.com/electricity/series-parallel): Compare total resistance and current distribution side by side.
- [RC Circuit](https://physandbox.com/electricity/rc-circuit): Charging and discharging capacitor with exponential curves.
- [RC Filter (LP / HP)](https://physandbox.com/electricity/rc-filter): Bode gain vs frequency, f_c at −3 dB; live sine V_in and V_out.
- [RL Circuit](https://physandbox.com/electricity/rl-circuit): Series RL: τ = L/R, i(t) rise & decay, v_L; DC steady i = V/R.
- [RLC Series (AC)](https://physandbox.com/electricity/rlc-circuit): Resonance peak, |Z|, phase. I(f) curve and live V & I waves.
- [Resistor Color Code](https://physandbox.com/electricity/resistor-color-code): Click bands to see resistance, or enter value to see bands.
- [Magnetic Field](https://physandbox.com/electricity/magnetic-field): Bar magnets and current-carrying wires with field line visualization.
- [Electromagnetic Induction](https://physandbox.com/electricity/electromagnetic-induction): Move magnet through coil. Faraday's law visualized.
- [Transformer Equivalent Circuit](https://physandbox.com/electricity/transformer): Turns ratio with magnetizing branch, leakage reactance, copper/core losses, load regulation, and efficiency versus load.
- [Induction Motor Torque-Speed Curve](https://physandbox.com/electricity/induction-motor-torque-speed): Slip, synchronous speed, breakdown torque, starting current, stable operating point, and V/f control sketch.
- [Mutual Inductance](https://physandbox.com/electricity/mutual-inductance): Coupled L₁, L₂ with k and M; sinusoidal primary drives secondary current through R₂.
- [Mass Spectrometer (Sector)](https://physandbox.com/electricity/mass-spectrometer): Velocity selector v = E/B then magnetic sector: different m/q bend to different radii (2D schematic).
- [Resonant Transformer (Tesla-style)](https://physandbox.com/electricity/tesla-resonant-transformer): Coupled RLC tanks; tune f_drive near secondary ω₀ to grow V on high‑Q side (linear coupled ODEs).
- [Wheatstone Bridge](https://physandbox.com/electricity/wheatstone-bridge): Four arms, G between mid nodes; V_B − V_C and null when R₁R₄ = R₂R₃.
- [Dipole in Uniform E](https://physandbox.com/electricity/dipole-uniform-field): τ = −pE sin θ, U = −pE cos θ; damped rotation; optional AC on E.
- [Ferrofluid (Stylized)](https://physandbox.com/electricity/ferrofluid-stylized): Purple metaball pool, spikes, field-line hints — visual only, not MHD.
- [Cyclotron (Schematic)](https://physandbox.com/electricity/cyclotron-schematic): B uniform, oscillating E in gap; spiral growth; ω_c = (q/m)B in sim units.

### Optics & Light

- [Eye: Myopia & Hyperopia](https://physandbox.com/optics/eye-myopia-hyperopia): Reduced eye + glasses; retina blur cue; presets and suggested ΔP.
- [Fresnel vs Fraunhofer](https://physandbox.com/optics/fresnel-fraunhofer): Slit diffraction: N = a²/(λL); Cornu spiral; Fresnel integral vs sinc².
- [Three Polarizers (paradox)](https://physandbox.com/optics/three-polarizers): P₁–P₂–P₃ Malus chain; crossed P₁⊥P₃ plus P₂ at 45° lets light through.
- [Airy Disk & Rayleigh Limit](https://physandbox.com/optics/airy-disk-resolution): Circular aperture Fraunhofer pattern; first dark ring; two-point resolution.
- [Optical Bench (sandbox)](https://physandbox.com/optics/optical-bench): Up to 4 elements: thin lenses, vertical mirrors, wedge δ; paraxial ray trace.
- [Telescope & Microscope (2 lenses)](https://physandbox.com/optics/telescope-microscope): Kepler / Galileo / microscope presets; paraxial rays, M, f_obj/f_eye hint.
- [Whispering Gallery (Rays)](https://physandbox.com/optics/whispering-gallery): Circular mirror: shallow chords refocus acoustic energy opposite the source (geometric optics).
- [Fiber Bragg Grating](https://physandbox.com/optics/fiber-bragg-grating): λ_B = 2 n_eff Λ; Lorentzian toy reflectivity vs λ and probe wavelength.
- [Laser Speckle](https://physandbox.com/optics/speckle-pattern): Random phased waves: |Σ e^{i(k·r+φ)}|² grain pattern (qualitative).
- [Bragg’s Law (X-ray)](https://physandbox.com/optics/bragg-law): 2d sin θ = nλ vs θ; crystal planes and constructive reflection (schematic).
- [Kaleidoscope](https://physandbox.com/optics/kaleidoscope): N-fold rotation + optional mirror: one bead etches a symmetric mandala trail.
- [Coffee-Cup Caustics](https://physandbox.com/optics/coffee-cup-caustics): Parallel beam, circular cup segment, reflected-ray density on a table line.
- [Soap Film (Minimal Surface)](https://physandbox.com/optics/soap-film-minimal): Non-planar wire: discrete Laplace relaxation on z; iridescent mesh — Plateau sketch.
- [CD / Grating Rainbow](https://physandbox.com/optics/cd-grating-spectrum): Lines/mm and incidence: scalar grating orders painted as λ-colored fans.
- [Moiré (Line Gratings)](https://physandbox.com/optics/moire-pattern): Two translucent rulings: spacing, tilt, shift — spatial beats and envelopes.
- [Fiber: Numerical Aperture](https://physandbox.com/optics/fiber-numerical-aperture): NA from n_core, n_clad; acceptance angle and bend loss cue.
- [Fiber Bend & TIR Loss](https://physandbox.com/optics/fiber-bend-loss): Curved core: outer-wall incidence vs θ_c; tight R leaks (meridional sketch).
- [Evanescent Wave (TIR)](https://physandbox.com/optics/evanescent-tir): Beyond θ_c: penetration depth scale in the rarer medium vs λ.
- [Retina & Diffraction](https://physandbox.com/optics/retina-diffraction-limit): Order-of-magnitude θ ~ λ/D and cone spacing scale.
- [Pinhole Camera](https://physandbox.com/optics/pinhole-camera): Camera obscura: similar triangles, image size h_i = h_o · v/u.
- [Rainbow in a Droplet](https://physandbox.com/optics/rainbow-droplet): Snell in a sphere: primary (1× internal reflection ~42°) and secondary (2× ~51°, reversed spectrum); scan b/R and n.
- [Rayleigh Sky (blue)](https://physandbox.com/optics/rayleigh-sky): Scattered intensity ∝ λ⁻⁴; compare blue vs red and qualitative sky gradient.
- [Inferior Mirage (hot road)](https://physandbox.com/optics/inferior-mirage): n(y) gradient above hot asphalt: rays bend — sky light looks like a wet patch.
- [Reflection](https://physandbox.com/optics/reflection): Flat, concave, and convex mirrors with auto-drawn ray diagrams.
- [Refraction](https://physandbox.com/optics/refraction): Light crossing boundaries. Snell's law with angle measurements.
- [Total Internal Reflection](https://physandbox.com/optics/total-internal-reflection): Find the critical angle. Fiber optics analogy.
- [Lens Simulator](https://physandbox.com/optics/lenses): Converging and diverging lenses with ray diagrams and image formation.
- [Simple Eye (Thin Lens)](https://physandbox.com/optics/simple-eye-model): P_total = P_eye + P_glasses; 1/v = P − 1/u vs retina; blur cue.
- [Thin Lens Equation](https://physandbox.com/optics/thin-lens-equation): Hyperbola d_i(d_o), pole at d_o = f, schematic + 1/d check.
- [Prism & Dispersion](https://physandbox.com/optics/prism-dispersion): White light through a prism creating a rainbow spectrum.
- [Thin-Film Interference](https://physandbox.com/optics/thin-film-interference): Wedge fringes: n, d, θ; cos²(δ/2) colors and I(λ) at mid thickness.
- [Newton’s Rings](https://physandbox.com/optics/newton-rings): Spherical lens on flat glass: air-film thickness fringes; optional π phase on reflection.
- [Anti-Reflection Coating](https://physandbox.com/optics/anti-reflection-coating): Air–film–glass at normal incidence: R(λ) from two-interface interference; quarter-wave design.
- [Michelson Interferometer](https://physandbox.com/optics/michelson-interferometer): I(Δ) = V cos²(πΔ/λ); tilt fringes; coherence length envelope.
- [Mach–Zehnder Interferometer](https://physandbox.com/optics/mach-zehnder-interferometer): Two-beam recombination; I ∝ cos²(πΔ/λ) vs one-arm OPD; schematic + fringe plot.
- [Fabry–Pérot Cavity / Etalon](https://physandbox.com/optics/fabry-perot-etalon): Airy transmission T(L), finesse F, free spectral range FSR = λ/(2n), linewidth ΔL ≈ FSR/F; animated cavity-length scan.
- [Gaussian Beam & q-Parameter](https://physandbox.com/optics/gaussian-beam-q): Waist w₀, Rayleigh range z_R, Gouy phase ζ(z), and ABCD propagation through a thin lens; w(z) and q at observation plane.
- [Fourier Optics 4f System](https://physandbox.com/optics/fourier-optics-4f): Object → lens → Fourier plane → spatial filter (low/high-pass) → lens → image; 2D FFT with circular aperture in spectrum.
- [Zernike Wavefront Aberrations](https://physandbox.com/optics/zernike-aberrations): Defocus, astigmatism, coma, spherical Zernike modes on a circular pupil; PSF via FFT, Strehl ratio and Maréchal estimate.
- [Optical Tweezers (Gradient Trap)](https://physandbox.com/optics/optical-tweezers): Gaussian trap U(r), stiffness k ∝ P/w₀², overdamped Langevin bead with Stokes drag and thermal Brownian jitter; ⟨r²⟩ trace.
- [1D Photonic Crystal Bandgap](https://physandbox.com/optics/photonic-crystal-1d): Alternating n₁/n₂ bilayers; TE transfer-matrix T(λ), R(λ); stop bands where |Tr(M_period)/2|>1 vs period Λ.
- [Sagnac (Ring) Interferometer](https://physandbox.com/optics/sagnac-interferometer): Δφ ∝ Ω·A/λ for counter-propagating beams in a rotating loop — optical gyro idea.
- [Brewster Angle](https://physandbox.com/optics/brewster-angle): tan θ_B = n₂/n₁; R_p→0; θᵢ+θₜ=90°; Fresnel R_s, R_p vs θᵢ.
- [Fermat's Principle](https://physandbox.com/optics/fermat-principle): OPL = n₁AP+n₂PB vs hit point; minimum = Snell path.
- [Chromatic Aberration](https://physandbox.com/optics/chromatic-aberration): Cauchy n(λ); thin-lens f(λ); paraxial rays R/G/B.
- [Diffraction](https://physandbox.com/optics/diffraction): Single and double slit with interference patterns.
- [Color Mixing](https://physandbox.com/optics/color-mixing): Additive (RGB) and subtractive (CMY) interactive color mixing.
- [Polarization (Malus)](https://physandbox.com/optics/polarization): Two polarizers. Rotate θ, see I = I₁ cos²θ and extinction.
- [Birefringence (Calcite sketch)](https://physandbox.com/optics/birefringence-calcite): Two scalar Snell paths for n_o and n_e — double refraction cartoon (not full optic-axis walk-off).
- [Holography (Recording principle)](https://physandbox.com/optics/holography-principle): Toy 2D interference |E_ref + E_obj|²: fringes encode phase; readout diffraction not simulated.

