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Check out my geometric algebra software https://grassmann.crucialflow.com https://grassmann.crucialflow.com The Grassmann.jl package provides tools for doing

Check out my geometric algebra software https://grassmann.crucialflow.com https://grassmann.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. Additionally, the universal interoperability between different sub-algebras is enabled by AbstractTensors.jl, on which the type system is built.

skratlo 7y ago
Holy cow, dude, I'm going to use this one to make people feel dumb. Craziest cryptononsense I've ever read!
- dang 7y ago
  Please don't be a jerk in comments here. It's against the spirit of the site (and also its rules). https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html
ComplexSystems 7y ago
What does the differential operator do and how does it relate to geometric algebra?
- ajkjk 7y ago
  There's a lot of connections between projective geometry and geometric algebra (well-- at least exterior algebra. Not sure about 'geometric', because I don't know what the geometric product means). If you implement _oriented_ projective geometry in homogenous coordinates (so a point (x,y) is a vector (x,y,1)), then the meet and join operators are implemented as ∧ and ∨. You can make a little dictionary: Join = ∧, Meet = ∨. Vector = point, Bivector = line, Trivector = area, etc. The figure spanned by points (a,b,c) = a ∧ b ∧ c. The boundary of the figure = ∧^(k-1) of the metric (a,b,c), equal to ∂(a,b,c) = a ∧ b + b ∧ c + c ∧ a. I have a very amateur blog that I never publicize about this stuff and I had a long post about this, but I've taken it down for now to rework it, or I'd link it here. Suffice to say there's a lot of connections and I feel like there are even more here that haven't been discovered yet. The book "Oriented Projective Geometry" by Stolfi has a lot of this, although it doesn't explicitly talk about geometric algebra or the wedge product -- but it uses all the same symbols. I'm on the lookout for a better reference that bridges the gap. [I have so far not figured out what the exterior derivative means in projective geometry, besides being dual to ∂; I believe that if derivative operators are just dual to basis vectors, then d is literally just dual to ∂. Not sure. I also have no idea what the geometric product means, and tend to be skeptical of it for that meaning.]
ajkjk 7y ago
Do you have a reference for all the operations you've implemented here? I've found random papers that make reference to ∨, d, and ∂ wrt geometric / exterior algebra, but I haven't really found a comprehensive summary of them, so I've been trying to figure it out myself.

skratlo 7y ago

Holy cow, dude, I'm going to use this one to make people feel dumb. Craziest cryptononsense I've ever read!

dang 7y ago
Please don't be a jerk in comments here. It's against the spirit of the site (and also its rules). https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html

ComplexSystems 7y ago

What does the differential operator do and how does it relate to geometric algebra?

ajkjk 7y ago
There's a lot of connections between projective geometry and geometric algebra (well-- at least exterior algebra. Not sure about 'geometric', because I don't know what the geometric product means). If you implement _oriented_ projective geometry in homogenous coordinates (so a point (x,y) is a vector (x,y,1)), then the meet and join operators are implemented as ∧ and ∨. You can make a little dictionary: Join = ∧, Meet = ∨. Vector = point, Bivector = line, Trivector = area, etc. The figure spanned by points (a,b,c) = a ∧ b ∧ c. The boundary of the figure = ∧^(k-1) of the metric (a,b,c), equal to ∂(a,b,c) = a ∧ b + b ∧ c + c ∧ a. I have a very amateur blog that I never publicize about this stuff and I had a long post about this, but I've taken it down for now to rework it, or I'd link it here. Suffice to say there's a lot of connections and I feel like there are even more here that haven't been discovered yet. The book "Oriented Projective Geometry" by Stolfi has a lot of this, although it doesn't explicitly talk about geometric algebra or the wedge product -- but it uses all the same symbols. I'm on the lookout for a better reference that bridges the gap. [I have so far not figured out what the exterior derivative means in projective geometry, besides being dual to ∂; I believe that if derivative operators are just dual to basis vectors, then d is literally just dual to ∂. Not sure. I also have no idea what the geometric product means, and tend to be skeptical of it for that meaning.]

ajkjk 7y ago

Do you have a reference for all the operations you've implemented here? I've found random papers that make reference to ∨, d, and ∂ wrt geometric / exterior algebra, but I haven't really found a comprehensive summary of them, so I've been trying to figure it out myself.