Pascal's simplex is a generalization of Pascal's triangle into n dimensions, just as multinomial theorem is a generalization of binomial theorem. In Pascal's triangle, binomial coefficients are arranged in the close-packing arrangement for circles. In Pascal's pyramid, trinomial coefficients are arranged in a close-packing arrangement for spheres (there are multiple in 3 dimensions). This is also true for 1-dimensional spheres (aka line segments). My intuition is that this pattern would continue infinitely, but I know that doesn't mean anything, especially as a layperson.
I should also point out that each the close-packing arrangement for spheres has cross-sections that look like the close-packing arrangement for circles, which has cross-sections that look like line segments stacked together, which is the close-packing arrangement in 1 dimension.
Are these apparent patterns just a coincidence?
If I've gotten anything wrong, please let me know.