### Gravity & Orbits

- [Schwarzschild Orbit Precession (Rosette)](https://physandbox.com/gravity/schwarzschild-orbit-precession): Schwarzschild geodesic in the φ-form d²u/dφ² + u = 1/L² + 3u² (G = c = M = 1) integrated by RK4. The closed Newtonian ellipse is replaced by an orange precessing rosette with apsidal advance Δφ ≈ 6πM/[a(1 − e²)] per orbit — the same mechanism that produces the historic 43″/century perihelion shift of Mercury. Horizon r = 2M and ISCO r = 6M annotated.
- [ISCO & Photon Sphere (V_eff)](https://physandbox.com/gravity/isco-photon-sphere): Schwarzschild effective potential V_eff(r) for massive (timelike) and photon (null) test particles in geometric units. Sliding angular momentum L collapses the stable / unstable circular pair into the innermost stable circular orbit r_ISCO = 6M (the inner edge of accretion discs); for photons the unstable photon sphere r = 3M defines the inner ring of black-hole shadow images.
- [Einstein Ring & Paczyński Microlensing](https://physandbox.com/gravity/einstein-ring-microlensing): Point-mass thin lens (weak-field GR): lens equation β = θ − θ_E²/θ gives two images θ_± = ½(β ± √(β² + 4θ_E²)) with magnifications μ_± = ½[(u² + 2)/(u√(u² + 4)) ± 1], u = β/θ_E. Animated source transit at impact parameter u₀ over timescale t_E renders the canonical symmetric Paczyński light curve and the full Einstein ring θ_E = √(4GM·D_LS/(c² D_L D_S)) at perfect alignment.
- [Gravitational Wave Binary Chirp (Inspiral)](https://physandbox.com/gravity/gw-binary-chirp): Leading-order post-Newtonian inspiral of a compact binary: f(τ) ∝ τ^(−3/8), strain h(t) ∝ M_c^(5/3) f^(2/3) / D_L. Tune component masses m₁, m₂ and luminosity distance D_L; live h(t) and f(t) traces with the orbiting bodies on the side. The chirp mass M_c = (m₁m₂)^(3/5)/(m₁+m₂)^(1/5) is the very quantity LIGO/Virgo measures from the early inspiral; the frequency freezes at the Schwarzschild ISCO.
- [Shapiro Time Delay (4th GR Test)](https://physandbox.com/gravity/shapiro-time-delay): A radio signal grazing the Sun picks up an excess one-way travel time Δt ≈ (2GM/c³) ln[(r_E + r_E cos α)(r_R + r_R cos β)/b²] on top of the Newtonian light-time. Cassini, Mariner and Viking presets, with the round-trip delay readout in microseconds and an animated bent-photon path against a straight Newtonian baseline. The Cassini 2003 conjunction constrains |γ_PPN − 1| < 2 × 10⁻⁵ — the strongest weak-field GR test to date.
- [Three-Body Figure-Eight](https://physandbox.com/gravity/three-body-figure-eight): Equal masses: Chenciner–Montgomery choreography in 2D (RK4, periodic orbit).
- [Restricted 3-Body (map)](https://physandbox.com/gravity/restricted-three-body-map): CRTBP: escape vs collision vs chaos proxy; μ slider.
- [Multistage Rocket (Tsiolkovsky)](https://physandbox.com/gravity/multistage-rocket): Δv per stage; sum vs single-stage with same total propellant.
- [Orbital Debris & Kessler (toy)](https://physandbox.com/gravity/orbital-debris-kessler): LEO shell: n = N/V, collision rate ∝ N²; optional fragment cascade.
- [Gravity-Assist Fly-By](https://physandbox.com/gravity/gravity-assist-flyby): Planet frame |u_out|=|u_in| rotated by δ; star frame v = V + u — Δ|v| from moving planet.
- [Orbit Simulator](https://physandbox.com/gravity/orbit-simulator): Fixed central mass: tune speed and launch angle for circular, elliptical, or escape trajectories.
- [Solar System](https://physandbox.com/gravity/solar-system): Interactive scaled model with time controls and orbital data.
- [Gravity Sandbox](https://physandbox.com/gravity/gravity-sandbox): Place masses and watch N-body gravitational interactions unfold.
- [Kepler's Laws](https://physandbox.com/gravity/keplers-laws): Elliptical orbits with equal-area sweeps visualized.
- [Escape Velocity](https://physandbox.com/gravity/escape-velocity): Launch from different planets and see trajectory results.
- [Gravitational Lensing](https://physandbox.com/gravity/gravitational-lensing): Massive objects bending light. Visual distortion effects.
- [Lagrange Points L1–L5](https://physandbox.com/gravity/lagrange-points): CRTBP effective potential; L1–L5; Coriolis test particle.
- [Earth–Moon Tides](https://physandbox.com/gravity/earth-moon-tides): Equilibrium tide bulges; orbit speed; ~12.4 h spacing note.
- [Binary Star (circular)](https://physandbox.com/gravity/binary-star): COM orbits; r₁,r₂; Kepler T² ∝ a³/(M₁+M₂).
- [Roche Limit](https://physandbox.com/gravity/roche-limit): Fluid d ≈ 2.456 R_p (ρ_p/ρ_s)^(1/3); vs orbit distance.
- [Space Elevator Tether](https://physandbox.com/gravity/space-elevator): 1D tension vs height; peak near GEO (normalized model).
- [Hohmann Transfer](https://physandbox.com/gravity/hohmann-transfer): Coplanar circles r₁,r₂; transfer ellipse; Δv₁, Δv₂ from vis-viva.
- [Geostationary Orbit](https://physandbox.com/gravity/geostationary-orbit): ω²r = GM/r² from sidereal vs solar day; Earth-fixed view with sub-satellite longitude.
- [Orbital Decay (Atmosphere)](https://physandbox.com/gravity/orbital-decay-atmosphere): Toy perigee drag: shrinking, circularizing ellipse; ISS lifetime intuition.
- [Mercury Perihelion Precession](https://physandbox.com/gravity/mercury-perihelion-precession): GR Δω per orbit vs Newton; ~43″/century readout; amplified animation.
- [Earth–Moon Barycenter Wobble](https://physandbox.com/gravity/earth-moon-barycenter-orbit): Heliocentric path of Earth’s center: barycentric ellipse plus lunar epicycle (exaggerated).
- [Oberth Effect](https://physandbox.com/gravity/oberth-effect): Same prograde Δv at peri vs apo on one ellipse; higher ε when burning deep.

### Astronomy & The Sky

- [Light Clock & Time Dilation](https://physandbox.com/astronomy/light-clock-time-dilation): Two side-by-side light clocks — one at rest, one moving at v = βc — show why a moving clock ticks slower. The photon zig-zag traces a longer Pythagorean hypotenuse cT/2 vs the rest hypotenuse cT₀/2, giving T = γT₀ with γ = 1/√(1 − β²) live. The Pythagorean diagram and the running tick ratio make the special-relativistic time-dilation derivation visual rather than algebraic.
- [Length Contraction (Lorentz)](https://physandbox.com/astronomy/length-contraction): A ruler of proper length L₀ in S′ flies past the lab S at v = βc and is measured to be L = L₀/γ — only in the direction of motion. Animated fly-by with on-board ruler ticks and a dotted L₀ baseline makes the contraction L = L₀ √(1 − β²) immediately readable; same factor explains why GeV cosmic-ray muons reach the ground despite their ≈ 2 μs lifetime.
- [Minkowski Spacetime Diagram (Lorentz Boost)](https://physandbox.com/astronomy/minkowski-spacetime-diagram): Interactive (x, ct) spacetime diagram: a Lorentz boost at v = βc rotates the (x′, ct′) axes inward by atan(β) toward the 45° light cone — relativity of simultaneity, time dilation and length contraction become pure geometry. Click events with Shift / Alt to read the invariant interval Δs² = (cΔt)² − (Δx)² and its time-/light-/space-like classification.
- [Relativistic Doppler Effect](https://physandbox.com/astronomy/relativistic-doppler): Source emits at rest frequency f₀ and moves at v = βc; observer at angle θ measures f_obs = f₀ √(1 − β²) / (1 − β cos θ). Animated lab-frame wavefronts, observer at any angle, and a 380–700 nm spectrum strip showing the apparent colour shift of a 555 nm reference line — including the purely relativistic transverse Doppler (θ = 90°) red-shift f₀/γ.
- [Relativistic Energy–Momentum Hyperbola](https://physandbox.com/astronomy/relativistic-energy-momentum): Energy–momentum relation E² = (pc)² + (mc²)². The green hyperbola E = √(p² + 1) is bracketed by the Newtonian parabola E ≈ 1 + p²/2 (low-momentum limit) and the photon-like asymptote E = pc (ultra-relativistic). Pick β, p or T as the independent slider and watch all four quantities (β, γ, p, T) lock together — the operational core of accelerator and cosmic-ray physics.
