Complex numbers
Lab: what complex numbers are for
Connect formulas and pictures: add waves of one frequency, transform a shape and follow a single orbit towards fractals.
Learning-path index · Read the article · Stage 7: equations and polynomials
Article: fractals, geometry and nature · Open the fractals lab
Model and result
Oscillations and transformations progress continuously. Orbits advance one index per second. Use the slider, rows or left/right arrows and Home/End on the plot.
Complex plane
The selected step
Iterations and coordinates use floating point: rounding matters near boundaries. The mathematical criterion is strict: |z|>R, not |z|≥R.
Time projections: x₁, x₂ and sum
Calculation table
Select a row to inspect its step. The table scrolls horizontally on mobile.