## December 26, 2022

### Guillotine Partitions and the Hipparchus Operad

#### Posted by John Baez

If you dissect a square into $n$ similar rectangles, what proportions can these rectangles have? Folks on Mathstodon figured this out for $n \le 7$, 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 (3)

## 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 (24)

## 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 $R$ has a commutative associative addition, an associative multiplication that distributes over addition, an element $0$ with $r+0 = r$ and $0r = 0 = r0$ for all $r \in R$, and an element $1$ with $1r = r = r1$ for all $r \in R$.

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

Is the free idempotent rig on $2$ 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

#### 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.

## 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)