Infinity Cube

Infinity Cube

Infinity cube^[1] is a kind of mechanical puzzle toy with mathematical principles, the shape is similar to a ${\displaystyle 2\times 2}$ Rubik's cube. It can be opened and put together from different directions, thus creating a visually interesting effect.

Construction[]

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 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 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:

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

See also[]

• Paper model
• Fidget Cube
• 15 puzzle

References[]

External links[]

Wikimedia Commons has media related to Infinity Cube.

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