When I was a kid, I was obsessed with an idea about making a wheeless vehicle...

If you sit on a chair with your feet off of the ground, and thrust or jerked your arms and legs back towards your body, you could cause the chair to scooch forward a few inches under you. If you did that over and over again, you scooch across the room a few inches at a time. I then reasoned that if you sat inside of a box, and did that, you get to the box to scooch forward, or even sideways depending on which way you jerked your weight. I then thought that you should be able to make a machine do what your body was doing, and the machine, sitting inside of a box, would cyclically jek weights and scooch across the floor little by little. Then I wondered if you could shorten the stroke and cycle it faster and faster until the jerks and scooches started blending together into a smooth motion. And finally, you would have a machine in a box sitting on the floor that would just drive around without wheels.

Later I realized that this is basically what a vibratory conveyor does for moving parts and food around a sorting machine. If you flipped the machine upside down, then the machine should be able to move around the floor using asymmetric vibrations, kind of like an unbalanced washing machine that tends to walk away from the wall.

The final mind-job came whit it occurred to me that you could tilt the machine back 90°, and it would attempt to jerk or leap up instead of forward. So the final question became, could you get enough friction from the air to cause an impossibly lightweight machine to hover or 'fly' without wings or expelling propulsion, just by vibrating asymmetrically and scooching through the air.