A few years ago a Harvard mathematician, L. Mahadevan, proposed that if we could get a surface to undulate similar to that of a ray along the seafloor it was mathematically possible to have our own flying carpets. Last fall Princeton grad Noah Jafferis put that idea to the test and designed a small model. Using plastic sheets that expand or contract dependent on electric charge delivered to several areas on the sheet, it could produce the wave motion necessary to stay afloat.
This prototype has a limited range due to being tethered by the wires to deliver charge, but Jafferis hopes to include some sort of power source soon. On a larger scale, it isn’t very feasible; a 50 square foot sheet would be required to get a human a millimeter off the ground. Jafferis thinks that the lower gravity on Mars is a more practical application. Once we get over the whole getting to and living on Mars, that is.