It takes a bit to appreciate, agreed. You have to have suffered through the standard methods. I might actually appreciate it more if I work on some applications. I'm thinking about making a computer application to get a full understanding.
@@XylyXylyX Coding is an extremely powerful tool for understanding material. Take advantage of ganja.js and its online tools for visualizations. It really simplifies the application.
The Wikipedia article states that any multivector can be expressed as a product of vectors and that rotations involving an even number of vectors are proper rotations, while rotations involving an odd number of vectors are improper rotations.
After learning more about rotors and their definition based on bivectors, it occurs to me that bivectors of different shape with same orientation and magnitude cannot be considered equal for this to produce consistent results. Also the geometric proof that a double reflection of a vector based on a bivector produces 2X the angle of the bivector is simple and easy to demonstrate, although Wikipedia has unnecessarily complicated and incorrectly notated this in the diagram included in their rotor article. I had sought a basis for the definition of spinors and this is it, although they do differ from rotors: "spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product" (taken from the Wikipedia article).
@@alecchalmers3744 Bivectors do not have a shape. There are no "bivectors of different shapes". They have a plane, "area", and orientation only. It's geometrically convenient to think of bivectors as pairs of vectors when you're spelling out how to use rotors, or how to calculate the magnitude from the area they bound, but the bivector itself retains no information about the vectors that were imagined to define it. There are an infinite number of pairs of vectors that can form any given bivector. This confusion is why I don't like to talk about bivectors' magnitude as an "area". It gives the impression that there is something conceptual in bivectors that the math doesn't really capture. The magnitude isn't really useful for anything geometric, anyway; when you use bivectors as planes, you almost always want them to be normalized.
If you want to think of reflecting only across planes, you need PGA, or projective geometric algebra. It's very consistent across dimensions; the lowest grade elements always bisect the space, and the products of those are always their intersections. Thus, in 3D, grade 1 are planes, grade 2 are lines, grade 3 are points. They also do the point-at-infinity as one of the grade 1 bases, e0, which works by squaring to 0 and yields translations when used as part of a rotor. It sort of rubs me the wrong way that the typical GA "reflection" through *v* ignores the magnitude of *v* but I have no clue why you are insisting on fiddling with hyperplanes. A line reflection is a perfectly good, well defined reflection and we do not need to talk about planes and normals and higher dimensions. This was a highly unorthodox and overcomplicated detour.
I've heard the praise for the sandwich operation but after seeing how it works and how powerful it is, the praise certainly makes sense.
It takes a bit to appreciate, agreed. You have to have suffered through the standard methods. I might actually appreciate it more if I work on some applications. I'm thinking about making a computer application to get a full understanding.
@@XylyXylyX Coding is an extremely powerful tool for understanding material. Take advantage of ganja.js and its online tools for visualizations. It really simplifies the application.
The Wikipedia article states that any multivector can be expressed as a product of vectors and that rotations involving an even number of vectors are proper rotations, while rotations involving an odd number of vectors are improper rotations.
After learning more about rotors and their definition based on bivectors, it occurs to me that bivectors of different shape with same orientation and magnitude cannot be considered equal for this to produce consistent results. Also the geometric proof that a double reflection of a vector based on a bivector produces 2X the angle of the bivector is simple and easy to demonstrate, although Wikipedia has unnecessarily complicated and incorrectly notated this in the diagram included in their rotor article. I had sought a basis for the definition of spinors and this is it, although they do differ from rotors:
"spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product" (taken from the Wikipedia article).
@@alecchalmers3744 Bivectors do not have a shape. There are no "bivectors of different shapes". They have a plane, "area", and orientation only. It's geometrically convenient to think of bivectors as pairs of vectors when you're spelling out how to use rotors, or how to calculate the magnitude from the area they bound, but the bivector itself retains no information about the vectors that were imagined to define it. There are an infinite number of pairs of vectors that can form any given bivector.
This confusion is why I don't like to talk about bivectors' magnitude as an "area". It gives the impression that there is something conceptual in bivectors that the math doesn't really capture. The magnitude isn't really useful for anything geometric, anyway; when you use bivectors as planes, you almost always want them to be normalized.
R-1[abcd]R = a_Rb_Rc_Rd_R, but how do we know that this corresponds to a rotation?
Because it is the mathematical equivalent of rotating *each* of the individual part.
@@XylyXylyX So if we have a simple trivector, a signed volume, to rotate it, we just need to rotate each of its three defining vectors?
And then take the geometric product of the rotated vectors.
@@charlesschmidt4272 Yes, that is the idea.
If you want to think of reflecting only across planes, you need PGA, or projective geometric algebra. It's very consistent across dimensions; the lowest grade elements always bisect the space, and the products of those are always their intersections. Thus, in 3D, grade 1 are planes, grade 2 are lines, grade 3 are points. They also do the point-at-infinity as one of the grade 1 bases, e0, which works by squaring to 0 and yields translations when used as part of a rotor.
It sort of rubs me the wrong way that the typical GA "reflection" through *v* ignores the magnitude of *v* but I have no clue why you are insisting on fiddling with hyperplanes. A line reflection is a perfectly good, well defined reflection and we do not need to talk about planes and normals and higher dimensions. This was a highly unorthodox and overcomplicated detour.