#cliffordalgebra — Public Fediverse posts

I’m interested in #GeometricAlgebra (and #ExteriorAlgebra, #CliffordAlgebra, #ExteriorProduct and #WedgeProduct). I’m trying to work up an intuition for a few things. Thoughts on this welcome!

* It’s very intuitive that adding vectors means something and is useful. Join pencils end to end and now you have the pencil of their path. I have much less intuition that adding #bivectors is useful.

* In 3D, the wedge product of two vectors is an oriented area (bivector). And the wedge of an area and a vector is a volume. This makes sense. But the wedge of two areas in 3D is zero. Always (right?!). Is it even a well typed operation?

* If I have an area (say, some solar panels) and a direction (say incident sunlight), I think I can wedge them to get collected light (as a pseudo scalar). How do I know to wedge here instead of dot product? What’s the intuition for that so that the question becomes absurd?

* Is there a most simplest toy problem for playing with these to work up intuition? I think maybe solar panels (with area and orientation) and incident light (with intensity and direction) is reasonable? Because it just about makes sense to add oriented panels. And possibly even directed incident light?

Thanks!

This is an introduction to #GeometricAlgebra, an alternative to traditional #VectorAlgebra that expands on it in two ways:
1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension.
2. It defines a product that’s strongly motivated by #geometry and can be taken between any two objects. For example, the product of two vectors taken in a certain way represents their common plane.

This system was invented by #WilliamClifford and is more commonly known as #CliffordAlgebra. It’s actually older than the #VectorAlgebra that we use today (due to #Gibbs) and includes it as a subset. Over the years, various parts of #CliffordAlgebra have been reinvented independently by many people who found they needed it, often not realizing that all those parts belonged in one system. This suggests that Clifford had the right idea, and that #GeometricAlgebra, not the reduced version we use today, deserves to be the standard “vector algebra.” My goal in these notes is to describe #GeometricAlgebra from that standpoint and illustrate its usefulness. The notes are work in progress; I’ll keep adding new topics as I learn them myself.