Infinity cube

An Infinity cube is a kind of mechanical puzzle toy with mathematical principles. Its shape is similar to a 2×2 Rubik's cube. It can be opened and put back together from different directions, thus creating a visually interesting effect.

An Infinity cube made of dice being played with

An animation showing different moves and states of the Infinity cube (click to animate)

Construction

The hinges are coloured, the glow denoting the next hinges to be employed. From the 6 cycle state of the Infinity cube, you can see that its group representation is isomorphic to ${\displaystyle \mathbb {Z} _{6}}$.

The principle of the infinity cube is simple and can be made by hand with simple paper cutting and pasting. First make 8 small cubes, then arrange the small cubes in a 2 by 2 by 2 way, and tape 8 edges together. When combined, there are 28 small squares exposed and 20 small squares hidden inside.

Mathematics

Like the Rubik's Cube, the various states of the Infinity Cube can be represented as a group, but the Infinity Cube has far fewer permutations than the Rubik's Cube.

Rubik's Cube group have ${\displaystyle 43{,}252{,}003{,}274{,}489{,}856{,}000\,\!={\frac {12!}{2}}\cdot 2^{12-1}\cdot 8!\cdot 3^{8-1}}$ permutations^[1][2] and isomorphic to the below group, where ${\displaystyle A_{n}}$ are alternating groups and ${\displaystyle \mathbb {Z} _{n}}$ are cyclic groups:

${\displaystyle (\mathbb {Z} _{3}^{7}\times \mathbb {Z} _{2}^{11})\rtimes \,((A_{8}\times A_{12})\rtimes \mathbb {Z} _{2}).}$

The largest group representation for Infinity Cube only contains 6 elements, and can be represented as:

${\displaystyle \mathbb {Z} _{6}}$

See also

• Paper model
• Fidget Cube
• 15 puzzle