The clicked number is c. Start at z = 0. Square z, add c, and repeat. The small graph follows z; the large plane keeps the colors of the c values you have tested.

Black means the path crossed |z| = 2, which proves escape. Red means it has not escaped within the tested steps. A finite red test does not prove convergence or membership: bounded paths can repeat or behave more intricately.

Each new background sample is tested for up to 256 iterations. The first 300 samples fill a coarse grid; further samples refine it. Refine picture targets the current display’s pixel density, using short background batches. Slower browsers can stop and resume refinement. A colored tile represents its center sample, not a proof about every number inside it.

A clicked path starts with 10 iterations. Continue by 10 or 50. Calculations stop at |z| ≥ 1,000,000 or 10,000 iterations; earlier escape is already marked black. The table shows the most recent 60 values. The path graph fits points within radius 12 and marks later off-scale steps.