Self-avoiding space-filling folding curves in dimension 3

Various examples of folding curves in $R^2$ have been considered: dragons and other square curves, terdragons and other triangular curves, Peano-Gosper curves based on hexagons. They are self-avoiding. They form coverings of $R^2$, by one curve or by a small number of curves, which satisfy the local isomorphism property. They were used to define some fractals. We construct an example with similar properties in $R^3$.