Insights into Mathematics | The geometry of the Dihedrons (and Quaternions) | Famous Math Problems 21c | N J Wildberger @njwildberger | Uploaded 3 years ago | Updated 1 day ago
The Dihedrons are a sister algebra to the Quaternions. They were first explicitly introduced and named by James Cockle in 1849 as split-quaternions. But because of the important connections with the dihedral group D_4, we would like to introduce the name "Dihedrons" as this indicates that they are not in any way secondary to the much better known Quaternions. In fact a good case could be made that mathematically they are even more fundamental than the Quaternions --- on account of the crucial facts that they are really the 2x2 matrices, defined over a general field, and that the fundamental quadratic form on them is just the determinant!!

In this video we delve into the rich geometry of the Dihedrons, but we want to position this adjacent to the simpler, and more familiar, Quaternion story. There are many similarities, but also important differences. The geometry is really a relativistic one --- and suggests once again that the geometry of Special Relativity could have been discovered by pure mathematicians long ago.

Some of the geometry here connects with generalized dot products and cross products or vector products. These topics have been systematically investigated by Gennady Notowidigdo in his PhD thesis here at UNSW, and hopefully we will be telling you more about our work together in the near future. In the meantime, you can check out Gennady's YouTube channel at youtube.com/user/ganot2 There are some nice videos there, and some are closely related to the material here.

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