## Wednesday, August 4, 2010

### Tesseract

Tesseract is an analog of cube in higher dimensions. A fourth dimensional cube is called Hinton's cube or hypercube.

Imagine the formation of a 3D cube. It can be done by folding 6 squares (2D cross) in 3rd dimension as shown in the figure.

A 3D cross can be formed by keeping 8 cubes as shown in figure.

By folding this 3D cross in 4th dimension we can get a 4D hypercube. By doing the exact opposite ie. by unraveling a 4D hypercube u get a 3D cross. So this is one of the methods to interpret a 4D object.

But we cannot visualize the folding of the 3D cross. To understand why, we have to take the analog of flat-lander (2D being) living on a plane. He can see the 2D cross. But when the 2D cross is folded, a square moves up into 3rd dimension and out of the plane. So the square would disappear from his world. With each folding the squares keeps on disappearing. Finally only one square would be left in his plane. This one square would depict a 3D cube for the flat-lander.

Similarly when the 3D cross is folded, each cube will start disappearing from our world. Finally only one cube would be left depicting a 4D tesseract.