Infinity Cube facts for kids

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

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Infinity Cube Facts for Kids. Kiddle Encyclopedia.