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There's a fascinating history of how out of the four founding fathers of modern multi-dimensional mathematics, three preferred geometric algebra, and only one p
by jiggawatts 3y ago
There's a fascinating history of how out of the four founding fathers of modern multi-dimensional mathematics, three preferred geometric algebra, and only one preferred vector and matrix algebra. The three all left academia or died young, the one dissenter lived to a ripe old age and was a vocal proponent of his preferred approach -- because it's more useful for statistics. He wasn't a physicist.
All of these weird "triple products", limitations to 3D space, etc... vanish in geometric algebra. You can use the exact same formulas in 2D space, 3D space, 4D spacetime, or 18 dimensional spaces with degenerate dimensions if you please.
There's this obstinate refusal to just admit that the maths that's ideally suited to solving statistical problems may not be ideal for physics, robotics, or optics. Instead, physicists insist on re-inventing the good stuff over and over, badly, with different names, each time.
All of the following are just geometric algebra in disguise, or various "subsets" of a geometric algebra, or a geometric algebra operation that got renamed:
- Cross products
- Triple product
- Exterior algebra
- Complex numbers
- Quaternions
- Octonions
- Pauli matrices
- Gamma matrices
And on, and on, and on.
As a random example: the way we represent 3D rotations using a vector in 3D doesn't work in 2D, because a 2D rotation vector would point out of the plane. It also doesn't work in 4D or higher dimensions. It just happens to "work" in 3D not because 3D is natural, preferred, or special, but because the broken maths happens to not be completely broken in one case due to a simple coincidence.
In geometric algebra, instead of using a vector, a bivector is used, which is like a surface patch. Notice that you can have a surface in 2D, so rotations in GA work in 2D. You can also have a surface in 3D, so rotations work in 3D. And in 4D (including both 4D space and 3+1D space-time!), and 5D... and all of rest, with the same formula!
PS: Geometric algebra also prevents gimbal lock, doesn't store redundant values, and has better numerical precision. Its transformations can be interpolated unlike (famously!) matrix transformations, which can't. In robotics or computer graphics this means everyone uses quaternions instead, which are... drumroll... the "even subset" of a geometric algebra.
- Jenz 3y ago…and in turn “geometric algebra” is the physicist’s name for a subset of the even more general mathematicians Clifford algebra (which of course is a subspace of the tensor algebra over the underlying vector space…)
- jiggawatts 3y agoClifford Algebra is basically just Geometric Algebra, except with complex numbers shoved into where there ought to be real numbers... because no Mathematician could ever resist doing so. As a programmer / physicist, the analogy I use is that if I had to represent an 8-dimensional parameter as an array, I would write: double foo[8]; but a mathematician would be unable to resist writing: complex foo[4]; They're the same, but the latter is more complex (hah!) for no real (haha!) benefit.
- aleph_minus_one 3y ago> They're the same, but the latter is more complex (hah!) for no real (haha!) benefit. When you consider problems in real geometry, this is plausible. But on the other hand, not all mathematics is real geometry. For some applications in mathematics, the complex-geometric perspective is more natural. P.S. Of course p-adic (and in particular 2-adic) geometry is even cooler than anything real or complex ... ;-)
- jiggawatts 3y agoIt's type confusion caused by duck-typing. Just because can you can add, subtract, multiply, and divide complex numbers doesn't mean they're a natural substitute for real numbers in all scenarios. Sometimes you have '2n' real numbers, not 'n' complex numbers. If mathematicians designed computer software, their data structures would look like: struct Person { char name; complex name_height; }
- eigenspace 3y ago> It's type confusion caused by duck-typing No it's not. This is such a typical and braindead HN take.
- aleph_minus_one 3y ago> doesn't mean they're a natural substitute for real numbers in all scenarios. I claimed the exact opposite: whether real or complex numbers as basic objects are more natural depends on the problem that you consider.
- pa7x1 3y agoWhenever this kind of topic arises there is always someone that has worked a bit on 3D geometry, that comes to school mathematicians and physicists on what they are missing out for not using geometric algebra. Usually with a weird tinge of conspiracy theory and cultish behavior mixed in. They are not missing out, they are aware of geometric algebras and the work of David Hestenes. They instead prefer to use exterior algebra and Clifford algebras as a generalization. There is no conspiracy theory or large-spread ignorance by the physics/mathematics community. Geometric algebra feels like introducing a quirky notation to talk about the same concepts, with no clear added conceptual or computational benefit. So most people ignore this approach. As I'm not particularly well-versed in the historical elements of this affair, nor I have much interest on them, I googled quickly and found this pertinent answer by John Baez: https://www.physicsoverflow.org/42912/geometric-algebra-geometric-calculus-all-that-its-hyped-be https://www.physicsoverflow.org/42912/geometric-algebra-geom...
- msla 3y ago“Symmetrical equations are good in their place, but ‘vector’ is a useless survival, or offshoot from quaternions, and has never been of the slightest use to any creature.” — William Thompson, 1st Baron Kelvin (better known as Lord Kelvin) Letter to G. F. FitzGerald (1896) as quoted in A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (1994) by Michael J. Crowe, p. 120
- jiggawatts 3y agoWait till I start telling you about how we should be using tau as a replacement for 2 pi! In all seriousness, we wouldn't accept this kind of thing in the standard library of a programming language. Imagine if due to a historical mistake, all arrays reported their length as 2x the number of items in them, and indices would similarly be multiplied by two. So the first item would be a[0], and then a[2], a[4], etc... Now imagine the entire software development world making excuses and justifications. "Oh, it's isomorphic, and shift-left-by-one is cheap!", or "sure, you can only use half of the system memory for any single array, but do you really need more than 2GB of data? 64-bit is common now anyway!" Physics is in a dire need of refactoring, but it has never had it happen. My opinion of studying it was that I didn't expect it to be the "history of physics". I expected to be studying the current state of the art, not the three-hundred-year journey of comedic errors getting there. Imagine starting Chemistry by studying alchemy for years and only then being told -- oh -- there's these things called atoms! Or trying to understand Linux by reading the commit history from day one. Etc...
- red_trumpet 3y agoWho are the "four founding fathers of modern multi-dimensional mathematic" your refer to?