<story><title>All Hail Geometric Algebra</title><url>https://crypto.stanford.edu/~blynn/haskell/ga.html</url></story><parent_chain><item><author>zrkrlc</author><text>I spent around two months being completely enamoured by geometric algebra until I realised it flies in the face of  some pretty strong trends of mathematics in the last couple of decades.&lt;p&gt;How come? Well, we’ve been moving towards more and more powerful generalisations and unifying concepts over the years. First, it was the various flavours of algebraic this-and-that, then category theory, then today, people are trying to do hot things with HoTT.&lt;p&gt;(of course math is much, much bigger than this, but this is the story I particularly have been watching unfold before me)&lt;p&gt;The trouble with geometric algebra is that it’s merely a special case of a much more general concept, namely tensor algebras. Tensor algebras are the most general algebra you can possibly get from things that behave like vectors; with GA you are stuck with vectors where the distances are measured by something squared as opposed to any other power (or some other crazy function like the taxicab metric).&lt;p&gt;Using geometric algebra feels like typeless programming because you’re combining everything into a single quantity called the multivector whose components aren’t straightforward to keep track of. Furthermore, you have to decide the dimension in advance because IIRC these types change when you add more dimensions (the formulas however do not, surprisingly).&lt;p&gt;That said, it’s really marvellous how compact things can get if you stay in 3+1 dimensions. Maxwell’s equations become literally a one-liner, and the horrible, horrible mess that is rotational dynamics becomes so much more tractable for mortals like me. One can literally do the entire physics undergrad curriculum using GA without breaking a sweat. Trouble is, you’d have to learn some basic algebraic theory and that’s a hard sell for people who are already afraid of math because “it’s too abstract”.</text></item><item><author>thechao</author><text>I cannot recommend Alan MacDonald&amp;#x27;s two books enough (Linear and Algebra; and, Vector and Geometric Calculus). They literally replace &lt;i&gt;reams&lt;/i&gt; of texts on complex numbers, quaternions, octonions, spinors, etc. under a single unifying concept. (They don&amp;#x27;t &lt;i&gt;replace&lt;/i&gt; those concepts; and, for sure, if you want to learn quantum you&amp;#x27;re going to need them; but they make getting your head around them so much easier.)&lt;p&gt;One issue is that most of the &amp;quot;online&amp;quot; references are &lt;i&gt;crap&lt;/i&gt;. MacDonald&amp;#x27;s presentation is a wonderful slow ramp: it starts with really basic linear algebra and then just slowly segue&amp;#x27;s into geometric algebra.&lt;p&gt;I firmly believe that in 100 years we just won&amp;#x27;t be teaching &amp;#x27;bare&amp;#x27; linear algebra, complex, quaternions, ... we&amp;#x27;ll just teach GA and then introduce the physical concepts separately.</text></item></parent_chain><comment><author>joppy</author><text>Tensor algebras are the most general algebras you can get (categorically inclined people would call them a “free object” in a suitable category), but because of this they are almost never useful on their own. We pretty much always study specialisations of tensor algebras, of which GA (or more generally Clifford algebras) are an example.&lt;p&gt;There are no interesting formulas in a tensor algebra, because the whole idea of it being “most general” (every other algebra is a quotient) is that it has no relations. It is only when passing to more specific algebras (symmetric, exterior, GA, ...) that the subject really becomes interesting.</text></comment>
<story><title>All Hail Geometric Algebra</title><url>https://crypto.stanford.edu/~blynn/haskell/ga.html</url></story><parent_chain><item><author>zrkrlc</author><text>I spent around two months being completely enamoured by geometric algebra until I realised it flies in the face of  some pretty strong trends of mathematics in the last couple of decades.&lt;p&gt;How come? Well, we’ve been moving towards more and more powerful generalisations and unifying concepts over the years. First, it was the various flavours of algebraic this-and-that, then category theory, then today, people are trying to do hot things with HoTT.&lt;p&gt;(of course math is much, much bigger than this, but this is the story I particularly have been watching unfold before me)&lt;p&gt;The trouble with geometric algebra is that it’s merely a special case of a much more general concept, namely tensor algebras. Tensor algebras are the most general algebra you can possibly get from things that behave like vectors; with GA you are stuck with vectors where the distances are measured by something squared as opposed to any other power (or some other crazy function like the taxicab metric).&lt;p&gt;Using geometric algebra feels like typeless programming because you’re combining everything into a single quantity called the multivector whose components aren’t straightforward to keep track of. Furthermore, you have to decide the dimension in advance because IIRC these types change when you add more dimensions (the formulas however do not, surprisingly).&lt;p&gt;That said, it’s really marvellous how compact things can get if you stay in 3+1 dimensions. Maxwell’s equations become literally a one-liner, and the horrible, horrible mess that is rotational dynamics becomes so much more tractable for mortals like me. One can literally do the entire physics undergrad curriculum using GA without breaking a sweat. Trouble is, you’d have to learn some basic algebraic theory and that’s a hard sell for people who are already afraid of math because “it’s too abstract”.</text></item><item><author>thechao</author><text>I cannot recommend Alan MacDonald&amp;#x27;s two books enough (Linear and Algebra; and, Vector and Geometric Calculus). They literally replace &lt;i&gt;reams&lt;/i&gt; of texts on complex numbers, quaternions, octonions, spinors, etc. under a single unifying concept. (They don&amp;#x27;t &lt;i&gt;replace&lt;/i&gt; those concepts; and, for sure, if you want to learn quantum you&amp;#x27;re going to need them; but they make getting your head around them so much easier.)&lt;p&gt;One issue is that most of the &amp;quot;online&amp;quot; references are &lt;i&gt;crap&lt;/i&gt;. MacDonald&amp;#x27;s presentation is a wonderful slow ramp: it starts with really basic linear algebra and then just slowly segue&amp;#x27;s into geometric algebra.&lt;p&gt;I firmly believe that in 100 years we just won&amp;#x27;t be teaching &amp;#x27;bare&amp;#x27; linear algebra, complex, quaternions, ... we&amp;#x27;ll just teach GA and then introduce the physical concepts separately.</text></item></parent_chain><comment><author>Koshkin</author><text>&amp;gt; &lt;i&gt;Maxwell’s equations become literally a one-liner&lt;/i&gt;&lt;p&gt;There still must be a limit to this. For example, I do not find &amp;quot;dF = 0&amp;quot; especially enlightening. It&amp;#x27;s almost like saying that &amp;quot;42&amp;quot; is the answer to the question about life, universe, and everything...</text></comment>