- [Pair Production Threshold & σ(E_γ)](https://physandbox.com/astronomy/pair-production-threshold): Pair creation channels γ + nucleus → e⁺e⁻ (Bethe–Heitler, σ ∝ Z²), the higher-threshold triplet γ + e⁻ → e⁻e⁺e⁻, and Breit–Wheeler γγ → e⁺e⁻ that limits TeV photons against the cosmic background. Live threshold marker, log-σ curve, and material presets (H, C, Al, Cu, Pb) make the 1.022 MeV / 4 m_e c² thresholds intuitive.
- [CMB Power Spectrum (Acoustic Peaks)](https://physandbox.com/astronomy/cmb-power-spectrum): Cosmic Microwave Background temperature D_ℓ vs ℓ with Sakharov peaks: tune Ω_b h², Ω_c h², n_s, A_s, τ, h and watch the parity flip between odd / even peaks, the Silk damping tail, and the Sachs–Wolfe plateau move. Pedagogical parametric ΛCDM model.
- [Big Bang Nucleosynthesis (BBN)](https://physandbox.com/astronomy/big-bang-nucleosynthesis): Light-element abundance curves H, ⁴He, D, ³He, ⁷Li vs cosmic time / temperature. Weak freeze-out, neutron decay gap, deuterium bottleneck → Y_p ≈ 0.245. Slide η₁₀ and N_eff over the classic BBN curves; observed values overlaid.
- [EM Calorimeter Shower (Heitler / Rossi)](https://physandbox.com/astronomy/em-shower-heitler): Toy electromagnetic cascade in a calorimeter: γ → e⁺e⁻, e⁻ → e⁻γ each X₀, with critical-energy cutoff E_c. Animate the branching tree, see N_max ≈ E₀/E_c, t_max ≈ log₂(E₀/E_c) live; presets for Pb, Cu, Si, air.
- [Bremsstrahlung & Synchrotron Radiation Pattern](https://physandbox.com/astronomy/bremsstrahlung-synchrotron): Polar plot of dP/dΩ for an accelerating charge: a ∥ v (linear / brems) and a ⊥ v (circular / synchrotron). Pull β = v/c toward 1 — Larmor donut collapses into a forward beam of half-angle ≈ 1/γ; ω_g, ω_c shown for given B.
- [Venus Phases (Galileo)](https://physandbox.com/astronomy/venus-phases-galileo): Heliocentric geometry: full disk vs crescent at inferior conjunction; Earth-view inset.
- [Stellar Aberration](https://physandbox.com/astronomy/stellar-aberration): Bradley: telescope tilt v/c ~ 10⁻⁴ rad; annual ~20.5″ — not parallax.
- [Milankovitch Cycles](https://physandbox.com/astronomy/milankovitch-cycles): e, obliquity, precession → toy high-latitude insolation curve; ice-age pacing context.
- [Supernova Light Curves](https://physandbox.com/astronomy/supernova-light-curve): Schematic Ia rise/decay vs II-P plateau; standard-candle note.
- [Apparent vs Absolute Magnitude](https://physandbox.com/astronomy/apparent-absolute-magnitude): m = M + 5 log₁₀(d/10 pc); distance modulus; flux ratios.
- [Spin–Orbit Resonance](https://physandbox.com/astronomy/spin-orbit-resonance): Moon 1:1 lock vs Mercury 3:2; schematic animations.
- [Pulsar Lighthouse](https://physandbox.com/astronomy/pulsar-lighthouse): Rotating beam cone, pulse profile; timing / ms pulsars context.
- [Meteor Shower & Radiant](https://physandbox.com/astronomy/meteor-shower-radiant): Earth crosses comet debris; radiant on a star field (schematic).
- [Cosmological Expansion (FLRW)](https://physandbox.com/astronomy/cosmological-expansion-flrw): a(t), z, χ and c/H vs time; flat Ω_m + Λ (toy ΛCDM).
- [Hubble Diagram (SN Ia & ΛCDM fit)](https://physandbox.com/astronomy/hubble-diagram-sn-ia): Apparent magnitude vs redshift; flat ΛCDM distance modulus; grid fit for H₀ and Ω_m.
- [Hydrogen Saha Equilibrium (recombination)](https://physandbox.com/astronomy/hydrogen-saha-recombination): n_p n_e / n_H vs T and ionized fraction X; steep Boltzmann knee ~3000 K scale (density dependent).
- [Chandrasekhar limit (Lane–Emden toy)](https://physandbox.com/astronomy/chandrasekhar-limit): White-dwarf M–R track for n = 3/2 cold degenerate electrons; UR n = 3 marks the ~1.44 M☉ scale (μ_e dependent).
- [Neutron star TOV: toy M–R (polytrope)](https://physandbox.com/astronomy/neutron-star-tov-toy): Schwarzschild TOV for polytropes n = 1 or 3/2; K normalized so the ρ_c scan peaks near ~2 M☉ (schematic only).
- [Galaxy Rotation Curve](https://physandbox.com/astronomy/galaxy-rotation-curve): Keplerian decline vs flat v(r); toy halo slider (dark matter motivation).
- [Stellar Life Cycle](https://physandbox.com/astronomy/stellar-lifecycle): Cloud → MS → giant/SN → WD / NS / BH vs initial mass (schematic).
- [Exoplanet Radial Velocity](https://physandbox.com/astronomy/exoplanet-radial-velocity): K from masses & P; sinusoidal V_r(t); M sin i.
- [Exoplanet Transit (light curve)](https://physandbox.com/astronomy/exoplanet-transit): Uniform disk overlap; R_p/R_*; impact b; F(t) vs period.
- [Exoplanet Transit (limb darkening u₁, u₂)](https://physandbox.com/astronomy/exoplanet-transit-limb-darkening): Quadratic limb-darkened disk; grid-integrated F(t); TESS-like solar preset vs uniform-disk page.
- [Sphere of Influence (Hill)](https://physandbox.com/astronomy/sphere-of-influence): r_H ≈ a (m/3M)^(1/3): schematic secondary orbit and Hill radius vs masses and a.
- [Measuring c (ToF toy)](https://physandbox.com/astronomy/speed-of-light-lab): c ≈ 2D/Δt round-trip; schematic path + Fizeau/Foucault context.
- [GPS & Relativity](https://physandbox.com/astronomy/gps-relativity): Weak-field + SR clock drift estimates vs altitude and orbital speed.
- [Nuclear Binding Curve](https://physandbox.com/astronomy/nuclear-binding-curve): Qualitative B/A vs A with fusion/fission context.
- [Seasons & Axial Tilt](https://physandbox.com/astronomy/seasons-axial-tilt): Obliquity ~23.4°: declination model vs day of year and noon sun altitude at latitude.
- [Single-Layer Climate (Toy)](https://physandbox.com/astronomy/single-layer-climate): S, α, ε: T_eff vs T_surface from gray-slab balance — intuition only.
- [Moon Phases](https://physandbox.com/astronomy/moon-phases): Sun–Earth–Moon geometry and synodic cycle; lit fraction from phase angle.
- [Solar Eclipse Geometry](https://physandbox.com/astronomy/solar-eclipse-geometry): Sun–Moon–Earth: angular sizes, umbra and penumbra cones, orbit tilt.
- [Lunar Eclipse Geometry](https://physandbox.com/astronomy/lunar-eclipse-geometry): Sun–Earth–Moon: Earth’s shadow on the Moon; total, partial, and penumbral (schematic colors).
- [Moon: Tidal Locking](https://physandbox.com/astronomy/moon-tidal-lock): Synchronous rotation: same lunar face toward Earth as it orbits.
- [Mars Retrograde Loop](https://physandbox.com/astronomy/mars-retrograde): Earth overtakes Mars: apparent backward loop among the stars (schematic).
- [Sidereal vs Solar Day](https://physandbox.com/astronomy/sidereal-solar-day): Why noon returns after slightly more than 360° of rotation (~4 min shorter sidereal day).
- [Axial Precession](https://physandbox.com/astronomy/axial-precession): Spin axis slowly cones; ~26 kyr cycle changes the pole star (schematic).
- [Ecliptic & Zodiac Band](https://physandbox.com/astronomy/ecliptic-zodiac): Celestial equator vs ecliptic, obliquity, Sun on the yearly path (symbols as map, not astrology).
- [Hertzsprung–Russell Diagram](https://physandbox.com/astronomy/hr-diagram): Schematic HR regions; drag a star to see main sequence, giants, white dwarfs.
- [HR Diagram: Evolutionary Track (Z = 0.02)](https://physandbox.com/astronomy/hr-diagram-evolution): Schematic ~1 M☉ track vs normalized age: MS, RGB, HB hook, AGB, WD cooling; play or scrub u.
- [Spectral Lines & Doppler](https://physandbox.com/astronomy/spectral-doppler): Absorption lines on a continuum shift with radial velocity (Δλ/λ ≈ v/c).
- [Cosmic Distance Ladder](https://physandbox.com/astronomy/cosmic-distance-ladder): Log distance scale with parallax, standard candles, and Hubble-flow cartoon rungs.
- [Black Hole Shadow (Schematic)](https://physandbox.com/astronomy/black-hole-shadow): Silhouette and stylized ring; Rₛ scales with mass — not full GR ray tracing.
- [Sunset Atmospheric Refraction](https://physandbox.com/astronomy/sunset-refraction): Geometric vs apparent horizon: lift of the solar disk from a layered atmosphere model.
- [Comet Orbit, Coma & Tails](https://physandbox.com/astronomy/comet-orbit-tails): Eccentric orbit, coma brightening near the Sun, ion and dust tails, solar wind toggle.
- [Aurora (Stylized)](https://physandbox.com/astronomy/aurora-stylized): Layered sine curtains; hue and drift — look only, not ionosphere physics.
- [Stellar Parallax](https://physandbox.com/astronomy/stellar-parallax): Earth orbit angle vs nearby star wobble on fixed background; π = 1/d(pc) arcsec, exaggerated.

