Unveiling the Connection: Rotation Groups & Hyperspheres

In summary, the conversation discusses a result about rotation groups, where the group of orthonormal rotations in a space of n dimensions is isomorphic to the group of geodesic translations in a positively curved space of n(n-1)/2 dimensions. The group of geodesic translations is further clarified as the mapping of geodesics between two points in a sphere to rotations in n dimensions.
  • #1
Tyger
398
0
My first post, about rotation groups..

A result about rotation groups.

To me this seems clear, simple and very intuitive, but in all the papers and books I've read on the subject I have never seen it presented. Maybe some of you have seen it, or maybe it is new. It is very simple to state:

The group of orthonormal rotations in a space of n dimensions, SO(n) is isomorphic to the group of geodesic translations in a positively curved space (hypersphere) of n(n-1)/2 dimensions.
 
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  • #2


Originally posted by Tyger
A result about rotation groups.

To me this seems clear, simple and very intuitive, but in all the papers and books I've read on the subject I have never seen it presented. Maybe some of you have seen it, or maybe it is new. It is very simple to state:

The group of orthonormal rotations in a space of n dimensions, SO(n) is isomorphic to the group of geodesic translations in a positively curved space (hypersphere) of n(n-1)/2 dimensions.

this is a nice thought. It may require a special clarification of what is meant by "group of geodesic translations" in order to make sense-----or this could be my private confusion and it is immediately understandable to everyone but me!

I think of the case n=3 where your theorem says
SO(3) is isomorphic to the geodesic translations of a sphere in 3 dimensions.
This seems right except that rotation around an axis is only a "geodesic translation" for points on the equator. So that one may have to extend the definition in some fashion.

Sorry about the vagueness, I just this moment saw your message and am replying directly.
 
  • #3
Maybe I better clarify my statement

Choose any two points in a sphere of n(n-1)/2 dimensions, draw a geodesic from one point to the other. Every such geodesic can be mapped to a rotation in a space of n dimensions.
 

1. What is the significance of rotation groups in understanding hyperspheres?

The concept of rotation groups is essential in understanding hyperspheres because it helps us visualize and manipulate these higher-dimensional objects. Rotation groups are a set of transformations that preserve the shape and orientation of an object, making them a useful tool for understanding the properties of hyperspheres.

2. How are rotation groups related to symmetry in hyperspheres?

Rotation groups play a crucial role in describing the symmetries of hyperspheres. Since rotation groups preserve the shape and orientation of an object, they can help us identify the symmetries present in a hypersphere. These symmetries are essential in understanding the behavior and properties of hyperspheres.

3. Can rotation groups help us visualize hyperspheres in higher dimensions?

Yes, rotation groups can help us visualize hyperspheres in higher dimensions. By using rotation groups, we can understand how the hypersphere would look when rotated in different dimensions, allowing us to gain insight into its properties and behavior.

4. Are there any real-world applications of rotation groups and hyperspheres?

Yes, rotation groups and hyperspheres have various real-world applications. In physics, they are used to describe the symmetries of physical systems, and in computer graphics, they are used to generate 3D images and animations. They also have applications in geometry, robotics, and engineering.

5. How can understanding rotation groups and hyperspheres benefit scientific research?

Understanding rotation groups and hyperspheres can benefit scientific research in many ways. It can help us better understand the properties and behavior of higher-dimensional objects, leading to new discoveries and advancements in various fields. It can also aid in solving complex mathematical problems and developing new mathematical models and theories.

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