Explainer Lab
Circles stacked on circles draw a square wave, and the Gibbs bump never disappears: an interactive lesson bench for the Fourier series and the heat equation.
Circles stacked on circles draw a square wave, and the Gibbs bump never disappears: an interactive lesson bench for the Fourier series and the heat equation.
Try Websites AgentBuild an interactive math lesson bench called Explainer Lab, with two lessons. The first lesson teaches the Fourier series: one circle is one sine wave, where radius is amplitude, rotation speed is frequency, and starting angle is phase. Attach a smaller circle to the pen tip of the previous one, keep stacking circle on circle, and watch a square wave emerge from the odd harmonics — a harmonics slider adds more terms and the sum gets squarer, yet the corner always keeps a bump: the Gibbs phenomenon, an overshoot of about nine percent that only gets narrower, never shorter. The second lesson teaches the heat equation: a point warms up in proportion to how much colder it is than its neighbors, so a metal plate with a hot center diffuses frame by frame toward equilibrium; clicking anywhere on the plate adds another heat source, and the page carries notes on diffusivity and boundary conditions beside the simulation. Single-file static page.
Adds sawtooth and triangle targets next to the square wave, so visitors can switch the goal shape and watch how the same stacked circles approximate each waveform differently.
Try Websites AgentBuild an interactive math lesson bench called Explainer Lab, with two lessons. The first lesson teaches the Fourier series: one circle is one sine wave, where radius is amplitude, rotation speed is frequency, and starting angle is phase. Attach a smaller circle to the pen tip of the previous one, keep stacking circle on circle, and watch a square wave emerge from the odd harmonics — a harmonics slider adds more terms and the sum gets squarer, yet the corner always keeps a bump: the Gibbs phenomenon, an overshoot of about nine percent that only gets narrower, never shorter. The second lesson teaches the heat equation: a point warms up in proportion to how much colder it is than its neighbors, so a metal plate with a hot center diffuses frame by frame toward equilibrium; clicking anywhere on the plate adds another heat source, and the page carries notes on diffusivity and boundary conditions beside the simulation. Single-file static page.
Restyles the bench as a light lecture handout on paper-white, while both lessons — the Fourier circles and the heat plate — keep working exactly as before.
Try Websites AgentBuild an interactive math lesson bench called Explainer Lab, with two lessons. The first lesson teaches the Fourier series: one circle is one sine wave, where radius is amplitude, rotation speed is frequency, and starting angle is phase. Attach a smaller circle to the pen tip of the previous one, keep stacking circle on circle, and watch a square wave emerge from the odd harmonics — a harmonics slider adds more terms and the sum gets squarer, yet the corner always keeps a bump: the Gibbs phenomenon, an overshoot of about nine percent that only gets narrower, never shorter. The second lesson teaches the heat equation: a point warms up in proportion to how much colder it is than its neighbors, so a metal plate with a hot center diffuses frame by frame toward equilibrium; clicking anywhere on the plate adds another heat source, and the page carries notes on diffusivity and boundary conditions beside the simulation. Single-file static page.
Adds a key-points summary card at the end of each lesson, so after dragging the harmonics slider or clicking the heat plate, readers can review what the demonstration showed.
Try Websites AgentBuild an interactive math lesson bench called Explainer Lab, with two lessons. The first lesson teaches the Fourier series: one circle is one sine wave, where radius is amplitude, rotation speed is frequency, and starting angle is phase. Attach a smaller circle to the pen tip of the previous one, keep stacking circle on circle, and watch a square wave emerge from the odd harmonics — a harmonics slider adds more terms and the sum gets squarer, yet the corner always keeps a bump: the Gibbs phenomenon, an overshoot of about nine percent that only gets narrower, never shorter. The second lesson teaches the heat equation: a point warms up in proportion to how much colder it is than its neighbors, so a metal plate with a hot center diffuses frame by frame toward equilibrium; clicking anywhere on the plate adds another heat source, and the page carries notes on diffusivity and boundary conditions beside the simulation. Single-file static page.
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