### Thermodynamics

- [Psychrometric Chart](https://physandbox.com/thermodynamics/psychrometric-chart): Moist-air HVAC chart: dry-bulb, RH, humidity ratio, dew point, wet-bulb approximation, enthalpy, and two-stream mixing.
- [Brayton Cycle (Gas Turbine)](https://physandbox.com/thermodynamics/brayton-cycle): PV: isentropic compress, isobaric heat in, isentropic expand, isobaric cool — jet/GT core cartoon.
- [Joule–Thomson Throttling](https://physandbox.com/thermodynamics/joule-thomson-throttle): Isenthalpic expansion: ideal gas ΔT = 0; toy μ_JT inversion for real gases.
- [van der Waals Isotherms](https://physandbox.com/thermodynamics/van-der-waals-isotherms): Reduced (P_r,V_r,T_r) curves; critical point; subcritical wiggle vs Maxwell plateaus (qualitative).
- [Bénard Convection (Rayleigh)](https://physandbox.com/thermodynamics/benard-convection): Heated-from-below layer: Ra vs Ra_c ~1708; schematic hex/roll pattern.
- [Black Body: Planck Spectrum](https://physandbox.com/thermodynamics/black-body-planck): B_λ(λ,T); Wien λ_max ∝ 1/T; Stefan–Boltzmann M = σT⁴ and numeric ∫πB_λ dλ.
- [Rankine Cycle (Steam)](https://physandbox.com/thermodynamics/rankine-cycle): T–s with vapor dome + schematic P–v: pump, boiler, turbine, condenser; x₄ and pressure sliders.
- [Refrigeration Cycle (Reverse Carnot)](https://physandbox.com/thermodynamics/refrigeration-cycle): PV loop like Carnot but reversed; COP_R & COP_HP; T_C/T_H in K; symbolic fridge sketch.
- [Vapor-Compression Refrigeration Cycle](https://physandbox.com/thermodynamics/vapor-compression-refrigeration-cycle): Compressor, condenser, expansion valve, evaporator: COP, cooling capacity, pressure ratio, and p-h sketch with states 1-2-3-4.
- [Fin Heat Transfer](https://physandbox.com/thermodynamics/fin-heat-transfer): Straight rectangular fin with convection: temperature profile, mL, fin efficiency ηf, effectiveness, and heat rate vs geometry.
- [Lumped Capacitance Cooling](https://physandbox.com/thermodynamics/lumped-capacitance-cooling): Newton cooling with Biot number, characteristic length, τ = ρVc/(hA), temperature curve, and validity check Bi < 0.1.
- [Pipe Friction & Moody Chart](https://physandbox.com/thermodynamics/pipe-friction-moody-chart): Reynolds number, relative roughness, Swamee-Jain/Colebrook-style friction factor, and Darcy-Weisbach head loss.
- [Open-Channel Flow & Hydraulic Jump](https://physandbox.com/thermodynamics/open-channel-hydraulic-jump): Rectangular-channel Froude number, critical depth, conjugate depths, energy loss, and subcritical/supercritical regimes.
- [Multilayer Wall Conduction](https://physandbox.com/thermodynamics/multilayer-wall-conduction): Three layers in series: R″ = L/k, q″ and U; T(x) sketch and preset plaster/brick/wool.
- [Adiabatic Cloud Parcel](https://physandbox.com/thermodynamics/adiabatic-cloud-parcel): Lift moist air: dry Γ, LCL, toy moist lapse; RH and cartoon cloud vs height.
- [Engine Cycles Compared (P–V)](https://physandbox.com/thermodynamics/engine-cycles-compare): One canvas: Carnot, Otto, Diesel, Stirling, Rankine sketch — switch cycles, same formulas as standalone labs.
- [Diesel Cycle (PV)](https://physandbox.com/thermodynamics/diesel-cycle): Air-standard: adiabat compress, isobaric heat, adiabat expand, isochoric out; η(ρ_c, β).
- [Maxwell’s Demon (Toy)](https://physandbox.com/thermodynamics/maxwell-demon): 2D billiards + gate: fast |v| crosses; ⟨|v|⟩ left/right + Landauer note.
- [Leidenfrost Effect (Toy)](https://physandbox.com/thermodynamics/leidenfrost-effect): T_plate vs ~200 °C threshold: vapor gap and lifetime curves — pedagogical, not measured boiling data.
- [Peltier & Seebeck (Schematic)](https://physandbox.com/thermodynamics/peltier-seebeck): Current pumps heat across junction; ΔT drives toy mV — same couple, two modes.
- [Joule Expansion](https://physandbox.com/thermodynamics/joule-expansion): Ideal gas into vacuum: Q = W = ΔU = 0; ΔS = nR ln 2 when volume doubles.
- [2D Ising Model](https://physandbox.com/thermodynamics/ising-2d): Square lattice Metropolis: kT/J vs |m| and E; T_c ≈ 2.27; optional field h.
- [Lorentz Gas (Billiard)](https://physandbox.com/thermodynamics/lorentz-gas): Point particle specularly reflected from fixed disks in a box: chaotic paths and MSD growth toward diffusion.
- [Site Percolation (2D)](https://physandbox.com/thermodynamics/percolation-lattice): Square lattice occupation p: clusters, left–right spanning, p_c ≈ 0.593; rough box-counting D̂ on the largest cluster.
- [Diffusion-Limited Aggregation (DLA)](https://physandbox.com/thermodynamics/diffusion-limited-aggregation): Random walkers stick on first contact with a seed cluster on a square lattice; branched fractal growth with literature mass dimension D ≈ 1.71 in 2D and a rough box-counting D̂.
- [Forest Fire (Cellular Automaton)](https://physandbox.com/thermodynamics/forest-fire): Toroidal lattice: trees regrow with probability p, lightning ignites trees with probability f, fire spreads to four neighbors; explore intermittent bursts and SOC-like phenomenology.
- [Schelling Segregation Model](https://physandbox.com/thermodynamics/schelling-segregation): Two agent types on a toroidal lattice with vacancies: unhappy agents swap into random empty cells when fewer than a fraction τ of occupied Moore neighbors share their type—mild local rules, global clustering.
- [Kuramoto Oscillators](https://physandbox.com/thermodynamics/kuramoto-oscillators): All-to-all coupled phases θ_i with intrinsic frequencies ω_i: order parameter r = |N⁻¹ Σ e^{iθ_i}| grows as coupling K crosses the synchronization window—classic mean-field rhythm transition.
- [Vicsek Model (Active Matter)](https://physandbox.com/thermodynamics/vicsek-active-matter): Self-propelled particles on a torus align headings with neighbors within radius R, add noise η, then drift at v₀; polarization V_a = |⟨e^{iθ}⟩| captures the flocking crossover with density ρ = N/L².
- [Wolfram Elementary Cellular Automata](https://physandbox.com/thermodynamics/wolfram-elementary-ca): 1D binary CA with Wolfram code R∈[0,255]: space–time raster, periodic or null boundaries, single-cell or random IC, rule-bit strip, and informal class hints for textbook rules (e.g. 30, 90, 110, 184).
- [Eden Growth (Lattice)](https://physandbox.com/thermodynamics/eden-growth): Square-lattice Eden model: each step occupies a uniformly random empty von-Neumann neighbor of the cluster; compact growth with a rough interface—track perimeter P vs 2√(πN) as a roughness proxy.
- [Ising 2D: Wolff Cluster Updates](https://physandbox.com/thermodynamics/ising-wolff-cluster): Ferromagnetic toroidal Ising (h = 0): Swendsen–Wang bond freezing with probability 1 − e^(−2βJ), build a same-spin cluster, flip it—fast mixing near T_c compared to single-spin Metropolis.
- [Hopfield Associative Memory](https://physandbox.com/thermodynamics/hopfield-network): 10×10 binary Hopfield network: Hebbian pattern storage, energy E = −½ Σ w_ij S_i S_j, and random asynchronous updates that flow downhill to fixed-point recall after noisy cues.
- [Bose–Einstein vs Fermi–Dirac](https://physandbox.com/thermodynamics/bose-fermi-distribution): Grand-canonical occupation f(E) at the same T and μ: FD, BE, and classical Maxwell–Boltzmann comparison.
- [Ideal Gas Simulator](https://physandbox.com/thermodynamics/ideal-gas): Bouncing particles in a box. See PV=nRT in action.
- [Gas Laws Interactive](https://physandbox.com/thermodynamics/gas-laws): Boyle's, Charles's, Gay-Lussac's laws with interactive piston.
- [Heat Transfer](https://physandbox.com/thermodynamics/heat-transfer): Conduction, convection, and radiation with temperature gradients.
- [Phase Diagram](https://physandbox.com/thermodynamics/phase-diagram): Temperature-pressure diagram with phase transitions.
- [Carnot Engine](https://physandbox.com/thermodynamics/carnot-engine): PV diagram animation with cycle steps and efficiency.
- [Maxwell–Boltzmann Distribution](https://physandbox.com/thermodynamics/maxwell-boltzmann): Histogram of |v| from Gaussian components vs the 3D Maxwell speed PDF; T and sample size.
- [Thermal Expansion](https://physandbox.com/thermodynamics/thermal-expansion): Linear ΔL = α L₀ ΔT; compare reference bar and heated/cooled length (schematic).
- [Brownian Motion](https://physandbox.com/thermodynamics/brownian-motion): Heavy particle with random kicks and friction; trail and running ⟨r²⟩ vs time.
- [Otto Cycle](https://physandbox.com/thermodynamics/otto-cycle): PV diagram: adiabatic compression/expansion and isochoric heat; η = 1 − r^{1−γ}.
- [Gas Mixing & Entropy](https://physandbox.com/thermodynamics/gas-mixing-entropy): Two species separated then mixed; ΔS = 2nR ln 2 for equal volumes and moles.
- [Stirling Cycle](https://physandbox.com/thermodynamics/stirling-cycle): PV: two isotherms and two isochores; ideal η equals Carnot with a perfect regenerator.
- [Wet Steam (T–s sketch)](https://physandbox.com/thermodynamics/wet-steam-ts): Vapor dome, horizontal isobar in two-phase region, quality x and superheat sketch.

