I am a scientist working quite a lot on discrete geometric problems, usually involving the parameterization of crystal structures, and got aware of and interested in geometric algebra this way. I read material on the internet, a few books, watched youtube videos, for years already. I find it so elegant and the glimpse of its potential power makes it so intriguing for me that I really want to use it for my own work.
But then I have a big problem. I somehow don’t find a proper way how to use it in my daily work. I try to find ways to use it, and ask myself ways of how I might apply it, but usually this means a day of researching the concepts again, getting distracted by the big picture in many ways, eventually without having a result of using geometric algebra, and as a consequence going back to the usual linear algebra calculations I got used to, which are so cumbersome to do, but eventually give you the right result. This is very frustrating and usually stops my interest in geometric algebra for a while. Until it crosses my way again, always triggering my fascination again.
Thinking a lot about the reasons of my failing attempts, I found, that it seems the problem to be not just getting used to the new concepts, like in learning just another new calculation method again, but rather it appears to me that exactly the abstract power of geometric algebra is blocking me from further progress. In one way I admire the coordinate free approach, but when I have to model something, there have to be the coordinates again at some point, and then it seems to get back to the same cumbersome bookkeeping style of work again but with too much of preliminary/intermediate abstractions added to it. And even without the coordinates it appears so abstract that I do not get the idea how to model my geometric problem in the first place, say to use the power of geometric algebra to find the most efficient model, to improve on the intermediate calculations, to get clean and concise solutions. It seems to me that geometric algebra might not even be very useful in rather simple geometric problems? But how to approach more complex problems, if simple problems are not the way to start with? I just fail to get it on paper.
I guess it is somehow related to the fact, that many ideas of geometric algebra appear unfamiliar in the context of simple problems at first. Like using projective or conformal geometric algebra right from the start. Or even just deciding, which of these to use for a given context. To know which abstract tool to invoke in a special situation. The best thing that happened to me so far was to understand parts of some conventional linear algebra calculation I had done in terms of geometric algebra concepts, but just in hindsight that was, and without making any difference really, to the ease of getting to a solution.
I don’t know if other beginners face the same problem? If so, I would like to know, how they tackle it. Or, if any people who are advanced users of geometric algebra experienced the same problem in their beginnings, and how they overcame it? I have the impression just reading more texts or watching more videos does not really help me.
Thanks and best wishes