>I don't see how any of this solves hard problems. You need to know a formula to compute what you want regardless of the underlying representation. The thing w

>I don't see how any of this solves hard problems. You need to know a formula to compute what you want regardless of the underlying representation. The thing with geometric algebra is that the function is the same regardless of what you're working with. In projective geometric algebra, joining two points into a line is the same function as joining a point and line into a plane. Intersections, projections, rejections, and transformations all have the same function, regardless of the objects involved. Reflecting across a plane, around a line, about a point, really any transformation, uses the same A * B / A, formula, regardless of whether B is a point, line, plane or transformation. You still need to learn it, but you only need to learn it once. >I also think the more abstract you get, the smaller portion of the population can understand and work with it. That's exactly the point. Geometric algebra is a large step down in abstraction from quaternions, yet explains them fully. The rules of geometric algebra build directly from vector algebra and could be taught in a single lecture, with several semesters worth of material in geometry, calculus and physics that follow from those simple rules. But the people who use quaternions already know quaternions, and the physicists who work with complex matrices already know complex matrices. They think they're standing on isolated towers of knowledge they had to scale by their own hard work, and see geometric algebra as a separate height that offers no advantage on its own, when it was actually a shallower ascent, that once climbed, reveals that what had appeared to be distinct areas of study were actually linked and could be understood together. Projective geometry and special relativity might seem like different subjects from the outside, or from within either, but when viewed from geometric algebra, they're just algebras with different signatures (whether the basis vectors square to 1, -1, or 0). A Lorentz transformation is little different than rotation around a line, and if you can work with one, your knowledge transfers directly to the other. All built from the building blocks of vectors, no complex matrices or generators required.