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  1.  
    Very cool!!

    I never used APL. I tried more the Prolog and Forth directions for explorations.

    Not until the '80s did I come across projective geometry (I had to because I was designing a 3D graphics chip) along with NURBS (non uniform rational B-splines) and similar monstrosities.
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      CommentAuthorAngus
    • CommentTimeJul 22nd 2021
     
    The problem I always had with the APL outer product as applied to a couple of N-dimensional matrices, was to figure out what the hell happened.
  2.  
    I believe the wedge product and the cross product are distinct. The cross product generates a vector orthogonal to both of the input vectors, so if you insist on staying in (say) 2D, then this isn't a good thing. I don't think the wedge (exterior) product does this.
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      CommentAuthorAngus
    • CommentTimeJul 22nd 2021
     
    The crossproduct of vectors does produce a vector orthogonal to the plane of the two inputs as you say. My understanding of the wedge product is that it is associated with a sort of rotation, which gives it a positive or negative value. I would associate that with a third dimension, but I guess that's not explicit.
  3.  
    The penny finally dropped about cross vs. wedge products. Although the coefficients are identical, that's where the similarity ends, because
    - the cross product is not (guaranteed to be) associative for three or more inputs: the wedge product is associative.
    - the cross product is only defined in 3D: the wedge product is defined in any number of dimensions.
    - the cross product (for vectors) always returns a traditional vector: the wedge product for two inputs returns a bivector (oriented plane element), and a trivector (oriented volume element) for three inputs. There are no matrices anymore!

    From all this it's clear that the cross product is a twisted abnormality of a much grander landscape.
    It was a stroke of genius to realise that a more general and useful geometric product would be the sum of the dot product and the wedge product (Grassman and then Clifford).
    The other genius move was to include the e0 basis whose square is 0, which allowed a smooth transition between spacetimes of positive and negative curvature passing through flat Euler (Minkowski) space, allowing efficient computer graphics to happen for the flat case (Gunn's PGA).

    I rewatched the physics talk at the original link I posted. Mind blown yet again. There's going to be a major shake-up in how the maths of physics is presented. Same goes for how computer graphics is done.
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      CommentAuthorAngus
    • CommentTimeJul 23rd 2021
     
    Posted By: Andrew PalfreymanIt was a stroke of genius to realise that a more general and useful geometric product would be the sum of the dot product and the wedge product


    Agreed. I'm still trying to get my head around the rest.
  4.  
    One thing that helped me to become sold on the approach was when I realised that real, physical realities naturally give rise to stuff that we were simply told to accept. Case in point is noncommutation and imaginary numbers. We can appreciate noncommutation as a physical thing when we consider the order in which rotations are performed. And yet this property directly gives rise to a basis in which the square of the basis must be negative. There's not much "imaginary" about that. It's baked into the geometry. [a,b] => ic.
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      CommentAuthorTrim
    • CommentTimeJul 24th 2021
     
    Putting a gravitational wave detector on the Moon could find new physics

    https://newatlas.com/physics/moon-gravitational-wave-detector/
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      CommentAuthorAngus
    • CommentTimeJul 24th 2021 edited
     
    Posted By: Andrew PalfreymanOne thing that helped me to become sold on the approach was when I realised that real, physical realities naturally give rise to stuff that we were simply told to accept. Case in point is noncommutation and imaginary numbers. We can appreciate noncommutation as a physical thing when we consider the order in which rotations are performed. And yet this property directly gives rise to a basis in which the square of the basis must be negative. There's not much "imaginary" about that. It's baked into the geometry. [a,b] => ic.


    OK now I'm choking on it. I am watching some lecturer tell me that a bivector has a corresponding dual vector whose length is equal to the AREA of the bivector. I have no difficulty with a bivector having an area instead of a length, but it violates all sense to make a length equal to an area. It would put you at the mercy of some arbitrary dimensioned constant. What gives?

    ETA I guess the constant is the pseudovector of the space. It's still wierd. How are you going to check your work by analysing dimensions?
  5.  
    I presume he misspoke - substitute "magnitude" for "length" maybe? In G3 we have 4 different kinds of magnitude, corresponding to the 4 different grades of multivector.
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      CommentAuthorAngus
    • CommentTimeJul 25th 2021 edited
     
    I think that is what he meant but he continued misspeaking in a manner worthy of a Nixon. I'll post the link when I find it again.

    The message seemed to be that in the particular setup he was discussing the numerical magnitude of the bivector equalled that of its dual vector. I don't see why that has any particular significance.


    ETA Here at 28:30
  6.  
    Coincidentally I am watching the same video, and |B| = |v| shocked me too. So I had a look at the definition of the magnitude of a bivector.
    https://en.wikipedia.org/wiki/Bivector#Magnitude
    from which it appears that it's a scalar (at least for "simple" bivectors i.e. 2D and 3D), so all is well.

    I think :)
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      CommentAuthorAngus
    • CommentTimeJul 25th 2021 edited
     
    The magnitude of anything is a scalar, isn't it? Both area (hectares) and length (metres) are scalars. What I'm concerned with is the dimension of the hectare, which is metres^2. It can be really useful to keep track of that.
  7.  
    You may recall me whining about this sort of thing a few months back when I pointed at (a + ib) and declared that you should not be able to add apples to pears (or indeed hectares and metres). I said that the use of the '+' symbol made me queasy. Yet in this GA world we have the geometric product producing these peculiar sums - e.g. a scalar plus a bivector. It's the same sort of thing.

    Perhaps it should be viewed as a set - a bag in which distinct object types are to be found.
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      CommentAuthorAngus
    • CommentTimeJul 25th 2021 edited
     
    Good point about the a.b + a × b product ( that'll have to do as I don't have a wedge on this thing.). I guess we just have to get used to it. It seems tbat it might go down easier with programmers than with physicists.

    ETA I mean the { a.b + (a x b) I } product. I just realised I could write "wedge" without having a wedge.
  8.  
    It's satisfyingly ancient Saxon. A good solid word.
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      CommentAuthorAngus
    • CommentTimeJul 25th 2021
     
    Posted By: Andrew PalfreymanIt's satisfyingly ancient Saxon. A good solid word.


    Well well! I'd have thought so too, but the Online Etymology Dictionary says
    ...of uncertain origin; perhaps related to Latin vomer "plowshare."
  9.  
    I find it comforting that relativity (specifically Lorentz transforms) can be expressed in terms of circuit theory. This means that considering warp drives evokes the smell of hot solder!
    https://www.youtube.com/watch?v=vOxV9hmXUZU

    ETA I have acquired a PDF of Geometric Algebra for Physicists by Duran & Lasenby. Shoot me a whisper if you're interested.