I'm a big fan of Eric Lengyel, and I never would have ever gotten my arms around Geometric Algebra if it hadn't been for his books and articles. How many people can do theoretical math AND code a state-of-the-art game engine? The guy is a walking, coding miracle.

So if I'm a little critical of the tone of the article, it comes from a place of love. There has been a very toxic, clickish vibe in Geometric Algebra circles, which have lead to some pseudo-disputes among those who should be natural allies.

One such is that Gunn, et al, prefer to represent a 3D vector using a dual basis (e.g. [a1, a2, a3]^T = a1*e32 + a2*e31+ a3*e12) whereas Lengyel prefers to just represent them as a1*e1 + a2*e2 +a3*e3. Some really unfortunately hostile back and forth arguing about which one is "the right way"--when in reality, it's a big-endian vs little-endian thing. One of the best parts of projective geometric algebra is that you can flip back and forth to the dual representation whenever you want to, according to what makes sense to you--and what makes the problem at hand easier to solve. Moreover, if you look at the actual calculations doing it one way vs doing it the other it's the same damn exact numbers being multiplied/added in the same damn way. It's not quite as silly as arguing about what font numbers should be printed with, but it's pretty close.

The tone of the article reflects wounds which are still pretty sore from these sorts of battles. So I understand. But sometimes the invective goes a bit to far.

An example of this is his critique of Gunn's initial cut of dualizaiton for PGA. The fact that e0 is not invertible is a big, fat wart, for sure. And frankly, I spent weeks trying to understand Gunn's workaround and its wierd lingo. J-map? What in the world is a J-map? I finally understood the concept, but I've never found out what "J" stands for :-) And Lengyel's treatment is much smoother and more coherent.

Yet, I don't think Gunn should be criticized for it at all. It was an act of courage for Gunn to come up with his janky J-map, and not let its janky-ness stop him and the rest of the field from moving forward. Sometimes that is exactly what is required in mathematics. For example, infinitesimals. For centuries, mathematicians from Archimedes to Newton found them indispensible, even though they had absolutely no coherent mathematical foundation. In point of fact, if you listen to, say, a Feynman lecture, you'll find that they are still indispensible today. But they didn't have any kind of mathematical foundation until the 1960's, when Robinson found a way to coherently axiomitize them.

I saw a video of Freeman Dyson once, where he was talking about how he was able to prove something which was a longstanding open problem. He described his proof as "very ugly" and then went on to say (with tongue partly in cheek) that you can judge how great a mathematician is by how many ugly proofs he creates :-) Because the first time something is proved, the proof is almost always very ugly. It's not until other mathematicians come in and find connections with other branches of math, and start being able to come up with more elegant proofs.

So let's celebrate the ugly, messy, janky-ness which is the reality of how mathematics is actually created, and the courage of the mathematicians to not rat-hole and bike-shed.

Eric Leyngel's presentation of projective geometric algebra is, IMHO, far more coherent and elegant than any other presentation. His books (and his source code) are a joy to read. For a newb like me, it is far easier and quicker to absorb. Isn't that good enough? Did he really have to go on to flame everybody else to a crisp? sigh like I said, he has been subjected to very unfair and toxic pillorying, and the wounds are still fresh, so like I said, I understand. But its very regrettable nevertheless.