March 15, 2016 at 02:11AM

Today I Learned: 1) A disjunctive sequence is an infinite sequence in which every possible finite subsequence occurs. An important point is that not all infinite nonrepeating sequences are disjunctive. Here's an example: consider the sequence that starts with 0011, then is followed by 00001111, then 0000000011111111, etc. This sequence is infinite. It does not repeat. However, there are plenty of finite sequences containing 0s and 1s that aren't in this sequence. Many real (irrational) numbers have digits that form a disjunctive sequence. In fact, they are so common that if you pick a random (and I mean truly random) real number, it will be a disjunctive irrational number. The most famous irrational number of them all, π, might or might not be disjunctive. Nobody knows. So next time you hear someone claim that every name, every word, and every work of art is contained in π in the right encoding, be skeptical. Many thanks to Andrés Ortiz Muñoz for bringing this fascinating fact to my attention! 2) I'm basically going to quote you an entire super-cute Science Friday segment on this one (thanks to Addgene's facebook page for sharing this!). How many digits of π is enough digits of π? You might feel like there is no such thing -- any finite number of digits is only a truncation of π, a phony invented by humans to approximate the real thing. Well, you're right, but it should be noted that 39 digits of π are sufficient to calculate the circumference of the universe to within the diameter of an atom. For now, that should do. 3) A technique for building diamonds of arbitrary (random) size in the abstract tile assembly model (aTAM) using only 11 tile types! My best design used 22. What a time to be alive! Thanks to Aileen Cheng for showing me her formulation.