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Seven-Dimensional Cross Product
- rtkwe 8y agoI'm always perplexed when just a raw wikipedia article is posted here with no context or commentary about it.
- Scene_Cast2 8y agoWell, if people upvote it... I guess it's more like "hey look, this article is really cool", without a direct application to most people's daily work.
- jetrink 8y agoAgreed, sometimes it's obvious why a particular article caught the poster's attention, but for this one, it would be nice if they gave some context. How did they hear about it? Is there an interesting fact hidden in the article that would not jump out to the uninitiated?
- teucris 8y agoNo idea why OP found it interesting, but this really caught my eye: “Given the properties of bilinearity, orthogonality and magnitude, a nonzero cross product exists only in three and seven dimensions.”
- isoprophlex 8y agoexactly this! these marvellous rarities are what makes mathematics so divinely interesting. why only 3 and 7? what's the relationship with quaternions and octonions? in a similar vein, https://en.m.wikipedia.org/wiki/Exceptional_object https://en.m.wikipedia.org/wiki/Exceptional_object
- deehouie 8y agoMy experience with HN is, there are some very thoughtful mathematicians engaging in deep research around here. Some of the posts here may make no sense to 99% of the readers ("why on earth should I care about 7-dim stuff, I just don't give a damn"), but it may just be the lightning strike needed for someone to make important breakthrough in certain area of mathematics.
- arbitrage 8y agoThis property of 3/7 dimensional cross product is not terribly insightful. It's taught to undergrads. It's not really a hallmark of advanced mathematical research.
- nilkn 8y agoThis 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.
- oarabbus_ 8y agoSee, 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.
- newen 7y agoCross 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.
- oarabbus_ 7y agoConvex optimization is also taught in high school. Organic chemistry, biochemistry, discrete math, probability, and statistics too. I fail to see your point.
- newen 7y agoMy point is that cross products are really really really simple and for some reason you have this notion that cross products are something complicated.
- Scene_Cast2 8y agoI feel like Geometric algebra / clifford algebra is something more people should know. It makes N-dimensional vector algebra easier and more intuitive.
- thelastbender12 8y agocould you recommend a good source to pick it?
- layoutIfNeeded 8y agoGeometric Algebra For Computer Science, An Object Oriented Approach to Geometry, published by Morgan Kaufmann Publishers http://geometricalgebra.net http://geometricalgebra.net David Hestenes: New Foundations for Classical Mechanics https://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792353021 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.htm http://www.euclideanspace.com/maths/algebra/clifford/index.h...
- nimish 8y agoLinear and geometric algebra Vector and geometric calculus Both by Alan macdonald, both very good.
- msla 8y agoEdit: Better link: https://www.av8n.com/physics/clifford-intro.htm https://www.av8n.com/physics/clifford-intro.htm https://www.av8n.com/physics/spacetime-welcome.htm https://www.av8n.com/physics/spacetime-welcome.htm
- jacobolus 8y agoWhat 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
- ngvrnd 8y agoIs this interesting primarily because of the noted correspondence with octonions?
- HelloNurse 8y agoIt'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".
- deehouie 8y agoThis' 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.
- nikofeyn 8y agoit's more that they're built on differential forms and the exterior derivative. for a reference, see section 4.6 and problem 19.13 of an introduction to manifolds by loring tu. this is also covered in gauge fields, knots and gravity by baez and muniain and in much detail in foundations of classical electrodynamics by hehl and obukhov.
- ajkjk 8y agoThe 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.
- schoen 8y agoIsn't the three-dimensional vector cross product also not uniquely defined, since the choice between the right-hand rule and left-hand rule is arbitrary?
- ajkjk 8y agoIt is certainly much more uniquely defined, since the direction of the result is determined. But yes, the cross product is defined relative to an orientation and changes coordinates accordingly. 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.
- kilovoltaire 8y ago"it is the only other non-trivial bilinear product of two vectors that is vector-valued, anticommutative and orthogonal"
- Tomminn 7y agoIn 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.