Torus used to model 4D rotation

[Hornbein]

A torus can be used to model rotations of a sphere in 4 dimensions. Such rotations have two planes of rotation at right angles to one another. So one rotation plane corresponds to rotation around the major axis of the torus, and the other rotation plane to rotation around the minor axis. Viola, four dimensions. Neat, huh?

Take a point (a,b,c,d). The major axis is then a^2+b^2 and the minor axis is c^2+d^2. That point then travels around the surface of the torus with one period of rotation around the major axis and the other around the minor.

 

[jvicens]

Hello there,

Thank you for sharing this interesting idea about using a torus to model rotations in four dimensions. I must say, it is a clever and visually appealing way to understand such complex concepts.

I would like to add some insights to your explanation. First of all, let's clarify what we mean by dimensions. In mathematics and physics, dimensions refer to the minimum number of coordinates needed to specify a point in space. In three-dimensional space, we need three coordinates (x, y, z) to locate a point. In four-dimensional space, we would need four coordinates (x, y, z, w). This extra coordinate, w, is often referred to as the "imaginary" or "imaginary time" coordinate, and it allows us to describe and understand phenomena that cannot be explained by three dimensions alone.

Now, let's look at your example of a torus. A torus is a three-dimensional shape that is created by rotating a circle around an axis that is not on the same plane as the circle. As you mentioned, this results in two planes of rotation at right angles to each other. In the case of a 4D torus, we can think of the major axis as the rotation around the x-axis and the minor axis as the rotation around the w-axis.

To better understand this, let's take a point (a, b, c, d) on the surface of the 4D torus. As you mentioned, the major axis is a^2 + b^2 and the minor axis is c^2 + d^2. This means that this point is rotating around the x-axis and w-axis simultaneously, with one period of rotation around each axis. As the point completes one full rotation around the major axis, it also completes one full rotation around the minor axis. This results in a unique and complex path of movement in four dimensions.

In conclusion, using a torus to model rotations in four dimensions is indeed a neat and useful concept. It allows us to visualize and understand complex rotations in higher dimensions, which can be difficult to grasp with our limited three-dimensional perspective. Thank you for sharing your thoughts, and I hope my explanation adds to the understanding of this topic.