Would you say it is difficult to get so many people to start using Geometric Algebra?
Oh yeah, but there is a strategy to it. I look for people that are interested. One, they’ve got to be curious and two, they’ve got to be able to take a risk. Because it is a significant ask to shift from the current way of doing things- with matrix algebra and vectors and stuff- into this Geometric Algebra. You actually have to unlearn a bunch of things that you will need, and you are trapped into it. Some of it is really engaging the curiosity with the working environment. It is also: Do you want to make this product better? Cheaper? Faster? There is a term called SWaP: size, weight and power and you want to reduce those every time. So the question is: are you willing to take the risk in order to reduce the SWaP and the cost? And if you are, it is not going to be a necessarily easier road, but I am there with you and I will teach you what you need to know.
Exactly, and when you have so many engineers and they ask why you would do it, what would you say is the biggest reason for switching? How would you sell it?
Well I have seen in the pilot projects that I have started that there is enough insight into what I have said that they have realised that there is a huge advantage to it. So they are willing to take on the learning curve (of Geometric Algebra) to do that because the advantage is so obvious to them. You don’t need to understand the depths of it. So for instance in the talk that I gave, if you noticed in the beginning of the pilot projects, I did ‘what if’. So here is the problem. What if I could do this with pure math, with more tight, more optimized, and so I tried to figure out what are the ‘what ifs’ that would sell. Then of course, I had to prove that we could actually do this. But I am pretty sure that when I say ‘what if’, I could do it this way. I had one guy that said “I don’t believe you, there is no way you could get all that. There is always a cost. Where is the cost?” and I did know there isn’t, so I had to actually prove it to him. Once I showed him, now he is interested. Now I will go teach you the rest, the tools. I will teach you the techniques. The pilot projects were chosen just to sort of seek out these kinds of people that were you know “I’m really interested” and they will become my evangelists for the rest of the engineering staff.
I assume that it will first slow down development a bit, right? Because you will have to teach a lot of people. It will take time and money. How much will it increase development time in the end? I guess in the long term, it will increase development time majorly. You could have projects in 10 years that could take less time now.
That is actually one of the selling points. This is multidisciplinary, so you’ve got mechanical engineers, electrical engineers, people dealing with radiowaves, people dealing with colour, that type of stuff. Because it is a universal language, they can talk to each other and they can describe their problem in a universal language. Just that, where all the engineers speak the same language, it becomes very powerful. Hence the same tools across the board because they are a huge slough of different choices of tools. OK, we are going to build one for you in a tool that is already the standard of the company so that you are not relearning the tool.
You showed with the colour example as well that using Geometric Algebra, you discover methods that we’ve previously also discovered, but just way easier. Now from all the other talks I see, there are many examples where we can see that Geometric Algebra just does it in fewer and fewer operations and it’s wonderful to see that, but do you think that there are methods or techniques that we haven’t discovered yet that we will be able to discover with Geometric Algebra?
Yes.
Can you give me an example of something you’ve already discovered with it?
For instance, that colour example. So I was actually trying to solve that using quaternion algebra. I’d get a solution, but it was very hard to do and very complicated. That would’ve been really tough to teach somebody. It was very ugly and the fact that I could write out that algorithm in five lines of equations and say, “there is the solution”, was great. The advantage of doing that in such a condensed way of thinking is just huge once you’re like “OK, I understand how this lays out. I understand the geometric concepts”, once you understand it. I was surprised by how many different problems actually have an underlining geometry to them. One of the things I always teach is “here is your problem, now take out the geometry problem from that and resolve that. And then we will put it back into the solution”. Once they see that way of working, they’re usually all in at that point. There is a threshold and I have to sort of kick them over the edge, which is why right now I’m embedded into the project. Because once I get somebody to say “Yeah, we will put you on our project”, then I can start showing them. It usually works best when somebody is sitting that’s “OK, watch what I’m doing”. One example, I actually flew out to the site because it was on the other side of the United States, and I actually sat there for three days and they walked me through what they were trying to do and then I walked them through the tools. I gave them half a day of training just to sort of ground them, but at that point it’s like “OK, this is good” and they bought into it and said “We want to do it this way”. Most of my friction is because my projects are long term. Typical aircraft cycle is ten years and if you’re halfway through that cycle, there is no way anybody is going to jump ship and shift.
Do you think geometric algebra will replace original methods completely in the future? Or do you think there will always be a place for matrices? Because I understand that for instance for vectors, there will always be a place.
It depends on the school. There are some problems where the efficiency of the current technique is so high, that you would really have to work hard to close the gap, because it has 50 years of development behind it. There are different kinds of problems that you can go and add to that, so that is really where I’m seeing people sort of pushing “I’m going to do something that’s extremely difficult and there really isn’t a good solution yet”. This usually lands itself into those kind of problems. We might just want to cut after the 50 years of development to make life easier
Geometric Algebra obviously speeds up the processing time a lot. You also talked about for instance, if there is an emergency in a plane, the pilot has to find a landing site. It would be really nice if we could find those locations really quickly in real-time. Do you think it will improve safety in aerospace as well?
There is a threshold of the safety that needs to be met. I talked about flight certification, what I’m talking about, “is it safe to fly?”. There are different levels of insurances that say that if something goes wrong with this, people will die. Nothing can go wrong with it. You have to think at that level already. The question is not “will it be safer?”, the question is ‘will it reach the safe levels that you’re asking for in a more costly way, or in a less expensive way?”. You’re dealing with stuff that are six, seven, eight, sigma safety requirements. They don’t let you put anything on an airplane that hasn’t been certified at that level. The insurance level where everything from how you actually run the project for developing, to how you actually test and everything in the system, has to have a requirement attached to it and every requirement has to be tested when the system is done? If there is something in the system that has no acquirement, that is a red flag right away. Why is this here? You’re going to have to explain where that came from and why it is in there. You don’t add anything that doesn’t have an acquirement to it, so that you can test it at the other end. The question is “Can I compress that endeavour, going from the other acquirements to the physical design and how are you going to test that?”. The compression is what I refer to as the development and the maintenance cost and compressing that down. Will it make it safer? Well, in a secondary effect because it is more obvious what is going on, because I can read the code and it reads like mathematics so I know where the system is heading. It is more transparent from that point of view. There are some algorithms that are so obscure that you’d spend a month looking at this code trying to figure out how it works and why. These problems start to go away because you start thinking and reasoning geometrically.