I think I've designed a concept of accelerating spaceship in any direction, reaching asymptotically the maximal speed in each direction. Until now there were designs of static speed spaceship or decelerating designs like you could reach with Geminiods. You might be accelerating with geminoids but they will break very fast. My design is based on universal helix. How did we reached until now any directional spaceship? we could convert SLs into universal helix which at some point turns and moves in orthogonal direction using construction of the universal helix + some *WSSs. So we need to convert SL to *WSS stream + horizontal universal helix -> vertical universal helix -> SL at any dx, dy we want, this can be done with any speed and for every tape as long as we like. And universal helix time spending is quadratic while it's construction is linear. So we could build a tape doing *WSS stream into it operations while sending the stream of operations to commit some work, some extra information process in GOL. Which brings me to acceleration: this is obviously possible to make some mechanism that will convert a stream of N *WSS to stream of N + 1 *WSS. And it's possible to convert the stream into the universal helix additional *WSS which makes it move quadratically faster. This is the only thing there is to know - there is some mechanism that converts the whole construction into recursive self construction which somehow growth linearly. The additional growth of data is somehow linear so it should be possible to design a solution in O(1). This means we can accelerate in every direction. So from N additional spaceships our jump is N^2 this means we are quadratically accelerating - and this is a prediction from the theory - that the acceleration is quadratic using this design. Maybe there is a way to control this acceleration? maybe we can generate something like human consciousness in GOL like in Greg Igan novels (for example permutation city), and it will sit inside a spaceship read paper drink coffee and will control the acceleration and deceleration of the spaceship. But currently the design is only quadratic, and I think it wasn't obvious that we can accelerate in every arbitrary direction.

http://www.conwaylife.com/forums/viewtopic.php?p=30246#p30246

EDIT It might be simpler to start from accelerating caterloopillars toward c/4.