Science topics: Geometry and TopologyGeometric Algebra
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Geometric Algebra - Science topic
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The question of the geometric structure of Clifford algebras is studied in Section 3.3 of the book
Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS
Given the widespread use of Clifford algebras in physics, this discussion may be of interest not only to mathematicians.
I recently asked in math stackexchange a question regarding 3D rotations in geometric algebra https://math.stackexchange.com/questions/3922021/angle-and-plane-of-rotation-in-3d-geometric-algebra.
I've added a new question regarding 4D or even n-D rotations and rotors. The reformulated question is as follows:
"In general, a rotor in 4D consists of a scalar, 6 bivectors and one four-vector (in 3D, a rotor is just composed of a scalar and 3 bivectors). Then, assume that we already know two 4-dimensional vectors and one is a rotated version of the other. How is it possible to derive an expression to compute the associated rotor? does it exist? If so, could a general expression for n dimensions be obtained?"
Basically, in the computer vision applications, recently there are more focus on mathematical approach. Deep learning is used in almost all tasks in computer vision. Using geometric algebra etc. how we can improve the state of arts methods in different computer vision tasks.
Hello everybody,
I was wondering whether the formula of a geometric mean can be solved to determine the weights. More specifically:
- I have the weighted geometric mean of two values. I know the two values and the geom. mean.
- The sum of the weights is 2, so if w1 = 1, then w2 = 1.
Changing the formula of the weighted arithmetic mean was fairly easy so I am surprised why conventional tools such as the online Wolfram Alpha tool seem to fail, so I presume that I failed somewhere along the way.
I am looking forwards to your thoughts on this!
Lukas
“If geometric algebra is so good, why is it that 140 years after William
Kingdon Clifford’s death it is still not universally taught and used?”
My answer to this question is contained in the attached pdf - file. I am hoping my new book, "Matrix Gateway to Geometric Algebra, Spacetime and Spinors", and new article will help break the logjam.
Garret Sobczyk
I wonder how this projective geometric algebra with arbitrarily introduced metric can be recasted in mother neutral algebra ( https://en.wikipedia.org/wiki/Universal_geometric_algebra) aproach which embeds all metrics in one coherent structure?
My interest is to use such approach for description of fluid essence (aether) flows of this wonderful Creation of our Creator.
Such flows would define space time metric and all other phenomena.
For a Vacuum Solution, as shown in Geometric Algebra for Electrical
and Electronic Engineers by IEEE, and other places, Electric Field is shown as a Helix. And M (though being Axial Vector), in terms of logical wave, it is again a Helxi by phase difference of 90 degrees.
If the propagation is z-direction, the x-z has Sin projection and y-z has Cos Projection for E. And Magnetic it is Cos and -Sin. We will take Em and Bm =1 for magnitudes. We take c = 1, e = 1, u = 1. So B = H.
One can model Y-Z Rotations has function of angular frequency and Z as displacement from momentum.
Ex = Sin (wt), Ey = Cos (wt), and Ez = t
Bx = Cos(wt), By = -Sin (wt), and Bz = t.
S = H x B gives
Sx = Hy x Ez - Hz x Ey = -t[Sin(wt) + Cos(wt)] = -t 2 ^1/2(Sin(wt + 45))
Sy = Hz x Ex - Hx x Ez = t[Sin(wt) - Cos (wt)] = t 2 ^1/2(Sin(wt - 45))
Sz = Hx x Ey - Hy x Ez = Cos^2 (wt) + Sin ^2 (wt) = 1
So with increasing "t", the Pyonting Vector will spread in X and Y direction but not remain constant in Z direction!
What is wrong with my understanding?
Suppose an element of even subalgebra of geometric algebra over 3D is written in two ways: a+b1B1+b2B2 +b3B3 and exp(acos(a)B). I want to differentiate it, say, by b1. It looks like differentiation of geometrical sum and exponent give very different results. Where is possible error?
Of course, one can use Clifford Algebra Cl(3,1) in electromagnetism. But then, one is not working with Gibbs vectors anymore (NOTE 1.1, 1.2) , but with multivectors (NOTE 2.1, 2.2, 2.3). Better yet, we can try to check with tensors. There is a physics fundamental reason (NOTE 1.31P) why tensors are better than Cl(3,1) when one transforms coordinates.
Also, every element of a geometric algebra, such as Cl(3,1) and spacetime algebra by Hestenes, can be identified with a tensor, but not every tensor can be identified with an element of a geometric algebra. In that sense, tensors are more general than pure grade multivectors. The rank of a tensor is not restricted by the dimension of the base vector space like the grade of a multivector.
This means that there is no way to represent any 2nd-rank tensor by a bivector, vector, and scalar. Or, in particular, no way that any 3×3 matrix could be represented by a 2-vector in 3D!
So, it is not a 3D here and 1D there, but a union of the two, in 4D, that is desirable in electromagnetism spacetime. One can transform length into time, and vice-versa.
Tensors allow that, Cl(3,1) does NOT do that. And electromagnetism may be at least 4D, maybe 6D or more, which tensors can represent, but Cl(3,1) would not.
Similar question: what's the relationship of tensor and multivector . The short answer is that all multivectors are tensors, but not all tensors are multivectors, so geometric algebra is not and cannot be isomorphic to tensor algebra.
Therefore, geometric algebras (Clifford Cl(3,1), Hestenes) seem insufficient for electromagnetism, in all possible generality. What is your view?
