At the risk of veering into crackpot territory, this operation is extremely profound and shows up in an absurd number of places. For example, imagine you like

At the risk of veering into crackpot territory, this operation is extremely profound and shows up in an absurd number of places. For example, imagine you like the number three, so you want to make a three letter alphabet (ternary, which has the optimal radix economy amongst integers) — the simplest way to do it is to connect the 3rd row of Pascal’s triangle (1, 2, 1) to the popcount of two bits. And lo, you have trits! Coincidentally, this maps 4D onto 3D in such a way that there are two different spins to the number one: 01 or 10. Hmmm…where have we seen “spin” before? It’s interesting to entertain the possibility that simple properties of the simplest possible strings of the simplest possible languages could explain all kinds of physical phenomena. And why wouldn’t they? Also, HAMTs are incredible, so if the only thing you could do with popcount was make HAMTs, it would already be incredible. Such a seemingly silly and mundane thing, how many hands are raised, how many coins flipped heads, how many voted for this or that candidate, I guess it’s just a tally function, but damn if I didn’t find myself astounded by the usefulness of popcount earlier this year. Not surprising to see it on hacker news. Great article, and timely. Now, will someone please tell me how popcount is involved in neural coding? Maybe it’s literally everywhere, even inside our own minds. Neurons vote for the presence of features, tally the votes, you can simplify your data substantially by not giving a bleep about whose hands are raised and instead only care about how many of them you can see. That’s also why the second law of thermodynamics is, as Arieh titled his book, plain common sense. The more coins you flip, the more ways you can flip the same popcount total (a dim event), even though there might be many different combinations of heads and tails in various possible coin flip sequences (specific event). So, yeah. Cheers to this one!