Math Visualization
Vectors, trigonometry, graphing, Fourier series, and fractals
- NewUniversity / research
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.
- NewUniversity / research
Morlet Wavelet (CWT)
Continuous wavelet transform with the complex Morlet wavelet: scaleogram |W(s,t)|, log-frequency axis, cone of influence, adjustable ω₀ a…
- NewUniversity / research
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.
- NewUniversity / research
Kalman Filter (1-D)
Recursive optimal estimation: noisy measurements, hidden truth, predict + update with Q and R; random-walk or constant-velocity model wit…
- NewUniversity / research
Kalman Filter 2D Tracking
4-state constant-velocity tracker: noisy (x,y) measurements, process acceleration noise Q, measurement variance R, and a live ≈2σ covaria…
- NewUniversity / research
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.
- NewUniversity / research
Finite-Volume Advection-Diffusion 2D
Conservative scalar transport on a 2D grid: face fluxes, upwind vs central interpolation, Peclet number, CFL, and numerical diffusion.
- NewUniversity / research
Conjugate Gradient Solver
SPD system Ax=b as quadratic minimization: contour geometry, CG vs steepest descent path, residual norm, and condition number.
- NewUniversity / research
Power Iteration Eigenvalue Convergence
Visualize dominant eigenvector convergence: spectral gap ratio, Rayleigh quotient, eigen residual, and normalized power iterates on the u…
- NewUniversity / research
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 fai…
- NewUniversity / research
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/√…
- NewUniversity / research
PCA / SVD Geometry
Covariance ellipse and SVD view of dimensionality reduction: principal components, explained variance, and rank-1 reconstruction error.
- NewUniversity / research
Gradient Descent Optimizers
Compare SGD, momentum, and Adam on a curved loss landscape; tune learning rate, curvature, stability, and iteration count.
- NewUniversity / research
Gaussian Mixture EM Algorithm
Expectation-maximization for a two-component Gaussian mixture: responsibilities, E/M steps, likelihood, and covariance ellipses.
- NewUniversity / research
Support Vector Machine Margin
Hard/soft-margin SVM sketch with C penalty, hinge loss, support vectors, decision boundary, and margin bands.
- NewUniversity / research
Bayesian Updating / Conjugate Priors
Beta-binomial Bayesian updating with prior/posterior curves, posterior predictive probability, and credible intervals.
- NewUniversity / research
Markov Chain Mixing
Three-state Markov chain with transition matrix, stationary distribution, total variation distance, detailed-balance cue, and spectral-ga…
- NewUniversity / research
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…
- NewUniversity / research
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 fo…
- NewUniversity / research
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 loc…
- NewUniversity / research
ΔΣ (1-bit) Modulator
First- and second-order discrete-time ΔΣ with ±1 quantizer: shaped quantization noise, sine test tone, boxcar reconstruction, and Hann-wi…
- NewUniversity / research
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-identi…
- NewUniversity / research
Mandelbrot Deep Zoom
Drag/wheel deep zoom into the Mandelbrot set with smooth continuous coloring and named landmarks.
- NewUniversity / research
Julia Set Explorer
Pick c by clicking the embedded mini-Mandelbrot or animate c along a circle; Fatou dust vs connected sets.
- NewUniversity / research
Newton Fractal
Basins of attraction for Newton iteration on zⁿ−1 with adjustable relaxation ω.
- NewSchool
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.
- NewUniversity / research
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 co…
- NewUniversity / research
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, an…
- NewUniversity / research
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…
- NewUniversity / research
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…
- NewUniversity / research
Multigrid Relaxation
1D Poisson error dynamics: Jacobi and Gauss-Seidel smooth high-frequency error, while a V-cycle uses residual restriction and coarse-grid…
- NewUniversity / research
Gaussian Process Regression
Interactive GP regression with RBF and Matérn kernels: tune length scale/noise, add observations by clicking, view posterior mean with un…
- NewUniversity / research
Rössler Attractor
RK4 integration of ẋ=−y−z, ẏ=x+ay, ż=b+z(x−c); period-doubling cascade as c grows.
- NewSchool
L-Systems (Turtle)
Lindenmayer string rewriting + turtle: Koch, Sierpinski, Hilbert, Heighway dragon, plant.
- NewSchool
Bézier & de Casteljau
Drag control points; live recursive linear-interpolation scaffolding evaluates B(t).
- NewSchool
Convex Hull (Graham & QuickHull)
Click to add points, drag to move; Graham scan with step playback or QuickHull divide-by-farthest; compare vertex sets.
- NewSchool
Delaunay & Voronoi
Bowyer–Watson triangulation and dual Voronoi tessellation; click to add seeds, drag to move.
- NewUniversity / research
Physarum Slime (Agents)
~4500 agents follow a deposited chemoattractant: deposit + diffusion + decay + 3-sensor steering grow path networks.
- NewSchool
Savitzky–Golay Smoothing
Noisy cosine vs SG(7,2) convolution — preserves peaks better than a wide boxcar.
