I'm going to try making plaited bread. This will involve some group theory.
Follow along with me!
First, here's some dough that I've left to rise and just knocked back

I've divided it into 8 by cutting in half three times

Now I've rolled the bits into strands of equal length.
OK, similar length.

Next I have to connect them up like this, which reminds me of that puzzle about the hydra that I can never solve.

Your man Paul says: "Step 1: place 8 under 7 and over 1 Step 2: place 8 over 5 Step 3: place 2 under 3 and over 8 Step 4: place 1 over 4 Step 5: place 7 under 6 and over 1"
I need to label these tentacles

That looks like this. I think. I might not have done exactly the right thing, but I've definitely done some overs and unders.
To answer a comment: the numbers refer to positions, not tentacles, so you relabel after each step

So the immediate question is: where did each strand end up?
The steps are, in permutation notation (I say where each strand ends up) :
(2,3,4,5,6,7,8,1)
(1,2,3,4,6,7,8,5)
(1,8,2,3,4,5,6,7)
(2,3,4,1,5,6,7,8)
(2,3,4,5,6,7,1,8)

You can work out the final permutation by applying those in turn, to get:

(8,4,5,6,7,1,2,3)

So things don't end up where they started.

So... how many times do I have to do this to get them back where they started?

(brief intermission while some parenting happens)

One dance recital, one full potty and a cuddle later, I'm back!

You can write that permutation, (8,4, 5,6,7,1,2,3) as a cycle: now (a, b, c) means "a goes to b goes to c goes to a"

That is:
(1,8,3,5,7,2,4,6)

This cycle includes every strand, so I don't get to do a fun bit of maths. I was hoping it'd be several disjoint cycles - cycles that don't share any strands.

This cycle is 8 elements long, so if I apply it 8 times I get back where I started.

Fans, I forgot to keep count. Let's say I did it a multiple of 8 times so everything's back where it started.

(ignore the big mush of I-got-bored at the end of this plait)

If the permutation *had been* a product of disjoint cycles, I could use a cool fact to work out how many times I need to apply it: the least common multiple of the lengths of all the cycles.

That's what I get for trying to apply maths in the real world...

We'll see if the bread is as good as the maths!

Some books about maths and baking:

The Proof and the Pudding by Jim Henle
press.princeton.edu/books/hardcove…

And How to Bake Pi by Eugenia Cheng
profilebooks.com/how-to-bake-pi…

That's it! I'll try to add a photo when it's baked.