Infinity cube

Infinity Cube

Infinity cube is a kind of mechanical puzzle toy with mathematical principles, the shape is similar to a $2\times 2$ 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 $\mathbb {Z} _{6}$.

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.

Mathematics

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

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

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

$\mathbb {Z} _{6}$

See also

• Paper model
• Fidget Cube
• 15 puzzle

External links

• Magic Folding Cube - Mathematische-Basteleien
• 翻轉方塊製作教學 - 安妮手作
• 翻轉方塊製作教學 - 雅菲