## Hyperbolic Space

Here are two computer-animated videos from Youtube that demonstrate some properties of knots, the relations between knots and space, and how we arrive at hyperbolic space. Easy to watch, and very informative.   Part Two:   Related posts: Klein Bottle Tesseracts: from 3 to 4 Dimensions. 0 to 6 dimensions and back – simple rotation. […]

## Turning a Sphere inside-out

How do you turn a sphere inside out, without punching a hole into it? It is possible,as you can see in the transformations in this fascinating video: Related posts: Mathematical Problems Klein Bottle Unsolved Millenium Problems Tesseracts: from 3 to 4 Dimensions. 0 to 6 dimensions and back – simple rotation. Hyperbolic Space The German […]

## 0 to 6 dimensions and back – simple rotation.

This is a simple computer-simulated rotation from a point, which has 0 dimensions, to a line (1 dimension), a square (2), a cube (3), all the way up to 6 dimensions, and then down. Multi-dimensional objects are much more complex than we can imagine.   Related posts: Tesseracts: from 3 to 4 Dimensions. Klein Bottle […]

## Tesseracts: from 3 to 4 Dimensions.

The short video clip below shows the 3D rotations of a 4D object. The deeper questions concern the nature of a “dimension”. How do we know what a dimension is, and if we live in a 3D universe, could we possible also exist in a higher-dimensional universe? it is best to start with simple examples in order to train the mind to think about these questions. The clip below shows a tesseract, which is the four-dimensional analog of a cube. (In geomery, it is called a regular octachoron or cubic prism.) The tesseract is to the cube as the cube is to the square.