Lenore and Windell at Evil Mad Scientist Laboratories made this mobius circuit as a way to create a true “single sided” circuit board. They used watercolor paper:

While it is possible to have your own flexible circuit boards commercially fabricated (at a remarkably high cost), we’re executing a much cheaper DIY version here. There are a number of ways that you might go about making your own flexible circuitry we’re using a paper strip as our substrate and conductive ink for wiring. One of the reasons for this is that our wiring can stay strictly on the one surface of the paper strip, without punching through it. (Maintaining topological integrity is a strong motivation.)

For our circuit, we decided to make an LED chaser that would light up LEDs in sequence along the length of the Mobius strip. To do this we chose a smallish AVR microcontroller, the ATtiny2313, which is a 20-pin device with 17 available output pins. With each output we drive a single red LED, and the whole thing can be powered by a 3V type CR2032 lithium coin cell.