Kindle Notes & Highlights
Either a trajectory is periodic, it retraces the same path again and again, or it fills up the whole table in equal measure. There is nothing in between. (This is known as the Veech dichotomy after the mathematician W.A. Veech.)
Therefore, if you shoot a ball in a random direction, your chance of producing a periodic trajectory is zero too.
Informally, the Kolmogorov complexity of a sequence x, abbreviated K(x), is defined to be the number of bits in the shortest computer program whose output is x.
So the real problem is that the Kolmogorov complexity is impossible to compute.
It turns out that we can never be sure we’ve found the shortest one.
Consider asking for “the first number that can’t be described using 12 words or fewer.” But we just described the number, and we did so using only 12 words!
So we conclude that there simply aren’t enough short programs to go around: If you choose an n-bit string x at random, then you’ll almost certainly have K(x) approximately equal n.
One can represent the product of 47 × 52 as the product (50 − 3) × (50 + 3) less 47, which yields (2500 − 9) − 47. The many individual digits that must be kept in mind using the customary multiplication algorithm have been eliminated by introducing a single new idea, the difference of two squares.
Rota, G. (1997a) The phenomenology of mathematical beauty. Synthese 111(2), 171–82. Rota, G. (1997b) The phenomenology of mathematical proof. Synthese 111(2), 183–96.
The most striking use of such a simple principle is probably Dirichlet’s theorem on Diophantine approximation: for any real number θ and any positive integer Q, there exist integers p and q with 0 < q ≤ Q such that
Rota states that “the beauty of a piece of mathematics is strongly dependent upon schools and periods of history.
We must avoid the “light-bulb mistake” which consists in believing that mathematical beauty is “appreciated with the instantaneousness of a light bulb being lit” and hence its appreciation is “an instantaneous flash” (ibid., 179).