### Biophysics, Fluids & Geoscience

- [Enzyme Inhibition Kinetics](https://physandbox.com/earth-life-fluids/enzyme-inhibition-kinetics): Competitive, noncompetitive, and uncompetitive inhibition: Michaelis-Menten curves, apparent Km/Vmax shifts, and a Lineweaver-Burk sketch.
- [Hardy-Weinberg Population Genetics](https://physandbox.com/earth-life-fluids/hardy-weinberg-population-genetics): Allele frequency p, p²/2pq/q² genotype frequencies, plus selection, mutation, and genetic drift toy trajectories.
- [Logistic Harvest & Maximum Sustainable Yield](https://physandbox.com/earth-life-fluids/logistic-harvest-msy): Logistic growth with constant harvest: MSY = rK/4, stable stock, unstable collapse threshold, and population trajectory.
- [Leslie Matrix Age-Structured Population](https://physandbox.com/earth-life-fluids/leslie-matrix-population): Fertility and survival matrix model: total population trajectory, dominant eigenvalue lambda, and stable age distribution.
- [SIR Vaccination Threshold](https://physandbox.com/earth-life-fluids/sir-vaccination-threshold): Herd-immunity threshold simulator: R_eff = R0(1-v), SIR outbreak curves, peak infected, and final size vs vaccine coverage.
- [Reaction-Diffusion Turing Patterns](https://physandbox.com/earth-life-fluids/turing-reaction-diffusion-patterns): Activator-inhibitor reaction-diffusion toy with spots, stripes, worms presets, diffusion ratio, and schematic dispersion relation.
- [Hodgkin–Huxley Action Potential](https://physandbox.com/earth-life-fluids/hodgkin-huxley): Squid-axon HH ODEs: membrane voltage V(t) and gating (m, h, n); step stimulus and Na⁺/K⁺ conductance traces.
- [Cardiac Action Potential](https://physandbox.com/earth-life-fluids/cardiac-action-potential): FitzHugh–Nagumo/Noble-style cardiac toy: fast upstroke, plateau-like repolarization, refractory recovery gate, and S1-S2 premature-stimulus capture vs block.
- [Calcium Waves / IP3 Oscillator](https://physandbox.com/earth-life-fluids/calcium-ip3-oscillator): Li-Rinzel-style IP3 receptor oscillator: cytosolic Ca, ER store, calcium-induced calcium release, SERCA refilling, flux traces, and a Ca–h phase portrait.
- [Gene Regulatory Toggle Switch](https://physandbox.com/earth-life-fluids/gene-regulatory-toggle-switch): Two mutually repressing genes with Hill cooperativity: bistable A-high/B-high attractors, phase-plane nullclines, hysteresis under inducer sweeps, and noise-driven switching.
- [Circadian Oscillator (Goodwin Clock)](https://physandbox.com/earth-life-fluids/circadian-oscillator): Goodwin-style molecular clock: delayed negative transcriptional feedback, mRNA/protein/repressor phase lag, period readout, and light-pulse phase advance or delay.
- [Pharmacokinetics: 1-/2-Compartment](https://physandbox.com/earth-life-fluids/pharmacokinetics-compartment-model): Linear IV PK simulator: bolus, infusion, and repeated dosing; central/peripheral concentration curves, terminal half-life, AUC, Cmax/Cmin, loading dose, and maintenance rate.
- [Oxygen-Hemoglobin Dissociation Curve](https://physandbox.com/earth-life-fluids/oxygen-hemoglobin-dissociation): Hill-equation Hb saturation vs pO2 with effective P50 shifts from pH, CO2, and temperature; arterial/venous points, Bohr effect, and O2 content.
- [Hodgkin-Huxley / FHN Neuron Network](https://physandbox.com/earth-life-fluids/hodgkin-huxley-neuron-network): Multi-node excitable network using FHN reductions of HH dynamics: chain/ring topology, sigmoidal synaptic coupling, propagation delay, pacer-driven waves, and bursting-like activity.
- [Cable Equation on an Axon](https://physandbox.com/earth-life-fluids/cable-axon): 1D passive cable ∂V/∂t = D∂²V/∂x² − (V−V_r)/τ + I; compare uniform fiber vs myelinated (higher D in internodes).
- [Coupled FitzHugh–Nagumo Neurons](https://physandbox.com/earth-life-fluids/fitzhugh-nagumo-coupled): Two excitable FHN units with diffusive coupling: phase plots, time traces, and synchronization vs coupling strength.
- [Nernst Potentials & Membrane](https://physandbox.com/earth-life-fluids/membrane-nernst): Nernst E for K⁺ and Na⁺ from inside/outside concentrations; optional Goldman–Hodgkin–Katz resting estimate.
- [Kelvin–Helmholtz Shear Instability](https://physandbox.com/earth-life-fluids/kelvin-helmholtz): Two parallel flows with slip U: Fourier-mode growth rate sketch and animated interface billows (linear toy model).
- [Taylor–Couette Flow (Criterion)](https://physandbox.com/earth-life-fluids/taylor-couette): Gap d, cylinders Ω_i, Ω_o, kinematic ν: Taylor number Ta and critical Ta_c for toroidal rolls — cartoon + formula.
- [Darcy Flow & Penetration Depth](https://physandbox.com/earth-life-fluids/darcy-flow): 1D head h(z,t): ∂h/∂t = K ∂²h/∂z²; recharge at surface, fixed head at depth — relaxation time and diffusion length.
- [Groundwater Well Drawdown (Theis)](https://physandbox.com/earth-life-fluids/groundwater-well-drawdown): Confined-aquifer Theis solution: cone of depression s(r,t) = (Q/(4πT)) W(u), u = r²S/(4Tt); transmissivity, storativity, observation well.
- [Flood Frequency (Gumbel)](https://physandbox.com/earth-life-fluids/flood-frequency-gumbel): Annual maxima ~ Gumbel EV1: return period T, exceedance probability, design flood x_T, and method-of-moments fit to a synthetic sample.
- [Soil Water Retention Curve](https://physandbox.com/earth-life-fluids/soil-water-retention): van Genuchten θ(ψ), effective saturation S_e, Mualem K(θ), field capacity, wilting point, and plant-available water.
- [Watershed Runoff Hydrograph](https://physandbox.com/earth-life-fluids/watershed-runoff-hydrograph): Rainfall hyetograph, initial abstraction, infiltration losses, triangular unit hydrograph convolution, peak discharge, and time-to-peak.
- [Quadrupole / Penning-Style Trap](https://physandbox.com/earth-life-fluids/penning-trap-chain): 2D motion in a hyperbolic electric potential Φ ∝ x²−y² with axial magnetic field B: epicycloid-like bounded orbits.
- [Layered Medium: P/S Fronts & Hodograph](https://physandbox.com/earth-life-fluids/seismic-layered-hodograph): P and S in a two-layer straight-ray toy: reflected legs and optional head branches when the lower layer is faster; hodograph plus minimum-time first arrivals (no conversions or amplitudes).
- [Geomagnetic Reversals (Toy Dynamo)](https://physandbox.com/earth-life-fluids/geomagnetic-dynamo-toy): Low-order α–Ω cartoon: axial dipole B(t) from coupled amplitudes with symmetry breaking — stochastic flips, not MHD.
- [Greenhouse: One vs Two Slabs](https://physandbox.com/earth-life-fluids/greenhouse-two-layer): Compare gray-atmosphere surface temperature for one IR slab vs two coupled slabs; albedo α and absorptivities.
- [0-D Energy Balance & Ice–Albedo](https://physandbox.com/earth-life-fluids/energy-balance-0d-ice-albedo): Budyko-style (1−α)S/4 = εσT⁴ with icy vs clear albedo, freeze/melt hysteresis when S is swept, and optional smooth α(T) relaxation.
- [Urban Heat Island Energy Balance](https://physandbox.com/earth-life-fluids/urban-heat-island): Two 0-D surface slabs: albedo, heat capacity, anthropogenic heat, radiative/turbulent cooling; urban–rural ΔT and day/night lag.
- [Daisyworld (Climate Regulation)](https://physandbox.com/earth-life-fluids/daisyworld): Black and white daisies compete on bare ground, shifting planetary albedo; compare T(L) with a lifeless rock and watch homeostasis in a toy Watson–Lovelock model.
- [ENSO Delayed Oscillator (Toy)](https://physandbox.com/earth-life-fluids/enso-delayed-oscillator): Scalar delay differential equation ẋ = a·x(t−τ) − b·x − c·x³ + seasonal forcing: explore El Niño– / La Niña–like swings vs damping in a Suarez–Schopf–style cartoon.
- [Rossby Waves on a Beta-Plane](https://physandbox.com/earth-life-fluids/rossby-waves-beta-plane): Linear planetary wave ψ = A cos(kx+ly−ωt) with ω = −βk/(k²+l²+Rd⁻²): streamfunction bands and geostrophic arrows show westward phase propagation.
- [Geostrophic Balance & Thermal Wind](https://physandbox.com/earth-life-fluids/geostrophic-thermal-wind): Pressure-gradient force balanced by Coriolis: isobars set geostrophic wind, while meridional temperature gradients produce vertical shear between 1000 and 300 hPa.
- [Atmospheric Stability / Parcel Diagram](https://physandbox.com/earth-life-fluids/atmospheric-stability-parcel): Toy sounding with environmental lapse rate, dry/moist parcel ascent, LCL, LFC, equilibrium level, CAPE, CIN, and Lifted Index.
- [Groundwater Contaminant Plume](https://physandbox.com/earth-life-fluids/groundwater-contaminant-plume): Advection-dispersion pulse in groundwater with longitudinal/transverse spreading, retardation factor R, and a monitoring-well breakthrough curve.
- [Infinite Slope Stability](https://physandbox.com/earth-life-fluids/infinite-slope-stability): Limit-equilibrium factor of safety for a shallow planar slide: slope angle, cohesion, friction, depth, and rainfall-driven pore pressure set the landslide threshold.
- [Earthquake Aftershocks: Omori + Gutenberg-Richter](https://physandbox.com/earth-life-fluids/earthquake-aftershocks-omori-gr): Modified Omori aftershock decay n(t)=K/(t+c)^p combined with Gutenberg-Richter magnitude-frequency curves, b-value, and a synthetic catalog.
- [Mantle Convection Cell (Toy)](https://physandbox.com/earth-life-fluids/mantle-convection-cell-toy): High-Prandtl-number mantle convection cartoon: Rayleigh-number vigor, thermal boundary layers, hot upwelling, and a cold subducting slab in one viscous cell.
- [Carbon Cycle (4-Box Model)](https://physandbox.com/earth-life-fluids/carbon-cycle-box-model): Atmosphere, ocean mixed layer, deep ocean, and land biosphere exchange linearly; add fossil emissions or GtC pulses and watch inventory split vs a toy airborne fraction.
- [Mid-Ocean Ridge: Magnetic Stripes](https://physandbox.com/earth-life-fluids/mid-ocean-ridge-magnetic-stripes): Symmetric stripes accrete at a spreading center as polarity flips — a 2D cartoon of Vine–Matthews–Morley marine magnetic anomalies.
- [Glacier Flow 1D (Glen’s Law)](https://physandbox.com/earth-life-fluids/glacier-flow-1d-glen): Shallow-ice thickness H(x,t) on a tilted bed: nonlinear flux q ∝ H^{n+2}|∂s/∂x|^{n−1}∂s/∂x plus accumulation; τ_b ≈ ρgH sinα readout.
- [River Meandering (Toy)](https://physandbox.com/earth-life-fluids/river-meandering-toy): Pinned centerline y(x,t): toy migration ∂y/∂t ≈ k₁ ∂κ/∂s + λ y_xx with κ ≈ y_xx — bend sharpening cartoon, not full sediment hydraulics.
- [Ekman Spiral in the Ocean](https://physandbox.com/earth-life-fluids/ekman-spiral): Wind stress, Coriolis f, eddy viscosity A_z: classical Ekman spiral of horizontal velocity with depth; deflection angle.

