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Differential geometric algebra foundations: Grassmann.jl Ascend

Crucial Flow Research · Youtube · 5 HN points · 1 HN comments
HN Theater has aggregated all Hacker News stories and comments that mention Crucial Flow Research's video "Differential geometric algebra foundations: Grassmann.jl Ascend".
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Differential geometric algebra foundations: Grassmann.jl AIAA Ascend 2020

Mathematics and computer algebra research for differential geometric algebras, with an interest in aerospace engineering applications. Foundations of the open source free software Grassmann.jl are based on differential geometric algebra and developed in the Julia language. Interests include applications of Grassmann.jl software to explore relationships between quantum computing, number theory, geometric flows, aeroacoustics, relativity, optics, and electrical engineering.

www.ascend.events
https://crucialflow.com

The Grassmann.jl package provides tools for doing computations based on multi-linear algebra, differential geometry, and spin groups using the extended tensor algebra known as Leibniz-Grassmann-Clifford-Hestenes geometric algebra. Combinatorial products included are ∧, ∨, ⋅, *, ⋆, ', ~, d, ∂ (which are the exterior, regressive, inner, and geometric products; along with the Hodge star, adjoint, reversal, differential and boundary operators). The kernelized operations are built up from composite sparse tensor products and Hodge duality, with high dimensional support for up to 62 indices using staged caching and precompilation. Code generation enables concise yet highly extensible definitions. The DirectSum.jl multivector parametric type polymorphism is based on tangent bundle vector spaces and conformal projective geometry to make the dispatch highly extensible for many applications. The universal interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built.

Grassmann.jl: https://github.com/chakravala/Grassmann.jl
Dream Scatter: https://music.crucialflow.com

This Grassmann package for the Julia language was created by github.com/chakravala for mathematics and computer algebra research with differential geometric algebras. These projects and repositories were started entirely independently and are available as free software to help spread the ideas to a wider audience. Please consider donating to show your thanks and appreciation to this project.
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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

Also, my implementation of geometric algebra in the Julia language, 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
That’s cool.

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

⬐ jacobolus
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.h... ; for example the recent https://www.garretstar.com/sobczyk09-mar-2020.pdf

Oct 10, 2020 · 3 points, 0 comments · submitted by DreamScatter
Oct 07, 2020 · 2 points, 0 comments · submitted by DreamScatter
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