vihart

Metachirality?

Added 2018-04-09 16:15:20 +0000 UTC

Well good humans, Metachirality part 1 is live! Thank you for sticking around and helping fund its creation!

This video is based on a true story. More specifically, I was told many years ago that Metachirality was a strange property of some higher dimensional symmetry groups and that only John Conway understands it and that even he doesn't "apprehend"[1] it completely. Which of course made me go "pfff I could totally understand it the best ever, just you wait."

Years later I came across the word again while doing some research on 4d symmetry groups (and by this time The Symmetries of Things had come out, which is my #1 recommended symmetry text), and after wrangling through the definitions for a while doubting whether I REALLY understood it or whether the words/algebra actually meant something else, it turned out it was so simple that I doubted my own understanding, thinking it must be harder than that.[2] Though it still has some subtleties and is connected to a whole pile of interesting kinds of chirality, especially in higher dimensions.

The easiest and should-be-canonical example, though, exists in three dimensions! (Spoilers: screw symmetry, where the symmetry itself comes in both right and left handed versions.)

For whatever reason, line groups are missing from Symmetries of Things, beyond the 2D Frieze patterns. Personally I find these groups fascinating both because they are the simplest to have some interesting properties and because of their relationship to space-time, where chiral things come not in pairs but in quartets because there's two separate kinds of mirrors... as any student learns not in math class but in music class! (Another topic for a followup video.)

Anyway! I hope you like the video and are looking forward to part 2 :D

[1] John, in reference to his attempts to "comprehend" the metachirality of an object:

'I now think "apprehend" is a better word for it. What I mean, after all, is to "capture" the concept so that I can interrogate it at leisure whenever I want!'

[2] It helps that I could "apprehend" John about it, with examples in hand.