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Recently I made a video presentation about geometric algebra, including the formulation of Maxwell's equations in GA: https://youtu.be/7hlDRLEhc8o https://youtu

Recently I made a video presentation about geometric algebra, including the formulation of Maxwell's equations in GA: https://youtu.be/7hlDRLEhc8o https://youtu.be/7hlDRLEhc8o Also, my implementation of geometric algebra in the Julia language, Grassmann.jl https://github.com/chakravala/Grassmann.jl https://github.com/chakravala/Grassmann.jl It helped me truly understand Maxwell's equations for the first time, understanding that it is not just some physical artifact but actually a natural foundational idea in pure mathematics applicable to physics.

taliesinb 6y ago
That’s cool. What do you make of the algebra of the dihedrons? https://youtu.be/lqH4BLHGsFw https://youtu.be/lqH4BLHGsFw . It’s a “sister algebra” of the quaternions.
- jacobolus 6y ago
  These “dihedrons” (which you might as well just call “2x2 matrices with real entries”) are isomorphic to the geometric algebra of the 2-dimensional Euclidean vector plane with signature (+, +), with 1 scalar component, 2 vector components, and 1 bivector component. Personally I find the basis 1, e₁, e₂, e₁e₂ and the notation of GA to be conceptually much clearer than the matrix entries and standard matrix sum/product to work with, but sometimes using a matrix representation is convenient in a computer. (2x2 real matrices are also isomorphic to the geometric algebra of the 2-dimensional pseudo-Euclidean vector plane with signature (+, -), under a different interpretation.) By comparison the quaternions are the even subalgebra of the geometric algebra of 3-dimensional Euclidean vector space with signature (+,+,+), consisting of only the 1 scalar and 3 bivector components. Or under a different interpretation are isomorphic to the full geometric algebra of the 2-dimensional vector plane with signature (-, -). They can be represented as Pauli matrices. For more on this see the papers and books of Garret Sobczyk, https://garretstar.com/secciones/publications/publications.html https://garretstar.com/secciones/publications/publications.h... ; for example the recent https://www.garretstar.com/sobczyk09-mar-2020.pdf https://www.garretstar.com/sobczyk09-mar-2020.pdf

taliesinb 6y ago

That’s cool. What do you make of the algebra of the dihedrons? https://youtu.be/lqH4BLHGsFw https://youtu.be/lqH4BLHGsFw . It’s a “sister algebra” of the quaternions.

jacobolus 6y ago
These “dihedrons” (which you might as well just call “2x2 matrices with real entries”) are isomorphic to the geometric algebra of the 2-dimensional Euclidean vector plane with signature (+, +), with 1 scalar component, 2 vector components, and 1 bivector component. Personally I find the basis 1, e₁, e₂, e₁e₂ and the notation of GA to be conceptually much clearer than the matrix entries and standard matrix sum/product to work with, but sometimes using a matrix representation is convenient in a computer. (2x2 real matrices are also isomorphic to the geometric algebra of the 2-dimensional pseudo-Euclidean vector plane with signature (+, -), under a different interpretation.) By comparison the quaternions are the even subalgebra of the geometric algebra of 3-dimensional Euclidean vector space with signature (+,+,+), consisting of only the 1 scalar and 3 bivector components. Or under a different interpretation are isomorphic to the full geometric algebra of the 2-dimensional vector plane with signature (-, -). They can be represented as Pauli matrices. For more on this see the papers and books of Garret Sobczyk, https://garretstar.com/secciones/publications/publications.html https://garretstar.com/secciones/publications/publications.h... ; for example the recent https://www.garretstar.com/sobczyk09-mar-2020.pdf https://www.garretstar.com/sobczyk09-mar-2020.pdf