So I was looking at my gorgeous Christmas tree, and it got me thinking: aside from the color scheme, what makes a tree beautiful? Clearly the placement of the decorations has something to do with it. But how should ornaments be placed?
Hypothesis #1 is that you want the ornaments in a visible pattern with even spacing. Young children are often attracted to this hypothesis, and it leads them to draw Christmas trees that look like this:
Ugh. That’s no good at all – too regimented, too uniform. Reacting against hypothesis #1 might lead one to hypothesis #2: that ornaments should be arrayed randomly on the tree, with randomness instantiated via a uniform distribution over tree space. But if you really placed each ornament as an independent draw from the distribution, you’d get something like this:
Ooh... strike two. The problem here is that true randomness with independent draws leads to more clustering than most people expect (hence their willingness to believe in “hot shooters” at the craps table). This leads us to hypothesis #3: placement should simulate randomness, but with a constraint on spacing – no ornament too near or too far away from another. The outcome would be something like this:
Ahh, that’s better! An arrangement like this could, in principle, be accomplished via actual random placement, but with independent draws replaced by some function of distance from prior draws. The more distant a given empty spot is from already placed ornaments, the greater its probability of being selected for the next ornament. And in reality, I suspect our minds are performing just such a process, or at least something similar, when we decorate our actual Christmas trees.