Uh… what does it have to do with these "factorization diagrams"? Maybe I'm missing something, but I don't even see why are they "worth checking out". As far as

Uh… what does it have to do with these "factorization diagrams"? Maybe I'm missing something, but I don't even see why are they "worth checking out". As far as I can see following your link, these are just arranging a number of dots into (pre-determined) shapes that are humanly recognizable. I.e., these are literally just some caveman technologies for writing a number before a more convenient (i.e. arabic) number system was invented. If anything, that combination of squares and triangles is less readable way to write 2²×3³×5, and it can be constructed only for "convenient" numbers (well, you can arrange into blocks 23×11×2 and 23×13×2 too, but good luck telling them apart). That's just silly, and tells me absolutely nothing about relationships between numbers. However, all numbers from 1 to 1 000 000 forming a distinct clusters when being mapped to 2 dimensions with UMAP… I don't know. It might be nothing (like a representation of something trivial, like an observation, that multiples of 100003 are less common than multiples of 3 in the set of first 1 000 000 integers), and all these clusters may just disappear (converge) as we go closer to infinity. But there's definitely something a bit eerie about the possibility of it not being "nothing". Normally, you wouldn't expect any patterns to form like that. Or, well, it may be more that a "nothing" but less than "interesting" for a mathematician — maybe there actually is some pattern that becomes more visible in this visualization, but it's already well-known among number theorists. I just have no idea, that's why I'm asking. It's just weird to see any clustering at all here. (And, yeah, BTW, there's no such thing as "correlation vs causation" in number theory.)