I was thinking about the Pythagorean theorem and considering ways to see the truth of it much more intuitively, like a visual proof.
However, not just any visual proof that happens to work, but more like seeking an enclosing context - information that is hidden when you just look at a right triangle - by which it genuinely appears elemental. It’s almost as if the Pythagorean theorem is kind of an optical illusion, the man behind the curtain confusing you about how something is possible because what’s really going on you can’t see.
It’s difficult but first I was thinking about how in some ways what I see as the essential relationship being expressed is more like a ratio between sides a and b. If you have a right triangle, you can explore shortening or lengthening either a or b, and c just follows as a consequence of that, the connecting line between them. Also, as you make a and b both longer, you create proportional, larger triangles to smaller ones. And, if a is short and b long, as a gets longer and b shorter, after they pass each other, now you are studying the same triangles as before, only the reverse image of them.
So, maybe what we’re looking at is sort of like… a much smaller representative space of all possible right triangles - it’s actually just if you choose any arbitrary length for b, we are actually just trying to examine the relationship between any positive value of a as it approaches infinitesimally small and infinitely large - with, well, b, or with c, you can look at it either way, but I think I find thinking of c as the relationship between a and b more intuitive.
This idea of “finding a hidden enclosing context” has many possible directions to be taken. One at this point could be thinking more about the angle of altitude BC, but that is conventional and non-intuitive. I think it’s more interesting to think about the ratio of a to any arbitrary fixed length or point on some orthogonal axis, I guess.
Of course, as a approaches 0, the length of c tends to b; and as a tends to infinity, c tends to a. But at what rate, in all the space between?
As for the idea of seeking a broader context, instead of expressing the “difference” between two vectors as something called an “angle”, I think in the more foundational viewpoint of linear algebra, we have something more like a cross product? I think a cross product is an attempt to quantify the relationship between vectors in multiple dimensions because it is just a matrix that stores all possible ratios of all pieces of information of each vector. It is much more clear and fundamental than the notion of “angle”: angle requires us to ask what that is, where it comes from - a cross product is a very simple arithmetic operation which does the same thing, it captures a certain aspect of relatedness, a way to compare vectors which have the issue that they can’t be directly compared (in terms of magnitude) because they are not two points along the same axis.
I need to think more but maybe if you could stop thinking of c as a kind of arbitrary distance between a and b and instead reformulated the Pythagorean theorem in terms of some third dimensional operation, maybe dot product was what I had in mind rather than cross product, not sure, you could show how the relationship between a and b is a more obvious arithmetic operation capturing a spatial relationship between them projected onto a third axis.
Sorry if my reasoning is extremely butchered here but I think there might be some aspect of this I can keep thinking about, and would like to. I would also be interested in other perspectives looking at right triangles as surface level forms resulting from some higher dimensional relationship.
