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**Physics, Astronomy & Cosmology / Re: What is the Hopf gauge?**

« **on:**14/03/2024 21:26:28 »

If you want to know what a monoid is, it's a simple (!) thing to understand.

I encountered them in language theory, where it's the concatenation of letters or characters from a set.

Then all the finite strings are in a set--the length 1, length 2, . . . strings form a set which is closed.

obviously length 1 strings are the alphabet itself.

When you walk around, your steps are strings in an alphabet of steps. Walking is the concatenation of steps, ordered by a relation between the two halves of your local symmetry. Monoids are everywhere.

And, your strings of steps are all accepted by a finite state machine, Logic is what you say it is, pretty much.

I need to correct something I said before about the twisted strip I call a graph with two loops. I realised that if I continue to cut along the strip, I'm only separating the middle from the loop which doesn't contract, it stays the same size except for getting thinner. That is actually a bit more interesting because now, I want to know what I did so the length of either loop is invariant, under this slicing part of it away. It's a partition that gives you a longer and longer flat but tapering ribbon. If I continue slicing it I'll approach a circle attached to a line, and the circle will have a fixed length.

And it's a graph with half-edges (if I say it is).

Moreover, it's a Mobius 'looped ribbon with a twist', and it's sliced or cut in a spiral. So it visits a set of points on a line across the strip in an ordered fashion. It does that with a straight cut too, but the order is different, An algebra of cutting through a set of points along a line emerges. Time for a bit of a rest though.

I encountered them in language theory, where it's the concatenation of letters or characters from a set.

Then all the finite strings are in a set--the length 1, length 2, . . . strings form a set which is closed.

obviously length 1 strings are the alphabet itself.

When you walk around, your steps are strings in an alphabet of steps. Walking is the concatenation of steps, ordered by a relation between the two halves of your local symmetry. Monoids are everywhere.

And, your strings of steps are all accepted by a finite state machine, Logic is what you say it is, pretty much.

I need to correct something I said before about the twisted strip I call a graph with two loops. I realised that if I continue to cut along the strip, I'm only separating the middle from the loop which doesn't contract, it stays the same size except for getting thinner. That is actually a bit more interesting because now, I want to know what I did so the length of either loop is invariant, under this slicing part of it away. It's a partition that gives you a longer and longer flat but tapering ribbon. If I continue slicing it I'll approach a circle attached to a line, and the circle will have a fixed length.

And it's a graph with half-edges (if I say it is).

Moreover, it's a Mobius 'looped ribbon with a twist', and it's sliced or cut in a spiral. So it visits a set of points on a line across the strip in an ordered fashion. It does that with a straight cut too, but the order is different, An algebra of cutting through a set of points along a line emerges. Time for a bit of a rest though.