I would say 3 resembles a minor-note. Like whispers and illusions.
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In the point of view of a non mathematician (definitely not the best at it), I think humans attribute significance to the number 3 perhaps because it’s balanced. 1 is, well, only one; two would lead to two opposite sides which in certain contexts would “fight” or disagree with each other, it’s either one or the other. 3, however, is more balanced. It’s harder for three opposing sides to cancel each other out, or the third side could be neutral. 4 could divide into two groups of two, 5 would also make smaller potentially opposing groups (2 and 3) and so on as you increase.
I always thought the best friend groups were trios, especially in discussion. You’re somewhat forced to all be engaged together, two friends only talking with each other would lead to the exclusion of the third friend.
If anyone here watched The Rings Of Power, there’s a scene where they momentarily mention the balance in creating three elven rings for three elves to share power (if I remember it correctly).
Either way I don’t necessarily believe three has any natural/universal power, I think we just found the number 3 practical and can coincidentally (or not) find it often, though of course there are always things beyond our understanding (or at least mine).
I think you’re getting at the same intuition I was talking about in the OP. As I noted in an earlier post, I don’t think this is a mathematical insight.
This made me think of the quote from Wang Bi in my OP.
Agreed, no universal power. As I noted, perhaps its a cognitive phenomena. Humans seem to have a tendency to separate unitary concepts into two opposing or complementary ones, e.g. good/bad, ugly/beautiful. I think it’s possible that’s a fundamental feature of human thought. There’s no reason that there can’t be something similar about three.
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I’ve made some objections
- Apophenia
- Confirmation bias
- How does 3 compare to \varphi
Your evidence set fails to support your thesis. A mathematically-trained person would know that. I’m going to be a little troll and say, yes there is something about 3 … nothing, nada, zip, nyx. to be precise 
He’s saying 3 is everywhere, in the same sense that bricks are, in Legoland. It’s, unfortunately, not true. 
Doubtless all this demonstrates the Holy Trinity.
What a lot of cobblers.
Quibbling isn’t my forte, but regarding the holy trinity, is it 1 or is it 3? Is 1 = 3?
I was clear in my OP that this was speculation based on intuition. I don’t need strong evidence for that.
As I noted, I don’t see this as a question of mathematics.
I don’t like you Marvin.
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I have it on authority to be so.
Not that of an authority in whom I trust, mind you.
Yes, as this is speculation I was also closing things off too quickly. My apologies.
Someone asked me, what’s the difference between religion and true religion? 
I’m not discounting the claim of 3-ness. Who knows, 3 could be a fundamental number, for all we know. However, there are inferential barriers, like apophenia, confirmation bias. Have you taken these into account? I don’t question intuition but wouldn’t it be nice if reason vindicated it?
Would you accept a gift?
f(x) = \begin{cases} 3, \text{ if } x \in \mathbb{R} \\ 3 \text{ if } x \cancel {\in} \mathbb{R}\end{cases}
Isaac Newton was a non-trinitarian.
Too, the holy spirit is largely ignored in theology. It only gets mentioned when as a mediator between god and us. So effectively 2, not 3. 