Infinity Cube

The dark colored line segments are the taped position, while the light black line segments are not taped. 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 squares, then arrange the small squares in a 2 by 2 by 2 way, and tape 8 edges together. When combined, there are 24 small squares exposed and 24 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 have far fewer variations than the Rubik's Cube, so the representation is also much simpler.

Rubik's Cube group have ${\displaystyle 43{,}252{,}003{,}274{,}489{,}856{,}000\,\!={\frac {12!}{2}}\cdot 2^{12-1}\cdot 8!\cdot 3^{8-1}}$ elements[2][3]，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:

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

• Paper model
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