# 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.

• Problems might just be slices at real valued intervals through the cube rotated iteratively until a confused state is reached and then solving it. Jan 8, 2019 at 1:58
• You might want to look up Thompson's groups $F$, $T$ and $V$, and also the Brin-Thompson groups $nV$. They are not exactly what you are looking for, but I think they might be close. Certainly, they are "slices" of the interval $[0, 1)$, and the groups $nV$ are slices of the $n$-cube $[0, 1)^n$. They also have rather pleasing graphical representations. Jan 8, 2019 at 11:55
• You have to say what you mean. For example, it is the cube $[-1,1]^3$, but rotations can be made fixing any subset of one of the axes: rotate the planes corresponding to that subset, fix the rest of the planes Jan 18, 2019 at 11:42
• I wonder how an infinite rubik's cube can have finite sides? Jan 21, 2019 at 3:14

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.
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.