The real problem is that it takes quite a bit of practical problem-solving experience to get used to a mathematical language, months or years of work, to really become fluent with it. The same is equally true of trigonometry, vectors, complex analysis, quaternions, calculus, matrices, spinors, group theory, differential forms, manifolds, etc. etc.

But with a mainstream bit of technology, you can go to any good university math department and sign up for a course on the subject. You can take your pick of 100 textbooks. You can read resources aimed at applying specific tools to any number of concrete practical uses. You can often assume that people with a math, science, or engineering degree are already familiar with the technology and understand how it works. You can go find open source implementations of tools built using that technology. Etc.

As a result, we don’t remember how hard all those tools were to learn, and how much work went into building up the conceptual ecosystem.

With a non-mainstream bit of technology, you get to choose from a small handful of resources, often aimed at an audience you aren’t properly a member of or aimed at uses you don’t care about and missing parts that you need. Often best approaches for solving particular problems haven’t been worked out yet. From one source to another, notation conventions and choices of model details can vary, making it hard to compare sources, notice errors, or communicate ideas. Concrete code implementations have seen little scrutiny and so are likely to be incomplete, inefficient, and buggy.

This is a problem with any difficult new bit of technology. Sometimes it takes decades or even centuries for a piece of mathematical technology to go from introduction to wide adoption. Sometimes they don’t make it or sit abandoned on the shelf for a while.

As far as I can tell mathematicians haven’t adopted Geometric Algebra for a simple reason: Mathematicians make their living proving new theorems, and GA is equally/less powerful (in a theoretical sense) than existing abstractions they have, so as a result isn’t particularly useful for their proofs. Its advantages are as a sort of “refactoring” of a bunch of existing technology, and it’s usually a career limiting move in any field to spend too much of your time refactoring, however advantageous it might be for humanity in the long run.

If you’re interested in physics applications, you should check out Hestenes’s book New Foundations for Classical Mechanics, which is pretty good. I’m particularly impressed with the completeness of the treatment of rotations and rigid body mechanics in chapters 5 and 7. General relativity is a field where Geometric Algebra already sees a fair amount of use in the academic literature. Hestenes has also written papers which sketch promising approaches to electrodynamics, quantum mechanics, etc., and the guys at Cambridge have been working on those a bit, but it’ll probably take a large group of others to really flesh it out.

> On the contrary, when I've tried to use it, the complexity of GA just makes it impractical. The problem with 4 or 5 dimensional GA for doing work in 3D, is that we end up dealing with 16 or 32 values per object (if you unify vectors, bivectors, etc), or else having five or six specialised forms of object plus their combinations (if you keep them separate), with potentially hundreds of specialisations of basic operation. This isn't just me, the GA libraries around have this bloat.

This is two tricky problems:

(1) In most cases Geometric Algebra solutions to problems don’t essentially require more computation than other approaches, but figuring out how to write efficient practical implementations (especially against our current hardware and software which is heavily optimized for solving matrix problems) is non-trivial and hasn’t really seen that much work yet.

(2) Existing programming languages really aren’t optimized syntactically for interacting with multidimensional objects in a natural way. Trying to cram a GA system into C++ or Java (e.g.) is just asking for a bloated and confusing syntactic nightmare. You see the same problem with trying to write code for doing matrix algebra in those languages, but at least we have a lot of experience with the domain and there are languages like Matlab or Julia or libraries like NumPy that make things much nicer. Nothing like those exist for Geometric Algebra, and it’s a non-trivial problem to solve.