Seven-Dimensional Cross Product
en.wikipedia.org
en.wikipedia.org
David Hestenes: New Foundations for Classical Mechanics https://www.amazon.com/Foundations-Classical-Mechanics-Funda...
Geometric Algebra on euclideanspace.com by Martin John Baker: http://www.euclideanspace.com/maths/algebra/clifford/index.h...
Vector and geometric calculus
Both by Alan macdonald, both very good.
And perhaps follow up with the GA for Computer Science book, or if you like (or want to learn about) Newtonian mechanics, try Hestenes’s book New Foundations for Classical Mechanics.
For people with a mathier background, I would recommend https://arxiv.org/abs/1205.5935
https://arxiv.org/abs/1401.2371 for ultimate contrarian math.
“Given the properties of bilinearity, orthogonality and magnitude, a nonzero cross product exists only in three and seven dimensions.”
why only 3 and 7? what's the relationship with quaternions and octonions?
in a similar vein,
But would that insight be plainly obvious to most? I highly doubt it. I have an MS in an engineering discipline, and this was not obvious to me until you said it. Maybe I'm just dumb (I kid - I would have picked up on this while in undergraduate or grad school, when I could really throw my weight around in mathematics, but like anything else, linear algebra is a muscle that wastes if you don't flex it) but really there should be some commentary associated rather than a raw article - the parent comment is totally right.
I could link HN to pentation or the Ackermann function; they're interesting, but useless without context.
-You define the cross product as the area of the parallelogram formed by two vectors,
-Draw a normal to it,
-Use some convention to decide on which side the normal should stick out such that when you reverse the two vectors being multiplied, the resultant vectors is multiplied by -1.
In more (3+n) dimensions there is a problem with this approach. The resulting normal vector can point in any one of (n+1) dimensions. The intuition here is try to draw of normal vector to a line in 3D space. You can do it in 2D space, but in 3D space there is a plane that it normal to the line.
So we need to decide what direction should the vector point in inside this 1+n dimensional space. It seems like any convention will do. We could solve the orientation problem in 3D after all. But it seems like any convention you try has the property that when you break vectors a and b into parts and perform the cross product operation on all pairs of parts (one part from each vector a_i x b_j) and then sum it up the result isn't equal to a x b.
This article is saying a working convention can be given in 7 dimensions (n=4), and no other dimensions. Which is nuts. If anyone has any insight as to why I'd love to hear it.
The cross product in physics is the wedge product. (which is also featured in 'geometric algebra', as another commenter mentioned, though I would contend that GA is the wrong approach). It produces area vectors from two vectors, and happily extends to any dimensions. For instance the electromagnetic field tensor is a bivector (F = d_u A_v - d_v A_u) which is basically a 4d curl of the 4-potential A, and has bivectorial components for each (uv) plane in spacetime.
That's one of many reasons why the wedge product is more natural. If I could have my way there would not be a single cross product in the entirety of physics.
I just said GA gives the most elegant formulation in another comment. What alternative do you suggest? Seeing that you mention the wedge product and bivectors, may you be thinking in something based in differential forms?
I'm fascinated by GA, although I have not had the chance to use it heavily yet. I am very interested if you know something better.
(The reason the geometric product is used in GA is that it's the extension of complex/quaternion multiplication to higher dimensions, and allows for inverting (non-zero-norm) multivectors . But I don't think that makes it worth including. It's not true a priori that inverting vectors is useful to physics, and it only really makes sense when they're being used as linear transformations -- in which case you can just use linear transformations explicitly.
Given the inner product, you can define the “hodge star”, which restores this information. Another way of restoring the information is to use a modification of the exterior algebra, where rather than having the product uv = u wedge v, we deform the product by setting uv = u wedge v + <u, v>. This is called a Clifford algebra [1], and when the input vector space is R^3 and the input inner product is the Euclidean dot product, this Clifford algebras is exactly the “geometric algebra”.
I prefer to read about Clifford algebras, because the language is much less sensationalistic than many of the articles I’ve seen on geometric algebra. It also generalises to not just inner products, but any symmetric bilinear form, such as the Minkowski form on R^4 which is used in the study of relativity.
A bivector x^y can be used as a generator of rotations, by contracting vectors with it: x.(x^y) = y, y.(x^y) = -x. If you define R(v) = v.(x^y), then general rotation is via the exponential map: e^(Rt)x = (cos t) x + (sin t) Rx.
(To be clear: exterior algebra as the algebra of the wedge product operation alone is not enough to do physics. The contraction operation is also necessary (and Hodge star, which is contraction with a particular volume element). But I don't think the geometric product as a standalone operation is useful compared to these components.)
You say the geometric product complicates things, but while you can derive any geometric algebra result simply defining this product (which does not have a difficult definition at all), in the approach you are suggesting you need to define not only the wedge product (which I do not find conceptually simpler than the geometric product) but also contractions. Both approaches are valid, and in certain contexts the differential forms one may be better, but I do not think you can claim it is simpler (when you need to define more concepts) or more fundamental (if you can define wedge product and contraction as a special case of the geometric product, I would say the geometric product is more fundamental).
Nevertheless, as I said I would be glad of learning some simpler tool than geometric algebra, and I am not an expert, so I will give you more details about my use case to see if you can come up with a good suggestion (it may happen that although differential forms are a better tool for physics in general, they are not for my particular problems).
I work in the field of crystallography. We have to constantly deal with crystal orientations (calculate misorientations between them, distributions of orientations in polycrystals, define properties and behavior in basis to these distributions, and so on). The field has traditionally used Euler angles (ugh!), but when new PhD students start developing computer models, obviously they run into problems, and then someone (usually me) suggests them to learn about quaternions.
The students gladly start looking into the subject, but reading definitions that talk about generalizations of complex numbers to four dimensions of stories about mathematical vandalism in bridges do not really help them to build an intuitive idea of how quaternions work, so I usually give them a brief and very informal lecture on geometric algebra (in a very simplified way, I avoid introducing the wedge product for example). After learning the geometric product, they see quaternions as a quite simple concept. We then do a few exercises, like rotating some vectors, calculating misorientations, defining symmetry operators,...
Now, you claim the geometric product complicates things, so I am intrigued. However, I do not see how the life of the students could be made any easier without it. I would have to explain what is the wedge product and contractions, rotations expressed as an exponential have the problem of trigonometric functions (and consequent rounding problems), and some simple operations like calculating the rotation between two vectors become problematic. The connection with quaternions is also much less obvious.
I have a very basic understanding of differential forms. I found fascinating the Discrete Differential Geometry introduction by Keenan Crane and have read some other introductory material, but that's about it. If you can point me in the right direction to learn how to best use it for my problems, I would appreciate it.
That said, this is only my current judgment of the situation, which may change someday. I'm waiting to be convinced that the geometric product is so useful. I just find it so very cludgy when, for instance, GA texts are filled with use of 'grade projection" : take a geometric product with all its terms and then project off the grade you're interested in. It seems like the wrong tool is being used, and almost every time there's a multivector of 'mixed grade', it's regrettable.
(Indeed, wedge products are used everywhere already, in hiding -- integration, determinants, matrix minors, commutators, cross products, rotation matrices, projective geometry, grassmann algebra, ...)
> it has no geometric interpretation
This is ridiculous. The geometric product is thoroughly “geometrical”.
A more accurate translation of your sentence is “I ajkjk have not personally thought enough about it to visualize a multivector”.
The most elegant formulation of Maxwell equations is obtained, in my opinion, using Geometric Algebra, which reduces them to a single and very simple equation.