Complex Exp
w = exp(z · exp(i · t · 0.3))
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The exponential
Read exp(z · exp(i · t · 0.3)) from the inside out. exp(i · t · 0.3) has modulus 1 whatever the clock reads, so the inner product only turns z about the origin, by 0.3 radians per second at Speed 1. Set it aside for a moment and the function is exp(z), whose two halves come apart cleanly: |w| = e^x depends on the real part alone and arg w = y on the imaginary part alone.
Why it is a plaid
The hue repeats every 2π ≈ 6.28 along the imaginary direction, in bands running parallel to the real axis. The plot is 8 wide at the opening Span of z, so a little over one full sweep of the colour wheel fits. The brightness bands fall where x/log(2) is a whole number, straight lines log(2) ≈ 0.693 apart crossing the hue bands at a right angle.
The relief is all on one side. At x = 4, |w| ≈ 54.6 and the height is 0.6 · log(55.6) ≈ 2.41. At x = −4, |w| ≈ 0.018 and the sheet lies flat on the ground. A cliff on the right, a plain on the left, and no zero anywhere: e^x is never 0, so the exponential never touches the ground and never has a pole either.
Restore the rotation and the whole plaid turns rigidly about the origin, once every 2π/0.3 ≈ 20.9 seconds, dragging the cliff around with it.
Try
- Top: the plaid seen flat on, with the two families of bands square to each other.
- Raise Iterations to 20 so the formula is fed its own output over and over. Every point whose orbit escapes drops to the outside tone, shaded by how soon it left.
- Halve the turn rate with exp(z · exp(i · t · 0.15)): a full revolution now takes 2π/0.15 ≈ 41.9 seconds.
- Compress the plaid with exp(2z · exp(i · t · 0.3)): the hue bands and the brightness lines both halve their spacing, and the cliff doubles in steepness.
- exp(1/z): the same striping wound around an essential singularity at the origin, where every value but 0 is taken over and over in any disc you care to draw.