### Math Visualization

- [STFT & Spectrogram](https://physandbox.com/math/stft-spectrogram): Slide a windowed FFT across the signal: chirps, two-tones, bursts. Tune window M, hop, type — see the time–frequency trade-off live.
- [Morlet Wavelet (CWT)](https://physandbox.com/math/wavelet-morlet): Continuous wavelet transform with the complex Morlet wavelet: scaleogram |W(s,t)|, log-frequency axis, cone of influence, adjustable ω₀ and scale range.
- [Butterworth / Chebyshev IIR](https://physandbox.com/math/iir-filter-design): Design Butterworth, Chebyshev I/II LP/HP filters: |H(f)|, phase, impulse response, and z-plane pole–zero plot via bilinear transform.
- [Kalman Filter (1-D)](https://physandbox.com/math/kalman-1d): Recursive optimal estimation: noisy measurements, hidden truth, predict + update with Q and R; random-walk or constant-velocity model with ±2σ band and innovations.
- [Kalman Filter 2D Tracking](https://physandbox.com/math/kalman-2d-tracking): 4-state constant-velocity tracker: noisy (x,y) measurements, process acceleration noise Q, measurement variance R, and a live ≈2σ covariance ellipse.
- [Lattice Boltzmann D2Q9 Flow](https://physandbox.com/math/lattice-boltzmann-d2q9): Interactive D2Q9 BGK solver: lid-driven cavity or flow past a cylinder, vorticity colors, Reynolds-number control, and bounce-back walls.
- [Finite-Volume Advection-Diffusion 2D](https://physandbox.com/math/finite-volume-advection-diffusion-2d): Conservative scalar transport on a 2D grid: face fluxes, upwind vs central interpolation, Peclet number, CFL, and numerical diffusion.
- [Conjugate Gradient Solver](https://physandbox.com/math/conjugate-gradient-solver): SPD system Ax=b as quadratic minimization: contour geometry, CG vs steepest descent path, residual norm, and condition number.
- [Power Iteration Eigenvalue Convergence](https://physandbox.com/math/power-iteration-eigenvalue-convergence): Visualize dominant eigenvector convergence: spectral gap ratio, Rayleigh quotient, eigen residual, and normalized power iterates on the unit circle.
- [Newton-Raphson Basins in 2D Systems](https://physandbox.com/math/newton-raphson-basins-2d): Map Newton basins for nonlinear F(x,y)=0 systems: initial-guess sensitivity, iteration counts, root attraction, and singular-Jacobian failures.
- [Monte Carlo Integration & Variance Reduction](https://physandbox.com/math/monte-carlo-integration-variance-reduction): Compare plain Monte Carlo, importance sampling, and stratified sampling for ∫f(x)dx, with convergence curves, standard error, and the 1/√N rate.
- [PCA / SVD Geometry](https://physandbox.com/math/pca-svd-geometry): Covariance ellipse and SVD view of dimensionality reduction: principal components, explained variance, and rank-1 reconstruction error.
- [Gradient Descent Optimizers](https://physandbox.com/math/gradient-descent-optimizers): Compare SGD, momentum, and Adam on a curved loss landscape; tune learning rate, curvature, stability, and iteration count.
- [Gaussian Mixture EM Algorithm](https://physandbox.com/math/gaussian-mixture-em-algorithm): Expectation-maximization for a two-component Gaussian mixture: responsibilities, E/M steps, likelihood, and covariance ellipses.
- [Support Vector Machine Margin](https://physandbox.com/math/svm-margin-visualizer): Hard/soft-margin SVM sketch with C penalty, hinge loss, support vectors, decision boundary, and margin bands.
- [Bayesian Updating / Conjugate Priors](https://physandbox.com/math/bayesian-updating-conjugate-priors): Beta-binomial Bayesian updating with prior/posterior curves, posterior predictive probability, and credible intervals.
- [Markov Chain Mixing](https://physandbox.com/math/markov-chain-mixing): Three-state Markov chain with transition matrix, stationary distribution, total variation distance, detailed-balance cue, and spectral-gap proxy.
- [LMS / NLMS Adaptive Noise Cancellation](https://physandbox.com/math/lms-adaptive-filter): Primary p = s + v with v a fixed unknown FIR of Gaussian reference x[n]. Watch an L-tap FIR adapt by LMS or NLMS so error e = p − wᵀx → s; running MSE and ‖w − h‖.
- [DCT & JPEG Quantization (8×8)](https://physandbox.com/math/dct-image-compression): 64×64 luma: 8×8 DCT with −128 shift, ISO luminance quant table scaled by JPEG quality, or zigzag AC truncation (keep K). Click a block for coefficient heatmaps and zigzag trace.
- [PLL (Phase-Locked Loop)](https://physandbox.com/math/pll-phase-locked-loop): Discrete-time analog-style PLL: multiplier PD e = K_d sin(φ_ref − φ_VCO), PI loop filter, VCO ω = ω_fr + K_v u; step ω_ref to explore lock, capture, and steady-state phase error.
- [ΔΣ (1-bit) Modulator](https://physandbox.com/math/delta-sigma-1-bit): First- and second-order discrete-time ΔΣ with ±1 quantizer: shaped quantization noise, sine test tone, boxcar reconstruction, and Hann-windowed error spectrum (last 1024 samples).
- [Polyphase L/M Resampling](https://physandbox.com/math/polyphase-resampling): Zero-stuff by L, Hamming-windowed sinc FIR at the high rate with min(π/L,π/M) cutoff, then decimate by M; spectra in/out and Noble-identity polyphase intuition.
- [Mandelbrot Deep Zoom](https://physandbox.com/math/mandelbrot-zoom): Drag/wheel deep zoom into the Mandelbrot set with smooth continuous coloring and named landmarks.
- [Julia Set Explorer](https://physandbox.com/math/julia-set): Pick c by clicking the embedded mini-Mandelbrot or animate c along a circle; Fatou dust vs connected sets.
- [Newton Fractal](https://physandbox.com/math/newton-fractal): Basins of attraction for Newton iteration on zⁿ−1 with adjustable relaxation ω.
- [Newton's Method (1D)](https://physandbox.com/math/newton-method-1d): Graph f(x), click x₀, iterate x − f/f′ with numeric derivative; pan and presets. Complex basins: separate Newton fractal sim.
- [Runge–Kutta Stability Regions](https://physandbox.com/math/runge-kutta-stability): Absolute-stability regions in the z = hλ plane for explicit Euler, RK2, and RK4; move λ and h, compare |R(z)|, and see why stiff modes constrain explicit time steps.
- [Heat Equation: Finite Differences](https://physandbox.com/math/heat-equation-finite-differences): 1D heat equation u_t = αu_xx with explicit FTCS and implicit backward Euler; tune CFL r = αΔt/Δx², watch explicit blow-up for r > 1/2, and compare numerical diffusion.
- [Advection Schemes: Upwind / LW / MacCormack](https://physandbox.com/math/advection-schemes): Linear advection u_t + cu_x = 0 on a periodic grid: compare upwind diffusion, Lax-Wendroff and MacCormack dispersive ringing, exact pulse translation, L2 error, and CFL ν.
- [Finite Element Poisson Solver (2D)](https://physandbox.com/math/finite-element-poisson-2d): Triangular P1 finite elements for -Δu=f on a square: assemble stiffness matrices, pin Dirichlet nodes or add natural Neumann flux, solve by CG, and view potential as a colored membrane.
- [Multigrid Relaxation](https://physandbox.com/math/multigrid-relaxation): 1D Poisson error dynamics: Jacobi and Gauss-Seidel smooth high-frequency error, while a V-cycle uses residual restriction and coarse-grid correction to remove low-frequency modes.
- [Gaussian Process Regression](https://physandbox.com/math/gaussian-process-regression): Interactive GP regression with RBF and Matérn kernels: tune length scale/noise, add observations by clicking, view posterior mean with uncertainty bands, and sample at max posterior variance.
- [Rössler Attractor](https://physandbox.com/math/rossler-attractor): RK4 integration of ẋ=−y−z, ẏ=x+ay, ż=b+z(x−c); period-doubling cascade as c grows.
- [L-Systems (Turtle)](https://physandbox.com/math/l-systems): Lindenmayer string rewriting + turtle: Koch, Sierpinski, Hilbert, Heighway dragon, plant.
- [Bézier & de Casteljau](https://physandbox.com/math/bezier-de-casteljau): Drag control points; live recursive linear-interpolation scaffolding evaluates B(t).
- [Convex Hull (Graham & QuickHull)](https://physandbox.com/math/convex-hull): Click to add points, drag to move; Graham scan with step playback or QuickHull divide-by-farthest; compare vertex sets.
- [Delaunay & Voronoi](https://physandbox.com/math/delaunay-voronoi): Bowyer–Watson triangulation and dual Voronoi tessellation; click to add seeds, drag to move.
- [Physarum Slime (Agents)](https://physandbox.com/math/physarum-slime): ~4500 agents follow a deposited chemoattractant: deposit + diffusion + decay + 3-sensor steering grow path networks.
- [Savitzky–Golay Smoothing](https://physandbox.com/math/savitzky-golay): Noisy cosine vs SG(7,2) convolution — preserves peaks better than a wide boxcar.
- [Markov Chain (Weather)](https://physandbox.com/math/markov-chain-weather): Sun/Rain two-state chain: P matrix, stationary π, empirical vs theory.
- [Gradient Descent (2D)](https://physandbox.com/math/gradient-descent): Level sets of f(x,y) and path (x,y) ← (x,y) − η∇f; bowl or elliptic well.
- [Minkowski Diagram](https://physandbox.com/math/minkowski-diagram): Light cone and boosted axes in 1+1D; γ from v.
- [Twin Paradox](https://physandbox.com/math/twin-paradox): Out-and-back worldlines; proper time τ = T/γ vs Earth time T.
- [Monte Carlo π](https://physandbox.com/math/monte-carlo-pi): Uniform samples in a square; 4·(in disk)/N estimates π.
- [Random Walk](https://physandbox.com/math/random-walk): 1D or 2D steps; trail and running mean ⟨r²⟩ vs diffusion intuition.
- [Random Walk (2D / 3D Lattice)](https://physandbox.com/math/random-walk-2d-3d): Nearest-neighbor SRW on Z² or Z³: ensemble mean ⟨r²⟩ vs time with y = t reference, histogram of r² across walkers, and Monte Carlo first-return times (recurrent vs transient).
- [Vector Addition](https://physandbox.com/math/vector-addition): Place vectors and see the resultant with head-to-tail animation.
- [Trigonometry Circle](https://physandbox.com/math/trig-circle): Unit circle with live sin, cos, tan values as you drag.
- [Function Grapher](https://physandbox.com/math/function-grapher): Enter f(x) and see instant plots with zoom and pan.
- [Fourier Series](https://physandbox.com/math/fourier-series): Build waveforms from sine waves. Add harmonics one by one.
- [FFT Magnitude Spectrum](https://physandbox.com/math/fft-spectrum): Paint or preset a 256-point signal; radix-2 FFT shows |X[k]| vs bin (DC to Nyquist).
- [2D Phase Portrait (ODE)](https://physandbox.com/math/phase-portrait-2d): Direction field and click-to-trace trajectories for planar systems: harmonic, damped, saddle, nodes, foci, pendulum (RK4).
- [Lissajous Curves](https://physandbox.com/math/lissajous): Beautiful patterns from two frequencies with adjustable ratio.
- [Harmonograph](https://physandbox.com/math/harmonograph): Two damped harmonic sums in x and y: decaying rosette trace vs Lissajous loops.
- [Spirograph (Trochoids)](https://physandbox.com/math/spirograph): Hypo- or epitrochoid: fixed R, rolling r, pen d; hue trail and period hints.
- [Sorting Algorithms (parallel)](https://physandbox.com/math/sorting-algorithms-viz): Bubble, insertion, merge, quicksort, and heapsort on the same shuffled permutation — five bar rows advance in lockstep so you can compare how each method moves values.
- [SIR Epidemic Model](https://physandbox.com/math/sir-epidemic): S + I + R = 1: βSI and γI; ℛ₀ ≈ β/γ, herd threshold 1 − 1/ℛ₀; RK4 time plot.
- [SEIR / SEIRS Epidemic Model](https://physandbox.com/math/seir-seirs-epidemic): S + E + I + R = 1: latent compartment σE delays infectiousness, recovery γI, optional waning ωR → S; ℛ₀ = β/γ; RK4 time plot.
- [Three-Species Food Chain (Hastings–Powell)](https://physandbox.com/math/three-species-food-chain): Plants x → herbivores y → predators z; Holling II fᵢ(u)=aᵢu/(1+bᵢu); logistic x; chaotic attractors when b₁ is varied (1991).
- [Tumor Growth (Gompertz / Logistic)](https://physandbox.com/math/tumor-growth-gompertz): V(t) → plateau K: Gompertz rV ln(K/V) or logistic rV(1−V/K); chemotherapy as linear kill −kV; RK4 vs untreated reference.
- [Keller–Segel Chemotaxis](https://physandbox.com/math/keller-segel-chemotaxis): ∂ₜn = Dₙ∇²n − χ∇·(n∇c), ∂ₜc = D_c∇²c + αn − βc; bacteria follow attractant; collapse at high χ; 96² grid.
- [MinD / MinE Oscillation (E. coli Rod)](https://physandbox.com/math/min-protein-oscillation-ecoli): 1D reaction–diffusion: membrane MinD u, fast MinE v; pole-to-pole oscillation; division plane at time-averaged MinD minimum.
- [Sandpile (SOC)](https://physandbox.com/math/sandpile-soc): BTW abelian model: add grains, ≥4 topples to neighbors; critical avalanches.
- [Flow Field Particles](https://physandbox.com/math/flow-field-particles): Synthetic v(x,y,t); advection with wrap; optional arrow grid.
- [Fractal Generator](https://physandbox.com/math/fractal-generator): Mandelbrot, Julia, Koch snowflake. Zoom infinitely.
- [Conway's Game of Life](https://physandbox.com/math/game-of-life): B3/S23 on a torus: paint cells, run, step — glider, LWSS, Gosper gun, pulsar, and more.
- [a → v → x](https://physandbox.com/math/kinematics-a-v-x): Integrate acceleration to velocity and position; stacked time graphs.
- [Taylor Polynomial](https://physandbox.com/math/taylor-series): sin, cos, or exp vs Taylor sum about center a up to order n.
- [Complex Phasor](https://physandbox.com/math/complex-phasor): exp(iωt) on the unit circle; Re, Im, and phase φ.
- [Chaos Game (Sierpiński)](https://physandbox.com/math/chaos-game-sierpinski): Random vertex + midpoint walk; the attractor is the Sierpiński gasket — try RGB by vertex.
- [Lagrange vs Cubic Spline](https://physandbox.com/math/lagrange-spline): Click knots: Lagrange polynomial vs natural cubic spline; Runge preset shows edge oscillations.
- [Least Squares Fit](https://physandbox.com/math/least-squares): Noisy linear data; fitted slope and intercept with residuals.
- [Linear Regression: OLS, Ridge, Lasso & R²](https://physandbox.com/math/linear-regression-metrics): Click/drag scatter points; fit y = β₀ + β₁x with OLS, Ridge (L2 on slope), or Lasso (L1 on slope). Spike Δy on the largest |x| point to see outlier sensitivity; compare SSE and R².
- [K-Means Clustering (Lloyd)](https://physandbox.com/math/k-means-clustering): Click to add points, choose k, randomize centroids, then step Lloyd iterations (assign to nearest centroid, update means). Optional Gaussian-mixture demo; watch within-cluster SSE decrease.
- [DBSCAN Density Clustering](https://physandbox.com/math/dbscan-clustering): Sliders for ε and minPts on a click-built point set: core / border / noise coloring, optional ε-disks around cores, demo with scattered outliers.
- [PCA in 2D (principal components & 1D projection)](https://physandbox.com/math/pca-dimensionality): Click-built cloud: covariance eigenvectors as PC1/PC2 arrows from the mean, optional orthogonal drops to the PC1 line, and a bottom strip of PC1 scores — the standard rank-one projection coordinate.
- [Decision Tree Classifier (2D toy)](https://physandbox.com/math/decision-tree-2d): Greedy axis-aligned splits on a click-labeled scatter: compare **Gini** vs **entropy** impurity, max depth, and min-samples-per-leaf; shaded rectangles show leaf decisions, dashed lines show recursive partitions.
- [Toy 2-Layer MLP + Backprop (XOR / spiral)](https://physandbox.com/math/backprop-toy-mlp): Click-labeled 2D data; **tanh** hidden layer + **logistic** output trained by **full-batch** gradient descent on **binary cross-entropy**. Heatmap shows **P(class 1)** evolving across epoch blocks — watch the **0.5 decision contour** wrap XOR or untangle spirals.
- [Convolution (pulses)](https://physandbox.com/math/convolution-demo): Two rectangular pulses; overlap length at τ = 0.
- [Euler vs RK4 (Pendulum)](https://physandbox.com/math/euler-vs-rk4-pendulum): Same nonlinear pendulum ODE and step h; Euler vs RK4 side by side.
- [Lotka–Volterra](https://physandbox.com/math/lotka-volterra): N′ = αN−βNP, P′ = δNP−γP; phase plane RK4; equilibrium dot.
- [Logistic Growth](https://physandbox.com/math/logistic-growth): dN/dt = rN(1−N/K); exact S-curve vs carrying capacity K.
- [Logistic Map Bifurcation](https://physandbox.com/math/logistic-map-bifurcation): x_{n+1}=rx_n(1−x_n): scan r, plot attractors — period doubling to chaos (Feigenbaum cascade).
- [2×2 Matrix & Eigenvectors](https://physandbox.com/math/eigen-2x2-grid): Grid deformation under M; real λ eigen-direction arrows.
- [Lorenz Strange Attractor](https://physandbox.com/math/lorenz-attractor): σ, ρ, β ODEs; sensitive butterfly in (x,z) projection — RK4 trace.

