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There is a process where you put in a space (number of dimensions) and a metric (Euclidean, for example) and produce a Clifford algebra. This is also sometimes
by joppy 8y ago
There is a process where you put in a space (number of dimensions) and a metric (Euclidean, for example) and produce a Clifford algebra. This is also sometimes called “geometric algebra”, although there are a lot of sensationalist posts using this name on the internet.
This Clifford algebra has an odd and even part, and the even part is an algebra in its own right, and is very useful for representing “rotations” in the space.
With the Euclidean metric, with one input dimension, the even Clifford algebra is just the reals. With two dimensions, you get the complex numbers. With three, the quaternions. However, since a Clifford algebra is always associative, you don’t get the octonions in this way.
Finally, if you put in four dimensions and the minkowski metric, the even Clifford algebra construction gives you a lovely algebra representing spatial rotations in three of the dimensions, and “boosts” along the time direction - exactly what you need to do calculations in special relativity.
- eggy 8y agoI am working my way through Understanding Geometric Algebra by Kenichi Kanatani up to the Grassmann Algebra chapter, so I can then pick up John Browne's Grassmann Algebra Volume 1 Foundations. Kenichi's book works from basic vector calculus, Hamilton, Grassmann and Clifford. Grassmann was not even a mathematician by trade, but a school teacher and a linguist. He translated the Rig Veda. His big mathematical work did not get the attention when written as it did posthumously. I know things with cool names like differntial forms and fiber bundle space [1]. I'm becoming quite the Grassmann fanboy! I am thinking if I can learn these algebras (Grassmann, Clifford, etc...) I will be able to deal with more abstract thinking in geometry and handle it more concisely and logically. Almost like when I use J programming language for math! [2] I think Quarternions don't hold after 3 dimensions, so what about Octonions? [1] https://galileospendulum.org/2012/04/26/everything-is-geometrical-hermann-grassmanns-algebra/ [2] jsoftware.com