The rule of three is also a decorating rule. You should always clump picture, vases, etc. in clumps of three. The reason always given is balance. Three esthetically produces the best visual balance. Maybe the universe has an esthetic sense?
China/Chinese, for example, emphasises balance via symmetry. Hanzi are balances by layout into 4 grids (even practice books have a 2x2 grid for a character) - yes I am aware some characters have 1, 3 or 5 particles however these also fit within this grid.
Design of architecture is based around 2s and 4s, leading to relatively 'square' stuff (easy example, see the layout of the Forbidden City) and, sometimes, within this, the concept of non-uniformity.
I think the rule-of-three is quite culturally specific.
Perhaps two points create an implied line that cuts across the table, where as three or more create a polygon that fits -within- the table with some balanced amount of space between the polygon sides and the table sides.
Alternatively (but perhaps similarly), it's the same reason why two-legged tables don't really exist.
> What makes two vases on a table less "balanced" than three?
Two leads to dialectic competition between them.
> What about a single vase?
Static world
> Four? Five?
Chaos
What I'll say on the matter is that both religious and esoteric texts place high importance on groupings like this. Certain texts may provide explanation for the generations of particles.
They are worth investigating with an independent mind, while assuming that what you're reading is incorrect.
Science verifies information. The issue with these texts is that it's very, very hard to verify anything being said. It's much easier to provide counterexamples to disprove it.
By using the phrase "independent mind," I mean: it's worthwhile investigating these texts, but go in with a truth-seeking mentality, not a disproving mentality.