- NewSchool
Markov Chain (Weather)
Sun/Rain two-state chain: P matrix, stationary π, empirical vs theory.
- NewSchool
Gradient Descent (2D)
Level sets of f(x,y) and path (x,y) ← (x,y) − η∇f; bowl or elliptic well.
- NewUniversity / research
Minkowski Diagram
Light cone and boosted axes in 1+1D; γ from v.
- NewSchool
Twin Paradox
Out-and-back worldlines; proper time τ = T/γ vs Earth time T.
- NewKids
Monte Carlo π
Uniform samples in a square; 4·(in disk)/N estimates π.
- NewSchool
Random Walk
1D or 2D steps; trail and running mean ⟨r²⟩ vs diffusion intuition.
- NewSchool
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-…
- FeaturedSchool
Vector Addition
Place vectors and see the resultant with head-to-tail animation.
- FeaturedSchool
Trigonometry Circle
Unit circle with live sin, cos, tan values as you drag.
- Kids
Function Grapher
Enter f(x) and see instant plots with zoom and pan.
- FeaturedSchool
Fourier Series
Build waveforms from sine waves. Add harmonics one by one.
- NewSchool
FFT Magnitude Spectrum
Paint or preset a 256-point signal; radix-2 FFT shows |X[k]| vs bin (DC to Nyquist).
- NewUniversity / research
2D Phase Portrait (ODE)
Direction field and click-to-trace trajectories for planar systems: harmonic, damped, saddle, nodes, foci, pendulum (RK4).
- School
Lissajous Curves
Beautiful patterns from two frequencies with adjustable ratio.
- NewKids
Harmonograph
Two damped harmonic sums in x and y: decaying rosette trace vs Lissajous loops.
- NewKids
Spirograph (Trochoids)
Hypo- or epitrochoid: fixed R, rolling r, pen d; hue trail and period hints.
- NewKids
Sorting Algorithms (parallel)
Bubble, insertion, merge, quicksort, and heapsort on the same shuffled permutation — five bar rows advance in lockstep so you can compare…
- NewSchool
SIR Epidemic Model
S + I + R = 1: βSI and γI; ℛ₀ ≈ β/γ, herd threshold 1 − 1/ℛ₀; RK4 time plot.
- NewSchool
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.
- NewUniversity / research
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).
- NewSchool
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.
- NewUniversity / research
Keller–Segel Chemotaxis
∂ₜn = Dₙ∇²n − χ∇·(n∇c), ∂ₜc = D_c∇²c + αn − βc; bacteria follow attractant; collapse at high χ; 96² grid.
- NewUniversity / research
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.
- NewSchool
Sandpile (SOC)
BTW abelian model: add grains, ≥4 topples to neighbors; critical avalanches.
- NewUniversity / research
Flow Field Particles
Synthetic v(x,y,t); advection with wrap; optional arrow grid.
- NewSchool
Fractal Generator
Mandelbrot, Julia, Koch snowflake. Zoom infinitely.
- NewKids
Conway's Game of Life
B3/S23 on a torus: paint cells, run, step — glider, LWSS, Gosper gun, pulsar, and more.
- NewSchool
a → v → x
Integrate acceleration to velocity and position; stacked time graphs.
- NewSchool
Taylor Polynomial
sin, cos, or exp vs Taylor sum about center a up to order n.
- NewSchool
Complex Phasor
exp(iωt) on the unit circle; Re, Im, and phase φ.
- NewKids
Chaos Game (Sierpiński)
Random vertex + midpoint walk; the attractor is the Sierpiński gasket — try RGB by vertex.
- NewUniversity / research
Lagrange vs Cubic Spline
Click knots: Lagrange polynomial vs natural cubic spline; Runge preset shows edge oscillations.
- NewSchool
Least Squares Fit
Noisy linear data; fitted slope and intercept with residuals.
- NewSchool
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 s…
- NewSchool
K-Means Clustering (Lloyd)
Click to add points, choose k, randomize centroids, then step Lloyd iterations (assign to nearest centroid, update means). Optional Gauss…
- NewSchool
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 o…
- NewSchool
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…
- NewSchool
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; sha…
- NewSchool
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**…
- NewUniversity / research
Convolution (pulses)
Two rectangular pulses; overlap length at τ = 0.
- NewUniversity / research
Euler vs RK4 (Pendulum)
Same nonlinear pendulum ODE and step h; Euler vs RK4 side by side.
- NewSchool
Lotka–Volterra
N′ = αN−βNP, P′ = δNP−γP; phase plane RK4; equilibrium dot.
- NewSchool
Logistic Growth
dN/dt = rN(1−N/K); exact S-curve vs carrying capacity K.
- NewUniversity / research
Logistic Map Bifurcation
x_{n+1}=rx_n(1−x_n): scan r, plot attractors — period doubling to chaos (Feigenbaum cascade).
- NewSchool
2×2 Matrix & Eigenvectors
Grid deformation under M; real λ eigen-direction arrows.
- NewUniversity / research
Lorenz Strange Attractor
σ, ρ, β ODEs; sensitive butterfly in (x,z) projection — RK4 trace.