Complex formula lab

Type real math notation — z and c only, no real/imaginary splitting

Presets

Formula

z → z² + c

Step formula, in terms of z and c

Available notation

z the current value (one complex number)
c this pixel's fixed starting point
i the imaginary unit (i² = -1)
+ - * / arithmetic, exactly as written
z^2 or z^3 powers
abs(z) distance from origin (a real number)
conj(z) flips the sign of the imaginary part
re(z) im(z) real / imaginary part alone
sqrt(z) square root

Coordinate space

Grid size

Center (real)

Center (imaginary)

Zoom width

c ranges over this window of the complex plane

Escape rule

Stop iterating once |z| exceeds Max iterations

Coloring

Palette (exterior - escaped points)

Color = how many iterations until |z| escaped.

Interior coloring (bounded points) Interior palette Interior max iterations

300 (separate from exterior's 80)

Bounded points keep iterating past the exterior budget, purely to refine interior detail. Does not change which points count as escaped vs. bounded — only the quality of color inside the bounded region. Higher values cost more render time.

Reveals structure inside the bounded region that plain black hides. Normalized to the range actually present in this view.

Ready

View

Rotate image

Paint by numbers

Number of color bands

9 bands

Quantizes the gradient into discrete bands so each pixel gets one number. Re-run after changing bands or re-rendering.

Plotter export

Number of pen colors

6 pens (+ 1 black bounded layer)

Physical size

inches, square

Position jitter

10% of cell spacing

Dot length

0.10mm

Mechanically a tiny stroke, not a stationary dot. Smaller reads as a clean point; larger blends into the pen's stroke width and can look bolder/painterly.

view: c spans [-1.5,1.5] + [-1.5,1.5]i

Hover over the grid to watch z's path for that point, step by step