title: Infinity cube
text: # 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.
  
  ## Construction
  
  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 $43{,}252{,}003{,}274{,}489{,}856{,}000\,\!={\frac {12!}{2}}\cdot 2^{12-1}\cdot 8!\cdot 3^{8-1}$ permutations and isomorphic to the below groupware $A_{n}$ 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:
  
      $\mathbb {Z} _{6}$
  