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Comment by Garlef | Hacker News Reader

While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge.

The basic object that the author seems to be interested in is that of an "algebra over a field" (https://en.wikipedia.org/wiki/Algebra_over_a_field).

Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied for a long time. (https://en.wikipedia.org/wiki/Division_algebra)

When studying math at german universities, alebras are something you'll encounter in your 2nd year (latest; but might already show up in 1st year analysis albeit with a different focus). Implicitly, division algebras show up a lot when students learn about field extensions, galois theory and the algebraic closure of a field (usually 3rd semester). A more general treatment of division algebras is not a common subject, though.

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It’s very intentionally written from the perspective of someone who knows linear algebra but maybe not that much, if any, abstract algebra, because that’s what the target audience is. Not university math students or graduates, but highschool graduates, CS/physics/engineering students and graduates, game/graphics programmers and so on.

Yeah. It's written to teach people who do not have (what some might consider) the standard prerequisite knowledge. I have no doubt the author understands algebra over a field. The author recommends teaching Geometric Algebra as a pedagogy at the end

The object the author is actually interested in is known as a geometric algebra. One often sees it discussed as an alternative theory for computer graphics or physics as it works well for expressing things like rotations.

See here: https://en.m.wikipedia.org/wiki/Geometric_algebra

I think it is probably not so helpful to merely think of it like a division algebra, and it is better to stay focused on the geometry. Curiously I find it easier to relate “actual” linear algebra to geometry than the thing people often call “linear algebra” that involves writing columns or rows or grids of numbers and manipulating them.

hey, by the way, to prove you right: please check the URL of the blog post

No, this is wrong. Geometric algebras aren't division algebras in general: they usually have zero divisors. Objects that live in a single grade are invertible, but composite objects don't always have multiplicative inverses.

As a concrete example, consider the elements 1 + x and 1 - x. Their product is 1 + x - x - xx = 1 + x - x - 1 = 0. So certainly 1 + x doesn't have an inverse, either.

Presumably you also meant to say "when X^2 = 1"

Yes, I meant that equation to be interpreted in the GA used in the article. But essentially all geometric algebras also have zero divisors, for similar reasons.

I think the author is interested in clifford algebras, or more specifically geometric algebras, rather than division algebras.

division algebras tend to be quite boring (if they are finite then they are just a finite field; if they are finite dimensional over an algebraically closed field then they are just the field itself. I guess the quaternions are an interested example in the non-algebraically closed case. but I think if you are over something other than R you're really just talking about a field extension)

clifford algebras are a sort of generalization of the exterior algebra one would have encountered in differential geometry and other spaces.

in fact it could be considered a "quantization" of the exterior algebra. as in "quantum groups". which is an entirely different part of maths. but that's not what this article is about.

I think using the language of geometric algebras / clifford algebras in physics as this article does versus the more traditional language is just a matter of taste.

Important correction: A division algebra is an algebra in which every non-zero element has an inverse. The dual numbers for instance are a Geometric Algebra which are not a division algebra because there are some non-zero dual numbers which don't have inverses. In fact, almost all Geometric Algebras fail to be division algebras.*

So your point about division algebras is not particularly relevant to the article.

* - Frobenius's theorem classifies all the finite-dimensional associative division algebras. They are: The real numbers, the complex numbers, and the quaternions. There are no others.

> It seems that this is written out of a perspective of some missing knowledge.

Well, the author talks about what "we learned in school", not university, so that checks out but only because you two have different audiences in mind.

Indeed. I learnt about vector dot and cross products, basic linear algebra (including diagonalisation, simple Markov chains and similar), partial differentiation, grad div and curl, the volume of a parallel piped and "all that Jazz" in high school, as a 17-18 year old. I learnt about the divergence theorem, Stokes's theorem, multivariate integration, integrating factors and higher order ODEs and simple PDEs, Fourier transforms and other integral transformations and similar in the first year of university (studying Physics).