### Engineering

- [Euler Column Buckling](https://physandbox.com/engineering/euler-column-buckling): Elastic column stability with P_cr = π²EI/(KL)²: choose end conditions, effective length factor K, first mode shape, and load ratio P/P_cr.
- [Mohr Circle & Stress Transformation](https://physandbox.com/engineering/mohr-circle-stress-transform): Plane stress σx, σy, τxy: Mohr circle, transformed stresses on a rotated element, principal stresses, τmax, and θp.
- [Fracture Mechanics: Griffith / K_IC](https://physandbox.com/engineering/fracture-mechanics-griffith-kic): Mode-I crack: K_I = Yσ√(πa), compare with K_IC, critical crack size, critical stress, and safe/unstable crack growth.
- [Thin-Walled Pressure Vessel Stress](https://physandbox.com/engineering/thin-walled-pressure-vessel-stress): Cylinder vs sphere membrane stresses: hoop σθ, longitudinal σz, von Mises stress, r/t thin-wall check, and yield safety factor.
- [Jeffcott Rotor Critical Speed](https://physandbox.com/engineering/jeffcott-rotor-critical-speed): Single disk on a flexible shaft: ω_n = √(k/m), unbalance response, whirl orbit, phase lag, and critical-speed crossing.
- [Beam Deflection: Unit Load Method](https://physandbox.com/engineering/beam-deflection-unit-load): Simply supported Euler-Bernoulli beam with point load P and UDL w: closed-form deflection vs virtual-work unit-load integral.
- [Hertzian Contact Stress](https://physandbox.com/engineering/hertz-contact-stress): Sphere or cylinder on a flat: effective modulus, contact patch, peak pressure p0, elastic approach, and subsurface shear estimate.
- [Fatigue S-N Curve + Miner Rule](https://physandbox.com/engineering/fatigue-sn-miner-rule): Basquin S-N curve with optional endurance limit, three cyclic load blocks, and Palmgren-Miner cumulative damage D = Σ n_i/N_i.
- [Vibration Isolation Transmissibility](https://physandbox.com/engineering/vibration-isolation-transmissibility): SDOF base-excitation isolator: transmissibility T(r,ζ), resonance peak, phase lag, and the isolation region above r = √2.
- [Heat Exchanger ε-NTU](https://physandbox.com/engineering/heat-exchanger-ntu-effectiveness): Parallel and counter-flow heat exchanger calculator: NTU = UA/Cmin, capacity ratio Cr, effectiveness, heat transfer, and outlet temperatures.
- [de Laval Nozzle Mach Number](https://physandbox.com/engineering/de-laval-nozzle-mach): Quasi-1D converging-diverging nozzle: area-Mach relation, choking pressure ratio, subsonic/supersonic branches, and a qualitative normal-shock mode.
- [Torsional Drivetrain Resonance](https://physandbox.com/engineering/torsional-drivetrain-resonance): Two-inertia torsional drivetrain: shaft stiffness and damping, twist angle, first natural mode, resonance response, and optional backlash deadzone.
- [Cart–Pole: LQR vs MPC vs PID](https://physandbox.com/engineering/cart-pole-lqr): Nonlinear cart inverted pendulum: linearize about upright φ = θ, discrete-time LQR from DARE (Q, R → K), finite-horizon LQ MPC with DARE terminal cost, and hand-tuned PID — same force limit, kicks, and plant as the mechanics cart-pole.
- [Bode & Nyquist from Pole–Zero Map](https://physandbox.com/engineering/bode-nyquist-pole-zero): Click the s-plane to place real or complex-conjugate poles (●) and zeros (×); G(s)=K∏(s−z)/∏(s−p). Live Bode magnitude/phase and Nyquist plot with −1 point; phase and gain margins from principal crossings (teaching heuristics).
- [Root Locus: 1 + K·G(s) = 0](https://physandbox.com/engineering/root-locus-gain): Same pole–zero plant editor as the Bode/Nyquist map: trace closed-loop poles as K sweeps log-spaced, asymptotes and real-axis locus sketch, Durand–Kerner roots at the current K; characteristic D(s)+K g₀ N₀(s)=0.
- [Particle Filter Localization (MCL)](https://physandbox.com/engineering/particle-filter-localization): 1D or 2D toy robot with fixed range beacons: predict with noisy odometry, update with Gaussian range likelihood, systematic resample on low ESS; weighted mean vs ground truth.
- [EKF SLAM (toy)](https://physandbox.com/engineering/ekf-slam-toy): Augmented state [x,y,θ, landmarks…]: noisy planar odometry predict, range & bearing updates with known IDs, map–robot covariance coupling; wrong map prior vs ground-truth Lissajous path.
- [MPC Pendulum Swing-Up (MPPI)](https://physandbox.com/engineering/mpc-pendulum): Sampling-based Model Predictive Control: K candidate torque rollouts over horizon H, MPPI cost-weighted update, bounded torque |u|≤u_max — swing up an inverted pendulum live and watch the planner replan.
- [A* / Dijkstra Pathfinding (Grid)](https://physandbox.com/engineering/astar-dijkstra-grid): Interactive 40×28 grid pathfinder: A* (f=g+h), Dijkstra, or greedy best-first; Manhattan / octile / Euclidean heuristics, 4- vs 8-connectivity, paint walls + weighted cells, watch open / closed sets expand.
- [Minimum Spanning Tree (Prim & Kruskal)](https://physandbox.com/engineering/minimum-spanning-tree): Random planar points, complete Euclidean-weighted graph: step through Prim from a root or Kruskal with union–find; compare total MST weight.
- [Maze Generators + A* (Grid)](https://physandbox.com/engineering/maze-generators): Perfect mazes on the same 40×28 cell lattice as the pathfinding lab: recursive backtracker, Wilson, Eller, or randomized Prim; then solve with Manhattan A* (4-neighbors). Paint walls or erase passages and move S/G.
- [RRT Path Planner (grid)](https://physandbox.com/engineering/rrt-path-planner): Same 40×28 wall map as A*: random samples, nearest-neighbor steer, goal bias, collision-checked edges; grow an RRT and compare summary stats with one-click Manhattan A* baseline.
- [Differential Drive Odometry](https://physandbox.com/engineering/differential-drive-odometry): ω_L, ω_R → v, ω with two-wheel kinematics; integrate pose and watch dead-reckoning drift from biased wheel radii / track and noisy rate readings.
- [Holonomic 2D Hovercraft MPC (MPPI)](https://physandbox.com/engineering/hovercraft-mpc-2d): Planar double integrator with ‖u‖₂ thrust cap: sampling-based MPPI steers to a draggable goal while soft-penalizing circular obstacles — rollout fan and best predicted path drawn live.
- [3-Link 3D Arm Inverse Kinematics (CCD)](https://physandbox.com/engineering/arm-3link-ik-3d): Continuation of two-link-arm-ik into 3D: 3 revolute joints (yaw + 2 pitches) solved with constrained Cyclic Coordinate Descent. Drag target in 3D or follow a helix / lemniscate / figure-8 trajectory.
- [Bicycle Model & Stanley Controller](https://physandbox.com/engineering/bicycle-stanley): Kinematic bicycle (rear-axle): δ = θ_e + atan2(k_e·e, v) Stanley path-following law. Pick oval, race-track, lemniscate, sine-road or S-curve and tune k_e, v, L; live cross-track e(t) and steering δ(t).
- [Stress–Strain & Hooke’s Law](https://physandbox.com/engineering/stress-strain-hooke): Qualitative σ–ε curve: elastic Hooke region, yield, strain hardening, necking, and fracture. Drag strain and tune E, σ_y, σ_u.
- [Quadcopter 2D (Pitch)](https://physandbox.com/engineering/quadcopter-2d): Side view: two rotors, PD on thrust split vs pitch — whiteboard quad slice.
- [Watt Governor (schematic)](https://physandbox.com/engineering/watt-governor): Flyballs and sleeve vs RPM; steam throttle gap — speed feedback cartoon.
- [Logic Gate Simulator](https://physandbox.com/engineering/logic-gates): AND, OR, NOT, NAND, XOR. Drag, drop, connect, see output.
- [Truth Table Generator](https://physandbox.com/engineering/truth-table): Build a circuit, auto-generate the truth table.
- [Bridge Builder](https://physandbox.com/engineering/bridge-builder): Place beams and joints. Apply load. See stress distribution.
- [Gear Train](https://physandbox.com/engineering/gear-simulator): Connect gears, adjust teeth count, see speed and torque ratios.
- [Planetary Gear Set](https://physandbox.com/engineering/planetary-gears): Sun, planets, internal ring: Willis equation, hold Sun/ring/carrier and compare speed ratios.
- [Beam Q, M & N Diagrams](https://physandbox.com/engineering/beam-mqn-diagrams): Simply supported beam: point load + UDL; shear, bending moment, and uniform axial diagrams.
- [AM / FM Modulation](https://physandbox.com/engineering/am-fm-modulation): Carrier + message: AM envelope vs FM phase; waveform and DFT spectrum snapshot.
- [Digital Modulation Eye Diagram](https://physandbox.com/engineering/digital-modulation-eye-diagram): NRZ / PAM-4 eye diagram with raised-cosine shaping, ISI, AWGN, timing jitter, and vertical opening estimate.
- [OFDM Subcarriers](https://physandbox.com/engineering/ofdm-subcarriers): IFFT subcarriers with cyclic prefix, multipath delay tap, one-tap ZF, and constellation distortion when CP is too short.
- [Planar Truss (triangle)](https://physandbox.com/engineering/planar-truss): Symmetric 3-bar truss: bar forces and reactions vs span, height, and apex load.
- [PID Controller (1D)](https://physandbox.com/engineering/pid-controller): Cart on a track: Kp, Ki, Kd and random velocity impulses toward set-point x = 0.
- [PID Tuning Sandbox](https://physandbox.com/engineering/pid-tuning-sandbox): Second-order plant step response: overshoot, rise/settling time, Ziegler–Nichols Ku/Pu hints, actuator saturation, and disturbance rejection.
- [Two-Link Arm IK (2R)](https://physandbox.com/engineering/two-link-arm-ik): Planar 2R manipulator: mouse goal, elbow-up / elbow-down inverse kinematics; joint angles live.
- [Four-Bar Linkage](https://physandbox.com/engineering/four-bar-linkage): Crank–rocker geometry with coupler-curve trace; link lengths and crank speed.
- [Cam & Follower](https://physandbox.com/engineering/cam-follower): Eccentric circular cam and knife-edge follower: lift and estimated velocity.
- [Finite State Machine](https://physandbox.com/engineering/finite-state-machine): Traffic-light Moore machine: timed green–yellow–red cycle or manual step; state graph.
- [ADC / DAC (Sampling)](https://physandbox.com/engineering/adc-dac-converter): Sine → samples → quantization → ZOH; Nyquist and optional aliasing demo.
- [Stepper Motor (4-phase)](https://physandbox.com/engineering/stepper-motor): Full-step vs half-step commutation; rotor snaps as coils A–B–A′–B′ sequence.
- [Thermostat vs PID](https://physandbox.com/engineering/thermostat-control): First-order room: on/off hysteresis vs continuous PID heater power.