NOTES:
1.1. The 3D vector cross product is not a vector, but a tensor. This is well-known and the reason to invalidate its use in Maxwell's equations, and physics equations. This is both a physical and a mathematical reason, commented below. But, would it not serve a restricted purpose adequately? No, it can create mistakes upon reference frame change, for example, simple mirroring, gives us wrong units in the SI MKS, worldwide, and this is all old news that have to be somehow continuously repeated, with a "life of its own," as a misconception in even current college books at competitive US universities.
1. 2. MATHEMATICAL REASON: The 3D vector cross product is not a closed operation in the 3D vector space, it produces a member that does not belong to the same set, the 3D vector space, although it may look like it in some cases.
1.3. PHYSICAL REASON: Both sides of an equation representing a physical relationship, such as A = B, must change equally when the frame of reference changes and the so-called inertial condition is obeyed, as already stated by Galileo, Newton, and Einstein, that the laws of physics are the same for all uniformly moving observers. But if one writes an equation using a 3D vector cross product, such as A= B x C, the left and right side may transform differently if the coordinate frame of reference changes, while still inertial. In the past, this was accommodated, not solved, by considering spurious things such as polar and axial vectors, and pseudoscalars. This is solved using tensors, which maintain the form A = B under inertial reference change.
2.1. Multivectors, in geometric algebras, or Clifford algebra, or Cl(n,1) algebras, or STA Hestenes algebra, solve the mathematical reason (1.2), creating a closed space in all operations. No geometric algebra operation is mathematically unsound.
2.2. Multivectors do not solve the physical reason (1.3), but tensors do. This is another reason, besides lack of isomorphism with tensors and no use of time as a coordinate in spacetime, that invalidates the use of geometric algebras in equations of physics, including electromagnetism.
2.3. With multivectors, If one eliminates the physically wrong results, by requiring an additional step of "filtering" through, let us say, a Hamiltonian, this will not produce those results that are physically valid but were ignored in the first place, using just multivectors. A sequence of filters cannot filter less than the first filter, well-known in physics, math, and engineering. This appears, more easily to see, in non-euclidean spaces, such as anything larger than, let us estimate in general, a few Planck lengths.
3. This thread arrived at a first conclusion, which is stated in the NOTES 1-2 above. There is no room to refuse to notice an obvious thing, or to re-explain here, the reasoning and references are available above, to anyone.
4. We are now moving on, to non-mathematical aspects of using Clifford algebras, even beyond the physical reason given in the NOTES above, where using Clifford algebras would be detrimental to special relativity. We are talking about time.
I found an old paper about geometric algebra from you:
Did the performance of your GA implementations impove?
Did you use different types, or just encode everything with a single multivector type?
How about now?
Need something specific to work on, particularly for beginner with little knowledge of geometric algebra. That is, something that can serve as a research hypothesis.
Please help.
A sort of mathematics capable of describing motions of universal fluidum is needed.
Is it a kind of geometric algebra, projective geometry?
If motions are building blocks, all phenomena of nature can be described by them.
If motions are building blocks, space-now metric is the result of it as well as the concept of space and time in now.
If motions are building blocks, then space and time are interconnected, interrelated. What is this relation?
My intuition tells me to use a kind of neutral algebra with natural dualities (linear, male, space and rotational, female, time).
The Wikipedia articles on these two subjects contradict one another.
Note: I refer to *the* Clifford algebra Cℓ_{0,3}(R), not to *a* Clifford algebra. The one with starting point R^3, the most common one in Geometric Algebra.
If I understand it correctly they are different. Both contain, as it were, two copies of the quaternions, but the multiplication tables are different.
The Clifford algebra has the right to be called a non-commutative algebra, as the phrase is usually understood, since multiplication is associative.
The algebra of the octonions is not associative but only satisfies a weaker property. Am I right? What is the connection of either with S^7 ?
We have scalar and cross product. Cross product works in 3D only. But why not define a torque in 2D? Or 4D?
Imagine now that we don't know anything about products of vectors. How to multiply? For two vectors a and b it would be the product ab. It is natural to expect distributivity and associativity, but not commutativity (cross product). So, let us decompose our new product (symmetric and antisymmetric part)
ab = (ab + ba)/2 + (ab - ba)/2
It is straightforward to show that antisymmetric part is not a vector (generally squares to negative real number)!
Now we have a simple rule: parallel vectors commute, orthogonal vectors anti-commute. For orthogonal vectors we have Pythagoras theorem without metrics!
In 3D (Euclidean) we have for orthonormal basis
eiej + ejei = 2 dij , dij is Kronecker delta symbol.
You see, it is Pauli matrix rule, ie, Pauli matrices became 2D matrix representation of orthonormal basis vectors in 3D.
This is geometric algebra (Grassmann, Clifford, Hestenes, ...).
My question is NOT to discuss geometric (or Clifford) algebras, it is about a general concept of number. How to multiply vectors?
If one accepts new vector product it changes everything! So, what are objections to such a concept (Clifford)? Could somebody suggest another multiplication rule?
Let R, S be rings and f : R -> S be an isomorphism.
If a in R s.t. a has property P, does f(a) have the same property?
Also, what about the anti-isomorphism and the properties of rings and its elements?
Can you give me a reference name has studied this topic?
For example n=250, the geometric mean is still maintaining validity?
