Grassmann.jl geometric algebra overview in Julia language

Grassmann.jl geometric algebra overview in Julia language
github.com/chakravala/Grassma...
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 include ∧, ∨, ⋅, *, ⋆, ', ~, 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. Additionally, the universal interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built.
crucialflow.com
Tools built on these foundations enable computations based on multi-linear algebra and spin groups using the geometric algebra known as Grassmann algebra or Clifford algebra. This foundation is built on a direct-sum parametric type system for tangent bundles, vector spaces, and also projective and differential geometry. Geometric algebra is a mathematical foundation for differential geometry, which can be used to simplify the Maxwell equations to a single wave equation due to the geometric product. Introduction of geometric algebra to engineering science disciplines will be easier with programmable foundations.In order to devise an expressive and performance oriented language for efficient discrete differential geometric algebra with the Grassmann elements, an efficient computer algebra representation was programmed. With this unifying mathematical foundation, it is possible to improve efficiency of multi-disciplinary research using geometric tensor calculus by relying on universal mathematical principles. Tools built on universal differential geometric algebra provide a natural geometric language for the Helmholtz decomposition and Hodge-DeRahm co/homology.

[@user-fl5nv7oh3z]

in my world I coined the concept of "extroverted autism" to characterize a personality that is deep into a subject, finds new worlds and tries to communicate this to the outer world, which is not ready yet to adopt. Such an individual is trying desperately to communicate and is seen speechless from the exterior. I have the strong feeling, what you do is very important! Keep going and don't struggle with stupidity ;-=)

[@CrucialFlowResearch]

I don't believe in psychology as a science, so i will consider your statement pseudo-science and therefore nothing but an opinion. I'm not desperately trying to communicate anything. I dont have autism, go find somebody else to apply your pseudoscience to. If you dont care about the topic of the video, you're not the target audience.

[@user-fl5nv7oh3z]

@@CrucialFlowResearch Sorry, just wanted to state that communication is problematic ;-) Sorry. I find your work fascinating, but it's far over my level. I just apply elementary math, like >90% of all human beings. How could I imagine GA when I learned: a vector is given by length and direction, so that if a vector changes over time, the difference is just another vector? But now I understand, that the difference could be seen as a bi-vector and with a few lines of code you can display vector fields etc. This is fantastic and anyway, I have problems to share this excitement with my neighbors.

[@CrucialFlowResearch]

@@user-fl5nv7oh3z it's obvious that you barely have studied these topics, it's good that you're excited about it, so before you share it with others, you need to refine and work on the math yourself. What you're probably referring to is a vector field generated by the exponential operator applied to a bivector. You know a few words, but you don't know enough to fully piece together these ideas yet. You probably need to study some calculus too, not just geometry, as this involves calculus.

[@user-fl5nv7oh3z]

@@CrucialFlowResearch You are right. Math alone is a huge field and you will not find a person knowledgeable of all the math. On the other hand, a simple question like: is a triangle defined by three numbers seen as the length of the sides equivalent to a triangle defined by three pairs of numbers equivalent? Even if the pairs of numbers are not coordinates in RxR with Euclidian norm? Or: what makes the line segments a triangle? Isn't it that geometry involves calculus only since Descartes? How can an external understand a concept if the internals use different notations?

[@CrucialFlowResearch]

@@user-fl5nv7oh3z a triangle doesn't require the measurement of length, it only requires the topology of 3 independent affine points, the notion of length is an additional feature imposed onto this for doing geometry by imposing a coordinate chart, and there are many choices of geometry, which require calculus to understand in the most general sense