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December 26, 2022

Guillotine Partitions and the Hipparchus Operad

Posted by John Baez

If you dissect a square into nn similar rectangles, what proportions can these rectangles have? Folks on Mathstodon figured this out for n7n \le 7, and I blogged about it here recently. But I was left feeling that some deeper structure governed this problem.

Various people on Mathstodon, including Steven Stanicki, David Eppstein and Rahul Narain, convinced me of the importance of a certain class of dissections called ‘guillotine partitions’. I started suspecting that these were connected to an operad I once blogged about here: the ‘Hipparchus operad’. And last night I put some of the pieces together… though there is still more to do.

Posted at 9:42 PM UTC | Permalink | Followups (2)

December 22, 2022

Dividing a Square into Similar Rectangles

Posted by John Baez

If you divide a square into some fixed number of similar rectangles, what proportions can these rectangles have? We’ve been having fun thinking about this on Mathstodon, and here is a progress report.

Posted at 6:15 PM UTC | Permalink | Followups (21)

December 21, 2022

Free Idempotent Rigs and Monoids

Posted by John Baez

I’ve been having a lot of fun on Mathstodon lately, and here’s an example.

A rig RR has a commutative associative addition, an associative multiplication that distributes over addition, an element 00 with r+0=rr+0 = r and 0r=0=r00r = 0 = r0 for all rRr \in R, and an element 11 with 1r=r=r11r = r = r1 for all rRr \in R.

A rig is idempotent if rr=rr r = r for all rRr \in R.

Is the free idempotent rig on 22 generators finite? If so, how many elements does it have?

Morgan Rogers raised this issue on the Category Theory Community server, and after a bit of progress I posed this as a puzzle on Mathstodon. By now three people there have independently figured out the answer.

Posted at 10:38 AM UTC | Permalink | Followups (6)

December 18, 2022

Adjoint School 2023

Posted by John Baez

Are you interested in applying category-theoretic methods to problems outside of pure mathematics? Apply to the Adjoint School!

Apply here. And do it soon.

  • January 9, 2023. Application Due.

  • February - July, 2023. Learning Seminar.

  • July 24 - 28, 2023. In-person Research Week at University of Maryland, College Park, USA.

Posted at 4:00 PM UTC | Permalink | Post a Comment

December 3, 2022

Neutrino Dark Matter

Posted by John Baez

I talked to Neil Turok at a café today. He used to be the head of the Perimeter Institute, but now he’s at the University of Edinburgh.

He coauthored a paper arguing that dark matter is very heavy right-handed neutrinos:

It’s very natural to add right-handed neutrinos to the Standard Model, and if they’re heavy they can make the observed left-handed neutrinos light via the ‘see-saw mechanism’. The problem is to keep them from decaying too fast!

Posted at 10:27 AM UTC | Permalink | Followups (23)