I made an infographic depicting every distinct possible symmetry of 2D periodic groups. There are 17 distinct symmetries, known as the Wallpaper groups. Here is a preview.
“Every 2D Periodic Symmetry”. You can find the full image on my flickr. Vector graphics available on request.
The king of all useless arguments
Arguments for the existence of God are, generally agreed, not very compelling, and irrelevant to any real political arguments associated religion. And the ontological argument is king of the junk heap, as it is especially unconvincing and irrelevant. The ontological argument attempts to prove God’s existence a priori, with all the certainty of a mathematical theorem. Most people would say this is overly ambitious, and doomed from the start.
The ontological argument presents a deep and enduring mystery: who actually says this stuff? Does anyone say this stuff? What’s their deal?
The number of people who advocate the ontological argument is nonzero, but it’s quite rare in practice. Let’s round down to zero and say nobody actually says this stuff. Theistic readers may proceed unthreatened, knowing that I’m talking about an argument that none of them actually advocate.
My husband described to me a version of the cosmological argument based on explanations. Everything in the world requires an explanation. Since the chain of explanations can’t go on forever, there must be an ultimate Explanation for it all. And that Explanation wants us to stop being gay. At least, that’s how I assume the rest of the argument goes. I don’t actually care.
We were joking about this, because the argument imagines that there is a cosmic pointer from each object to its explanation. But the very idea of an explanation is really quite squishy. What even is an explanation? It depends on social context.
So I dug up an old abandoned blog draft on the subject of explanations.
Back in the day, I used to post math puzzles, similar to how I post origami today. Today I’m revisiting a problem that I never fully solved.
Problem statement: There is a square patch of grass, with sides of unit length. Using the shortest length of fence, block all straight paths that cut across the grass.
I did not create this puzzle, but I am not sure of its original source. The riddle once appeared on the (now defunct) wu : riddles website, under the title “Samwise and Gandalf”. In this version of the puzzle, Lord Sauron has an underground telephone line that cuts a straight line somewhere through a square plot of land. Samwise doesn’t know where it is, so he intends to uncover it by digging up the perimeter. But Gandalf tells him that there is a more efficient solution. What’s the best Sam can do?
The impressive thing about this puzzle is that you can go through many iterations of finding better and better solutions. Afterwards, we’ll consider a hexagonal patch of grass, and other shapes.
Rock Paper Scissors is a game where two players simultaneously pick one of the three things in the title. Rock beats Scissors, Scissors beats Paper, Paper beats Rock, and if both players pick the same thing they tie. Rock Paper Scissors is important in game theory, because it is a toy model that helps understand a much broader class of games.
To understand the correct strategy in Rock Paper Scissors, we must understand the difference between pure strategies and mixed strategies. A pure strategy is deterministic, where a mixed strategy is random. There are only three possible pure strategies: pick Rock, pick Paper, and pick Scissors. There are infinitely many mixed strategies available, for example assigning 50% probability to Rock, 25% to Paper, and 25% to Scissors.
If you take a rating website, say IMDB, or Goodreads, and you sorted items purely by review scores, the stuff that would float to the top would be pretty obscure. That’s because the easiest way to maintain a perfect score is to have a very small sample size.
So, a math question: what is the statistically “correct” way to handle this?
In this analysis, I will assume there exists a “true” average review score, and we are trying to estimate it. The “true” average is the average that would be attained if there were a sufficiently large sample of reviewers. We’re not imagining that everyone in the world is reviewing the same book (for example, we don’t expect book reviews to reflect the opinions of people who don’t like reading books period). But we could imagine, what if there were a billion identical yet statistically independent Earths, and we averaged all their review scores. Obviously it’s very hard to come across a billion identical yet statistically independent Earths, and that’s why we use math instead.
This premise may be fairly questioned. I once discussed the philosophical problems with review scores, including questioning the very idea of taking averages. But here, I’m just focusing on the math for math’s sake. And, I really mean it, it’s hardcore math. If you don’t want math, just skip to the last section I guess.
Kittens Game is a clicker game that you can play in your browser. It makes a strong first impression, as it tempts you into choices that will kill off your kittens within twelve minutes. But I’m not here to review the game, I’m here to talk about spreadsheets!
Clicker games often support passive gameplay (e.g. leave it running overnight), active gameplay, or any combination of the above. On the very active end, you could try to optimize it, setting up spreadsheets to run calculations. So, I spent a thousand years tinkering with spreadsheets, and I liked it. There’s a story there, a mathematical story.