Math Visualization
Vectors, trigonometry, graphing, Fourier series, and fractals
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)
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
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)
Recursive optimal estimation: noisy measurements, hidden truth, predict + update with Q and R; random-walk or constant-velocity model with ±2σ band and innovations.
Lattice Boltzmann D2Q9 Flow
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
Conservative scalar transport on a 2D grid: face fluxes, upwind vs central interpolation, Peclet number, CFL, and numerical diffusion.
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
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
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
Compare plain Monte Carlo, importance sampling, and stratified sampling for ∫f(x)dx, with convergence curves, standard error, and the 1/√N rate.
LMS / NLMS Adaptive Noise Cancellation
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)
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)
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
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
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
Drag/wheel deep zoom into the Mandelbrot set with smooth continuous coloring and named landmarks.
Julia Set Explorer
Pick c by clicking the embedded mini-Mandelbrot or animate c along a circle; Fatou dust vs connected sets.
Newton Fractal
Basins of attraction for Newton iteration on zⁿ−1 with adjustable relaxation ω.
Newton's 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
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
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
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)
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
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
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
RK4 integration of ẋ=−y−z, ẏ=x+ay, ż=b+z(x−c); period-doubling cascade as c grows.
L-Systems (Turtle)
Lindenmayer string rewriting + turtle: Koch, Sierpinski, Hilbert, Heighway dragon, plant.
Bézier & de Casteljau
Drag control points; live recursive linear-interpolation scaffolding evaluates B(t).
Convex Hull (Graham & QuickHull)
Click to add points, drag to move; Graham scan with step playback or QuickHull divide-by-farthest; compare vertex sets.
Delaunay & Voronoi
Bowyer–Watson triangulation and dual Voronoi tessellation; click to add seeds, drag to move.
Physarum Slime (Agents)
~4500 agents follow a deposited chemoattractant: deposit + diffusion + decay + 3-sensor steering grow path networks.
Savitzky–Golay Smoothing
Noisy cosine vs SG(7,2) convolution — preserves peaks better than a wide boxcar.
Markov Chain (Weather)
Sun/Rain two-state chain: P matrix, stationary π, empirical vs theory.
Gradient Descent (2D)
Level sets of f(x,y) and path (x,y) ← (x,y) − η∇f; bowl or elliptic well.
Minkowski Diagram
Light cone and boosted axes in 1+1D; γ from v.
Twin Paradox
Out-and-back worldlines; proper time τ = T/γ vs Earth time T.
Monte Carlo π
Uniform samples in a square; 4·(in disk)/N estimates π.
Random Walk
1D or 2D steps; trail and running mean ⟨r²⟩ vs diffusion intuition.
Random Walk (2D / 3D Lattice)
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
Place vectors and see the resultant with head-to-tail animation.
Trigonometry Circle
Unit circle with live sin, cos, tan values as you drag.
Function Grapher
Enter f(x) and see instant plots with zoom and pan.
Fourier Series
Build waveforms from sine waves. Add harmonics one by one.
FFT Magnitude Spectrum
Paint or preset a 256-point signal; radix-2 FFT shows |X[k]| vs bin (DC to Nyquist).
2D Phase Portrait (ODE)
Direction field and click-to-trace trajectories for planar systems: harmonic, damped, saddle, nodes, foci, pendulum (RK4).
Lissajous Curves
Beautiful patterns from two frequencies with adjustable ratio.
Harmonograph
Two damped harmonic sums in x and y: decaying rosette trace vs Lissajous loops.
Spirograph (Trochoids)
Hypo- or epitrochoid: fixed R, rolling r, pen d; hue trail and period hints.
Sorting Algorithms (parallel)
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
S + I + R = 1: βSI and γI; ℛ₀ ≈ β/γ, herd threshold 1 − 1/ℛ₀; RK4 time plot.
SEIR / SEIRS Epidemic Model
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)
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)
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
∂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)
1D reaction–diffusion: membrane MinD u, fast MinE v; pole-to-pole oscillation; division plane at time-averaged MinD minimum.
Sandpile (SOC)
BTW abelian model: add grains, ≥4 topples to neighbors; critical avalanches.
Flow Field Particles
Synthetic v(x,y,t); advection with wrap; optional arrow grid.
Fractal Generator
Mandelbrot, Julia, Koch snowflake. Zoom infinitely.
Conway's Game of Life
B3/S23 on a torus: paint cells, run, step — glider, LWSS, Gosper gun, pulsar, and more.
a → v → x
Integrate acceleration to velocity and position; stacked time graphs.
Taylor Polynomial
sin, cos, or exp vs Taylor sum about center a up to order n.
Complex Phasor
exp(iωt) on the unit circle; Re, Im, and phase φ.
Chaos Game (Sierpiński)
Random vertex + midpoint walk; the attractor is the Sierpiński gasket — try RGB by vertex.
Lagrange vs Cubic Spline
Click knots: Lagrange polynomial vs natural cubic spline; Runge preset shows edge oscillations.
Least Squares Fit
Noisy linear data; fitted slope and intercept with residuals.
Linear Regression: OLS, Ridge, Lasso & R²
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)
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
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)
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)
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)
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)
Two rectangular pulses; overlap length at τ = 0.
Euler vs RK4 (Pendulum)
Same nonlinear pendulum ODE and step h; Euler vs RK4 side by side.
Lotka–Volterra
N′ = αN−βNP, P′ = δNP−γP; phase plane RK4; equilibrium dot.
Logistic Growth
dN/dt = rN(1−N/K); exact S-curve vs carrying capacity K.
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
Grid deformation under M; real λ eigen-direction arrows.
Lorenz Strange Attractor
σ, ρ, β ODEs; sensitive butterfly in (x,z) projection — RK4 trace.