See the equations you've been told to memorize — in 3D, in real time, fully interactive.
No textbook explains it better than watching it move. Every module is interactive — drag sliders, paint on the canvas, zoom into infinity.
All 12 visualizations are free in your browser. Scroll down to interact, or jump straight to a module below.
A strange attractor born from a three-variable ODE. Even an infinitesimal change in initial conditions produces a wildly different trajectory — the butterfly effect, made visible. Trail colored by instantaneous velocity magnitude.
eiθ = cos θ + i·sin θ — the most beautiful equation in mathematics, unwrapped into 3D. The Z axis represents θ; the helix is the path of eiθ. Its shadow on the XZ plane is cosine, on YZ plane is sine.
A 2D water-surface simulation solved with finite-difference time-domain (FDTD) integration. Click anywhere on the surface to drop a ripple, then watch interference patterns bloom.
Thermal diffusion on a 2D surface. Click and drag to paint heat; watch it spread. The inferno colormap maps temperature to color: black (cold) → purple → orange → white (hot).
Any periodic signal is a sum of rotating circles. N epicycles (each spinning at an integer multiple of the base frequency) trace the target waveform in real time.
See what a 3×3 matrix does to 3D space — live. Edit the matrix, pick a preset, and watch 1331 grid points morph smoothly. Eigenvalues, determinant, and SVD computed in real time.
A 3×3 matrix maps every point v ∈ ℝ³ to Mv. The grid of points shows the image of the standard cube. Eigenvectors (gold arrows) are the only directions preserved by the map. The determinant measures how volumes are scaled — negative means orientation flips, zero means the space collapses. The SVD decomposes any matrix into a rotation, a stretch, and another rotation: M = U·Σ·Vᵀ.
Fifty price paths evolve in real time, colored by their terminal value. The Itô correction σ²/2 is what separates the correct log-drift from the naïve one — a subtle but crucial term that underpins all of quantitative finance.
Itô's lemma tells us that for f(S) = ln S and dS = μS dt + σS dW: d(ln S) = (μ − σ²/2) dt + σ dW. The extra −σ²/2 term arises because ln is concave — Jensen's inequality at the stochastic level. It's not a rounding error; it's the fundamental price of randomness, and it's why real asset returns disappoint naïve expectations.
Watch how polynomial partial sums "hug" a function from the center outward, term by term. The colored dashed curve grows one power at a time; the shaded band shows where the approximation diverges.
Every polynomial of degree n has exactly n complex roots. Watch them glow as you drag the coefficient sliders — conjugate pairs stay mirror-symmetric across the real axis.
A point traces the unit circle as θ increases. Green, orange, and purple lines project its sin, cos, and tan values — then those same values unroll into waves along the Z axis.
Multivalued functions like √z become single-valued on a Riemann surface — the complex plane split along a branch cut and glued into a single connected 3D manifold. Each color band is one copy of ℂ.
M = {c ∈ ℂ : zn+1 = zn² + c stays bounded from z0=0}. Each Julia set Jc is the set of starting points z that stay bounded for a fixed c. Click any point on the Mandelbrot view to reveal its Julia set.