I made one of them, great fun. Started with some huge Russian paper capacitors and built it as a state machine using a PALCE22V10 gate array and 555 timers.

Charge the capacitors
Dump the charge across the spark gap
Look see what is left on the capacitors...
Fully charged = didn't spark, move down one step
Dead flat = shorted out move up one step
Residual = did spark, stay where you are

The every few seconds lift clear so the electrolyte can wash the soot away.

I made two mistakes...
Trying to pump paraffin electrolyte using a peristaltic pump, who'd have thought that paraffin would degrade just about every kind of silicone tubing to the point where it splits.

Should have used a high voltage initiator spark followed by a timed burst of 40 Volts, my erosion voltage was too high, a bit hard on the copper electrodes.