Infinite Rubik's Cube

Is there an infinite analog to the Rubik's Cube? What does its solution-algorithm look like? For illustration, consider the Rubik's cube with infinite tiles to a side, on all sides, with sides of finite length.

I made up two variants of infinite Rubik cube. One uses the space $\bigl(\Bbb{Q}\cap[-1;1]\bigr)^3$, which has countably many tiles. The other one uses the space $[-1;1]^3$, which has uncountably many tiles. In both cases, each face has its own colour and a twist is performed by rotating a slice which is orthogonal to any of the axes. The cube is solved if and only if all tiles on each face have the same colour.

If the Rubik cube is scrambled by finitely many twists, the cube can be discretised in such way that it is symetrical and each great tile has only one colour. After that, it can be solved as a Rubik cube with finitely many tiles.

The problems starts when the Rubik cube can be scrambled by infinitely many twists. Each scramble can be represented as a bijective function from $R$ to $R$ where $R$ is either $\bigl(\Bbb{Q}\cap[-1;1]\bigr)^3$ or $[-1;1]^3$. The task is to decompose such function to an infinite sequence of twists. There are uncountably many scrambles, so it might not be even algoritmically solvable.