This is not uncommon in the UK – but it is also not common either, and depends on exactly what A-level modules you did. My understanding is that it's quite rare to do exactly this in high school in the united states – but there, I think limits are taught much more heavily. I think having a clear, short statement of having "assumed knowledge" somewhere probably helps avoid these issues.

(I thought the article was excellent, and beautifully illustrated!)

I'll second this - my experience is pretty much the same (though I did some basic Fourier transforms at school), studying Physics at college. I did 3 Maths A levels along with Physics and Chemistry (and "General Studies" :) however, so I too may be atypical.

In the US, it’s common to refer to university as school.

Really? I was not aware. Do we know if the author is from the US though?

The author went to MIT, worked for NASA, and now lives in San Francisco https://twitter.com/mferraro89

It is unexceptional (indeed, expected) to get through an American undergraduate science or engineering degree without ever taking an abstract algebra course (much less the 2+ apparently expected of German pure math students).

But in any event, the top post here by Garlef is barking up the wrong tree. Division algebras, field extensions, and galois theory (per se) are not the tools to use for studying arbitrary-dimensional geometry. What you want is Clifford algebra (which Clifford himself, and later Hestenes, call “geometric algebra”) and then geometric calculus, which can be used on arbitrary manifolds, in non-metrical contexts, etc.

Basic geometric algebra should be taught to advanced high school students and all undergraduates studying any technical subject.

Math students looking for a math-style introduction to geometric algebra should try Chisolm (2012) https://arxiv.org/abs/1205.5935

Damn. They don't teach these stuff here atleast not in a computer science curriculum. What degree did you learn? Iss this generally taught in all German engineering courses?

German and french engineering school are pretty rough on math theory, for the better or the worse. Mostly because a lot of theory was born in these two countries.

Can confirm, before I was anywhere close to serious computer science, I basically did an undergraduate degree in mathematics / physics.

And it goes through some very advanced subjects in both. It basically prepares you to be an engineer in whatever field you choose, be it a structural engineer, or a computer scientist.

Can you tell me the topics they teach you there?

He is talking about math studies. Not much maths in engineering/CS in germany either.

But nobody is holding you back from visiting 'real' math lectures ;)

> Not much maths in engineering/CS in germany either.

That's simply not true. It depends entirely on the particular institution and its roots.

There are two origins of CS in German universities: electrical engineering and maths. At universities where CS originated as a subfield of maths, undergrad CS education is very similar to a maths undergrad to the point that most of the tests/mid-terms are basically identical between CS and maths.

If on the other hand CS came from the electrical engineering department, the focus is significantly less on maths and the lectures are very different indeed.

So you'd have to look at the history of each university and where the CS department originated to find out.

> There are two origins of CS in German universities: electrical engineering and maths.

Ah, I see, thanks for the information.

Which university would be an example for the latter?

Karlsruhe Institute for Technology is an example. Analysis 1+2 and Linear Algebra 1+2 is compulsory for every CS student (the exams are practically identical, just missing a single topic. You can always switch to the math lectures and get a few ECTS more). This is a great background for more advanced, theoretical lectures (CS+Math) but it's lacking electrical engineering lectures and especially there's no electrical engineering practice.

Seriously, which German university (not "FH") doesn't teach analysis and linear algebra in the first terms? I am really wondering.

TU Ilmenau has Analysis and Linear Algebra but that is only heard by mathematicians. There is a different class called "Math for engineers" which mechanics students, computer science students, ... hear which has different tests and a different focus but still teaches solving linear ODEs and stuff.

I think also in Germany you have to study pure math to see this. Where I study you even can avoid this by never taking Algebra 1, which is not obligatory (though imo it should be standard. There you learn why polynomials of degree >=5 have no closed form solutions)

> What degree did you learn?

Mathematics (I started a PhD but left academia in favour of starting a company). My focus was on category theory / algebra.

Speaking from personal experience, not all of us on HN have computer science degrees.

I studied mathematics in the UK and can confirm I learnt a lot about fields and other forms of linear and abstract algebras

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