
Seven-Dimensional Cross Product - murkle
https://en.wikipedia.org/wiki/Seven-dimensional_cross_product
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Scene_Cast2
I feel like Geometric algebra / clifford algebra is something more people
should know. It makes N-dimensional vector algebra easier and more intuitive.

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thelastbender12
could you recommend a good source to pick it?

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jacobolus
What is your background/perspective? For introductory motivation, check out
[https://www.shapeoperator.com/2016/12/12/sunset-
geometry/](https://www.shapeoperator.com/2016/12/12/sunset-geometry/)

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/1205.5935)

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thelastbender12
thank you, this looks interesting. The parent comment caught my eye since I'm
pretty comfortable visualizing linear algebra with 2D arrays but struggle
beyond that.

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rtkwe
I'm always perplexed when just a raw wikipedia article is posted here with no
context or commentary about it.

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nilkn
This is definitely a bit niche, but at first blush it's highly bizarre that
you can get a natural cross product operation in three dimensions... and not
again until seven. And then never again after that. It's a peculiar and
interesting (if not very useful) fact that is probably within reach of a
sizable chunk of the audience here.

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oarabbus_
See, now that you've said it, it really is quite interesting.

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.

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newen
Cross products are taught in high school. And I'm sure by college, a good
amount of people are aware that four dimensional cross products are not
possible. And like me, most of them probably thought four and higher
dimensional cross products don't exist. So when I saw 7-dimensional cross
product, I got pretty interested.

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oarabbus_
Convex optimization is also taught in high school. Organic chemistry,
biochemistry, discrete math, probability, and statistics too. I fail to see
your point.

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newen
My point is that cross products are really really really simple and for some
reason you have this notion that cross products are something complicated.

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Tomminn
In 3D the cross product can be understood as follows:

-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.

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kilovoltaire
"it is the only other non-trivial bilinear product of two vectors that is
vector-valued, anticommutative and orthogonal"

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deehouie
This' something really intriguing to me. Coming from a physics background,
this immediately takes me to electrodynamics where the pillar of half of
classical physics, namely Maxwell's equ, is built on cross product.

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ajkjk
The 7d cross product is almost certainly not useful for physics. It's not
uniquely defined! e_1 x e_2 can equal any of the other basis vectors.

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.

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yiyus
> I would contend that GA is the wrong approach

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.

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ajkjk
I think that the wedge product is fantastic and the geometric product is
basically useless -- it has no geometric interpretation, complicates things
massively, and conceals intuition wherever it is used. So I'd argue that
'exterior algebra' (which is the language used by differential forms, plus the
exterior derivative) is basically the right framework for physics.

(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.

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jacobolus
Not using the geometric product is sort of like working as a chef but not
allowing yourself any knives with blades between 3 inches – 15 inches long.
Only a paring knife, a hatchet, and a machete. Yes, it is _possible_ to
accomplish everything you wanted using a confusing mishmash of other
abstractions (the raw numerical data turns out basically the same), but it’s
really really inconvenient in comparison.

> _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”.

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ajkjk
I have no problem visualizing multivectors, and have studied them extensively.
What I have failed to find any intuition for is the general geometric product
of two multivectors (without restricting to a particular grade). If you have
one, I would love to hear it.

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ngvrnd
Is this interesting primarily because of the noted correspondence with
octonions?

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HelloNurse
It's a very notable algebraic structure because it "works" only in a specific
number of dimensions instead of belonging to an infinite family, but it
doesn't mean that it is more useful for practical or theoretical purposes than
boring wedge products. For starters, you'd need a specifically 7-dimensional
problem to solve; the 3-dimensional cross product is much easier to "sell".

