3 ms·

The Arithmetic of Braids (2022)

34 points by marysminefnuf 11mo ago

MarkusQ 11mo ago
Nice presentation. I would probably have written it as an additive group (and then shown that the two strand braid was isomorphic with ℤ) as a further bidirectional intuition anchor (both "look, this new thing relates to something you are familiar with" and "this thing you are familiar with generalizes in unexpected ways"). Definitely liked the exploration of how theorems arise from speculation, and don't spring fully formed straight from Zeus's forehead.
- geordiew 11mo ago
  the problem with additive notation is that the braid group isn't commutative when number of strands is > 2.
  - MarkusQ 11mo ago
    Excellent point. I tend to enjoy "garden path" pedagogy, leading the students to surprising twists and having them explore -- but it's important to lead and not mislead. So yeah, multiplicative notation is probably better.
fsckboy 11mo ago
this is pretty interesting but i don't think i can justify dropping everything to study it. but it did remind me of an adjacent "field", that guy who explores all the ways to lace shoes (i think this is the one, but whereas old search engines would find it right away, now the results are full of "i made this" or me-too spam) https://www.fieggen.com/shoelace/lacingmethods.htm https://www.fieggen.com/shoelace/lacingmethods.htm
- marysminefnuf 11mo ago
  What i do is keep a list of 3 things every day i learn about braids and it just kinda adds up after a little bit.
- srean 11mo ago
  I was expecting Mathologer's book when you mentioned shoelaces. Thanks for the link https://www.ams.org/books/mawrld/024/mawrld024-endmatter.pdf https://www.ams.org/books/mawrld/024/mawrld024-endmatter.pdf > whereas old search engines would find it right away One of my pet peeves is how Google makes it hard to get at the url to share directly from the search results. Yes there's the share link but that does not link to the original.
srean 11mo ago
Very nicely explained. I find the mathematics of weaving, knitting, braiding very dopamine squirt inducing. It seems this interest is shared by many here on HN because there have been many threads on them. @dang had once compiled a list. Let me repost one of mine where Richeson explores Maypole dancing through the lens of Braids. https://news.ycombinator.com/item?id=44225324 https://news.ycombinator.com/item?id=44225324
max002 11mo ago
Im bad at mathematics, but this looks great. Gonna use it in my app to make 'braidy' connections :) thank you very much!

34 points by marysminefnuf 11mo ago

Nice presentation. I would probably have written it as an additive group (and then shown that the two strand braid was isomorphic with ℤ) as a further bidirectional intuition anchor (both "look, this new thing relates to something you are familiar with" and "this thing you are familiar with generalizes in unexpected ways"). Definitely liked the exploration of how theorems arise from speculation, and don't spring fully formed straight from Zeus's forehead.

geordiew 11mo ago
the problem with additive notation is that the braid group isn't commutative when number of strands is > 2.
- MarkusQ 11mo ago
  Excellent point. I tend to enjoy "garden path" pedagogy, leading the students to surprising twists and having them explore -- but it's important to lead and not mislead. So yeah, multiplicative notation is probably better.

fsckboy 11mo ago

this is pretty interesting but i don't think i can justify dropping everything to study it. but it did remind me of an adjacent "field", that guy who explores all the ways to lace shoes (i think this is the one, but whereas old search engines would find it right away, now the results are full of "i made this" or me-too spam) https://www.fieggen.com/shoelace/lacingmethods.htm https://www.fieggen.com/shoelace/lacingmethods.htm

marysminefnuf 11mo ago
What i do is keep a list of 3 things every day i learn about braids and it just kinda adds up after a little bit.
srean 11mo ago
I was expecting Mathologer's book when you mentioned shoelaces. Thanks for the link https://www.ams.org/books/mawrld/024/mawrld024-endmatter.pdf https://www.ams.org/books/mawrld/024/mawrld024-endmatter.pdf > whereas old search engines would find it right away One of my pet peeves is how Google makes it hard to get at the url to share directly from the search results. Yes there's the share link but that does not link to the original.

srean 11mo ago

Very nicely explained. I find the mathematics of weaving, knitting, braiding very dopamine squirt inducing. It seems this interest is shared by many here on HN because there have been many threads on them. @dang had once compiled a list. Let me repost one of mine where Richeson explores Maypole dancing through the lens of Braids. https://news.ycombinator.com/item?id=44225324 https://news.ycombinator.com/item?id=44225324

max002 11mo ago

Im bad at mathematics, but this looks great. Gonna use it in my app to make 'braidy' connections :) thank you very much!