### Chemistry

- [Hückel π-MO (Butadiene & Benzene)](https://physandbox.com/chemistry/hueckel-pi-mo): Secular matrix H = αI + βA; eigen-energies and LCAO maps on the π skeleton.
- [MO Diagram: Homonuclear Diatomics (H₂–O₂)](https://physandbox.com/chemistry/homonuclear-diatomic-mo): σ/π ladder with Li–N vs O ordering; bond order from valence MOs; O₂ π* unpaired → paramagnetism.
- [Wigner Function (Coherent vs Squeezed)](https://physandbox.com/chemistry/wigner-function): Phase-space quasi-probability W(x, p) for a single-mode Gaussian quantum state: coherent |α⟩, displaced-squeezed D(α)S(ξ)|0⟩, and thermal. The 1σ ellipse rotates by half the squeeze phase θ/2 and shrinks below the vacuum floor along one quadrature — the basic picture of CV quantum optics.
- [Coherent State |α⟩ in a Harmonic Oscillator](https://physandbox.com/chemistry/coherent-state-oscillator): Animated Gaussian wavefunction of a coherent state |α⟩ in a 1-D harmonic well: rigid σ = 1/√2 packet whose centroid traces the classical orbit ⟨x⟩(t) = √2|α|cos(ωt − φ_α). Side-by-side phase space, |ψ(x,t)|², and ⟨x⟩(t) trace.
- [WKB / Bohr–Sommerfeld Quantization](https://physandbox.com/chemistry/wkb-bohr-sommerfeld): Semiclassical bound-state energies for arbitrary V(x): harmonic, quartic, Morse, double-well, asymmetric and square wells. Bisection on the action ∫√(2m(E−V))dx = (n+½)πℏ gives the WKB ladder; compare with the exact harmonic ladder ℏω(n+½).
- [Qubit Decoherence: Lindblad / T₁ T₂](https://physandbox.com/chemistry/lindblad-decoherence): Two-level Bloch master equation in the rotating frame with drive Ω, detuning Δ, T₁ relaxation and T₂ transverse decay. The Bloch vector spirals inside the sphere — the geometric picture of decoherence with live populations P(|0⟩), purity Tr ρ², and time traces of u_x, u_y, u_z.
- [Ramsey Fringes (Atomic Clock)](https://physandbox.com/chemistry/ramsey-fringes): Separated-oscillating-fields sequence π/2 — τ — π/2 with detuning Δ and coherence T₂. Sweep τ or Δ to see P(|1⟩) = ½(1 − cosΔτ · e^{−τ/T₂}) — fringe period 2π/Δ, exponential T₂ envelope; the Bloch sphere animates each stage. Foundational to atomic clocks and Ramsey interferometry.
- [CHSH Bell Inequality Test](https://physandbox.com/chemistry/chsh-bell-test): Monte-Carlo of the singlet state |Ψ⁻⟩ at four measurement angles (a, a′, b, b′): the running estimate Ŝ approaches the Tsirelson bound 2√2 ≈ 2.828 at the canonical Bell angles (0°, 90°, 45°, −45°), violating the local-realist limit |S| ≤ 2 — quantum entanglement made statistically visible.
- [Hong–Ou–Mandel Two-Photon Dip](https://physandbox.com/chemistry/hong-ou-mandel): Two indistinguishable photons enter opposite ports of a 50/50 beam splitter and bunch into the same output: coincidence probability P_c(δτ) = ½(1 − V·exp(−(δτ/τ_c)²)) (Gaussian) or Lorentzian. Drag the delay δτ to walk through the dip; live Monte-Carlo converges to the analytic curve. Visibility V = indistinguishability.
- [Kronig–Penney Bands & Brillouin Zone](https://physandbox.com/chemistry/kronig-penney-bands): Periodic δ-comb model of a 1-D crystal: cos(ka) = cos(qa) + (P/qa) sin(qa). Find allowed energy bands and forbidden gaps from the |·|≤1 corridor, then watch each band fold into the first Brillouin zone k ∈ ±π/a. Free-electron parabola overlaid for reference.
- [Phonon Dispersion: 1-D Mass–Spring Chain](https://physandbox.com/chemistry/phonon-dispersion-chain): Monoatomic ω = 2√(K/m)|sin(ka/2)| or diatomic acoustic/optical branches from alternating masses; ω(k) and group velocity v_g = dω/dk in the first Brillouin zone with animated lattice snapshot.
- [Debye vs Einstein Heat Capacity](https://physandbox.com/chemistry/debye-heat-capacity): Molar C_V(T): Debye integral → T³ law at low T and 3R Dulong–Petit at high T; compare with Einstein single-frequency model and toggle curves on one plot.
- [Anderson Localization (1D Tight-Binding)](https://physandbox.com/chemistry/anderson-localization-1d): Random onsite disorder W on a 1-D chain: diagonalize H, plot |ψ|² eigenstates, IPR vs energy, and localization length estimates (RMS and exponential fit).
- [Bloch Oscillations & Wannier–Stark Ladder](https://physandbox.com/chemistry/bloch-oscillations): 1-D tight-binding electron in uniform field E: semiclassical k(t), Bloch-periodic x(t), and finite-chain Wannier–Stark ladder with spacing ≈ eEa.
- [Graphene Tight-Binding Band Structure](https://physandbox.com/chemistry/graphene-tight-binding): Honeycomb π tight binding: E(k) along Γ–M–K–Γ, Dirac cones at K and K′ when Δ=0, gap 2Δ from sublattice stagger, and a Brillouin-zone energy map.
- [Quantum Hall Edge States & σ_xy Plateaus](https://physandbox.com/chemistry/quantum-hall-edge-states): Integer QHE: chiral edge channels and skipping orbits in a Hall bar, filled Landau levels vs μ, and quantized Hall conductance plateaus σ_xy = ν_f e²/h.
- [Drude–Sommerfeld Transport](https://physandbox.com/chemistry/drude-sommerfeld-transport): Free-electron Drude model: scattering time τ, mobility μ, σ(ω), mean free path ℓ = v_Fτ, skin depth δ(ω), and DC Hall σ_xx(B), σ_xy(B).
- [2D Box: Eigenstates & Degeneracy](https://physandbox.com/chemistry/box-2d-degeneracy): Particle in a 2-D rectangular infinite well: ψ_{n_x,n_y} ∝ sin(n_xπx/L_x)sin(n_yπy/L_y), E ∝ (n_x/L_x)² + (n_y/L_y)². Toggle a square box (L_x = L_y) to expose the (n_x, n_y) ↔ (n_y, n_x) accidental degeneracy and watch the doublets split as the box deforms.
- [Landau Levels in a Magnetic Field](https://physandbox.com/chemistry/landau-levels): Charged particle in a uniform B-field: equally-spaced Landau ladder E_n = ℏω_c(n+½) with cyclotron frequency ω_c = qB/m, magnetic length ℓ_B = √(ℏ/qB) and orbit radius r_n = ℓ_B√(2n+1). Animated cyclotron orbit + linear-in-B fan diagram; the n_B = qB/h degeneracy underlies the quantum Hall effect.
- [Maxwell–Boltzmann vs Eₐ](https://physandbox.com/chemistry/maxwell-boltzmann-ea): Translational energy density f(E); shaded fraction above activation energy; compare Arrhenius exp(−Eₐ/RT).
- [Water P–T Phase Diagram](https://physandbox.com/chemistry/water-phase-diagram): Qualitative fusion, sublimation, vapor pressure up to critical point — probe labeled regions (pedagogical curves).
- [Binary Phase Diagram & Lever Rule](https://physandbox.com/chemistry/phase-diagram-lever-rule): Isomorphous A–B T–x diagram: liquidus, solidus, tie line, and lever-rule phase fractions f_α and f_L for overall composition C₀.
- [Diffusion Couple (Fick's Second Law)](https://physandbox.com/chemistry/diffusion-couple-fick): Infinite A|B diffusion couple: error-function concentration profiles, diffusivity D, time t, and diffusion length 2√(Dt).
- [Adsorption Isotherms](https://physandbox.com/chemistry/adsorption-isotherms): Langmuir, Freundlich, and BET isotherms: surface coverage, monolayer capacity n_m, and multilayer rise near P₀.
- [Close Packing FCC / BCC / HCP](https://physandbox.com/chemistry/close-packed-lattices): Coordination numbers, maximal packing η, schematic ABC vs AB stacking beside a BCC cubic cell.
- [Chromatography Column](https://physandbox.com/chemistry/chromatography-column): Partition chromatography cartoon: Gaussian bands separate as retention on the stationary phase differs.
- [Michaelis–Menten Kinetics](https://physandbox.com/chemistry/michaelis-menten): v vs [S] saturation and Lineweaver–Burk line from slope Km/Vmax and intercept 1/Vmax.
- [DNA Replication (Schematic)](https://physandbox.com/chemistry/dna-replication-schematic): Fork, leading vs lagging strand, Okazaki fragments — static labeled cartoon.
- [Polymer Random Coil](https://physandbox.com/chemistry/polymer-random-coil): Lattice random walk R_g with Flory exponent ν slider (scaling hint vs Gaussian ν = ½).
- [Belousov–Zhabotinsky (Excitable)](https://physandbox.com/chemistry/belousov-zhabotinsky): Greenberg–Hastings excitable cellular automaton: target waves and rotating spirals as a qualitative BZ analogue.
- [Radioactive Decay & Chain](https://physandbox.com/chemistry/radioactive-decay): N(t) from half-life T₁/₂; optional parent → daughter → stable Bateman ODEs (RK4, normalized).
- [Le Châtelier Principle (Gas)](https://physandbox.com/chemistry/le-chatelier-equilibrium): N₂ + 3H₂ ⇌ 2NH₃: T, V, add/remove species; Q vs K and relaxing mole bars.
- [VSEPR Molecular Shapes (3D)](https://physandbox.com/chemistry/vsepr-shapes): Bonding vs lone pairs on a central atom: AXₙEₘ notation, electron geometry, and ball–lone-pair model.
- [Galvanic (Voltaic) Cell](https://physandbox.com/chemistry/galvanic-cell): Two half-cells, salt bridge, voltmeter; E°cell and Nernst E from ion concentrations (pedagogical E°).
- [Hess's Law (Enthalpy Paths)](https://physandbox.com/chemistry/hess-law): Two-step vs direct ΔH on an enthalpy diagram; sum must match declared overall ΔH.
- [Collision Theory (2D Particles)](https://physandbox.com/chemistry/collision-theory-2d): Hard-disk gas: elastic hits vs an activation speed threshold; T and Eₐ vs exp(−Eₐ/RT).
- [Crystal Field Splitting (Oct / Tet)](https://physandbox.com/chemistry/crystal-field-splitting): Δ_t = (4/9)Δ_o; HS vs LS d⁴–d⁷ from CFSE + pairing P; toy complex hue vs Δ_oct.
- [Unit Cell SC / BCC / FCC](https://physandbox.com/chemistry/crystal-unit-cell): Conventional cubic cells; yaw–pitch projection — lattice sites before basis detail.
- [Sequential Stern–Gerlach](https://physandbox.com/chemistry/stern-gerlach-sequential): Two SG devices: P(up on SG₂) = cos²(θ/2) or sin²(θ/2) after |±z⟩ filter.
- [Molecule Viewer (3D)](https://physandbox.com/chemistry/molecule-viewer): Common molecules in 3D. Rotate, zoom. Ball-and-stick models.
- [Periodic Table](https://physandbox.com/chemistry/periodic-table): Click element for properties, electron config, and uses.
- [Electron Configuration](https://physandbox.com/chemistry/electron-config): Fill orbitals visually with Aufbau principle animation.
- [Balancing Equations](https://physandbox.com/chemistry/balancing-equations): Interactive practice. Drag coefficients. Check balance.
- [pH Scale](https://physandbox.com/chemistry/ph-scale): Slider from 0 to 14 with common substances and color indicators.
- [Titration Simulator](https://physandbox.com/chemistry/titration): Add base to acid. pH curve in real time. Find equivalence point.
- [Reaction Rate](https://physandbox.com/chemistry/reaction-rate): Arrhenius k(T), order 0/1/2, [A](t) curves and particle tank.
- [Orbital Shapes (Schematic)](https://physandbox.com/chemistry/orbital-shapes): 2D |ψ|² colormap for s-, p-, and d-like angular patterns (pedagogical, not HF).
- [Particle in a 1D Box](https://physandbox.com/chemistry/particle-in-box): Superposition of n=1 and n=2: time evolution of |Ψ|² and ⟨E⟩ (model units).
- [Gaussian Wave Packet](https://physandbox.com/chemistry/gaussian-wave-packet): Free spreading: σ(t) from dispersion with ℏ = m = 1; uncertainty intuition.
- [Quantum Well Eigenstates (Box & HO)](https://physandbox.com/chemistry/quantum-well-eigenstates): Stationary ψ_n and |ψ_n|² for infinite square well or harmonic oscillator; optional phase animation (ℏ = 1).
- [Rectangular Barrier Tunneling](https://physandbox.com/chemistry/quantum-tunneling-barrier): Analytic transmission T(E) vs energy for a 1D barrier; tunneling below V₀ and resonances above (ℏ = m = 1).
- [Stern–Gerlach Beam (Cartoon)](https://physandbox.com/chemistry/stern-gerlach-beam): Silver-like beam through an inhomogeneous B-field: atoms deflect to two detectors illustrating S_z = ±ℏ/2.
- [Bloch Sphere & Rabi Drive](https://physandbox.com/chemistry/bloch-sphere-rabi): Two-level Bloch vector with du/dt = u × ω, ω = (0, Ω, Δ); u_x–u_z view, Rabi flopping when Δ = 0.
- [Hydrogen |ψ|² (xz slice)](https://physandbox.com/chemistry/hydrogen-orbital-probability): Coulomb hydrogen Z=1: |R_nl|² × |Y_lm|² colormap in the xz plane for textbook n,l states.
- [Hydrogen: Balmer / Lyman Lines](https://physandbox.com/chemistry/hydrogen-spectral-lines): Rydberg wavelengths 1/λ = R_H (1/n_f² − 1/n_i²); Lyman, Balmer, and Paschen series on a linear spectrum strip.
- [Heisenberg Product σ_x σ_p](https://physandbox.com/chemistry/heisenberg-uncertainty): Gaussian position and momentum widths with ℏ = 1; momentum “excess” over the minimum-uncertainty floor.
- [β⁻ Decay: Electron Spectrum](https://physandbox.com/chemistry/beta-decay-electron-spectrum): Continuous kinetic-energy spectrum up to endpoint Q (allowed-decay phase space); missing energy carried by the antineutrino.
- [Nernst Equation](https://physandbox.com/chemistry/nernst-equation): E = E° − (RT/nF) ln Q: sliders for E°, n, T, and reaction quotient.
- [Buffer Solution](https://physandbox.com/chemistry/buffer-solution): Henderson–Hasselbalch vs strong acid: pH curve as H⁺ is added (mole model).
- [Acid Dissociation α(pH)](https://physandbox.com/chemistry/acid-dissociation-alpha): α = 1/(1+10^(pKa−pH)); half-equivalence at pH = pKa; curves for α and 1−α vs pH.
- [Born–Haber Cycle (NaCl & CaO)](https://physandbox.com/chemistry/born-haber-cycle): Step ΔH° table and enthalpy ladder; lattice energy U = Σ(gas-ion steps) − ΔH_f°.
- [Frost Circle & Aromaticity (4n+2)](https://physandbox.com/chemistry/frost-circle-aromaticity): Inscribed n-gon on Hückel π energies; π count vs 4k+2 / 4k; Aufbau filling.
- [Henry's Law (Gas Solubility)](https://physandbox.com/chemistry/henry-law-gas-solubility): p = k_H c with k_H(T) from ΔH; c vs T at fixed p (ideal-dilute sketch).
- [Gray–Scott Patterns](https://physandbox.com/chemistry/gray-scott-rd): Reaction–diffusion u,v; coral / mitosis / worms / spirals; D_u, D_v, Δt.
- [Gibbs Free Energy](https://physandbox.com/chemistry/gibbs-free-energy): ΔG = ΔH − TΔS; sign vs spontaneity at constant p,T (no Q or K).

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