[computator]

As far as I can tell, the area of the tip of the pencil doesn't factor into any of his calculations. But why doesn't it? Is he assuming it's a zero area point like we do in geometry?

Taken to an extreme, if the tip were 1 cm² and flat, then it wouldn't ever tip over, just as a coffee cup doesn't tip over.

[gradschool]

Not only that, but if the tip is a single point then will the pencil ever be vertical? That would imply that the pencil's center of mass is perfectly aligned with the tip relative to the gravitational field despite both being located in a continuum. Can any mechanism or natural process bring about an intended zero probability event?

There are similar discussions to be had about arbiters, whose usual conclusion is that metastability is inevitably resolved by thermal noise. The article addresses that issue by postulating a vacuum and a very low absolute temperature, but how would the pencil be shielded from the gravitational effects of distant bodies in motion? Does this experiment have to take place in an empty universe?