It's fascinating to see the various ways of converting liquid fuel to horsepower. The Otto piston has long dominated, but is famously inefficient. Wankel was a nice challenge, but suffered from low compression and its oil squeegee valves burned more than they wiped.
The LiquidPiston guys set out solve the problems of both designs, and they may be getting close with this inverse Wankel approach. The picture above is of the older X1 design and the text at this link is distracting, but watch the video for a quick visualization:
Autoevolution on the LiquidPiston X Mini Engine
Here's a much better presentation of the technology:
Gizmag on the LiquidPiston X2 Engine
And here is a description of their early prototype:
MIT Technology Review of the X0 LiquidPiston
It will be interesting to see how this technology evolves.
Another weight and space efficient engine:
The Duke Axial
A better movie