Infinity cube

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.

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 groupware ${\displaystyle A_{n}}$ 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:

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

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Wikimedia Commons has media related to Infinity Cube.

• Magic Folding Cube - Mathematische-Basteleien
• How To Make A Paper INFINITY CUBE! on YouTube