They're created when a dielectric — a material that resists the flow of electricity — experiences a catastrophic failure. As such, they're often found in failed insulation when high-voltage equipment kills itself.
Hit any dielectric with a high enough voltage and you can generate one. The most impressive modern artistic examples are the Beam Trees made with a block of acrylic and a linear particle accelerator:
If you are unlucky enough to be struck directly by lightning, you may receive a Lichtenberg figure tattoo, sometimes called a 'Lichtenberg Flower' (Google Image Search at your own risk).
I don't have a ten million electron-volt particle accelerator, but I do have a car ignition coil. Here, I hit a Petri dish filled with tap water with 20,000 volts generated by the ignition coil running on a square-wave function generator. The central electrode is a few centimeters above the water's surface; the edge electrode is slightly submerged:
I don't have a ten million electron-volt particle accelerator, but I do have a car ignition coil. Here, I hit a Petri dish filled with tap water with 20,000 volts generated by the ignition coil running on a square-wave function generator. The central electrode is a few centimeters above the water's surface; the edge electrode is slightly submerged:
Anyone can build a high-voltage generator of this sort for under $25. If there's interest, I can post detailed designs for some of my generators. Not surprisingly, they're pretty dangerous when mishandled.
Other artists have also used dielectric breakdown to very cool effect.