> “…an abelian group is both associative and commutative…” If something is not associative it is not a group. An abelian group is a group which is commutative.
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Beautiful Abelian Sandpiles
21–30 of 31 posts
Re: Beautiful Abelian Sandpiles
#22Wikipedia has a picture/animation of the Identity for rectangular and square grids
Re: Beautiful Abelian Sandpiles
#23Very related (yet idiotically titled, as always) veritasium video https://youtu.be/HBluLfX2F_k?si=6lVPLvJNc2YH_4go
It's like reverse clickbait with him
https://m.youtube.com/watch?v=S2xHZPH5Sng
"Clickbait is Unreasonably Effective", 2021 - Veritasium's apologia for clicbait titles and and thumbnails, and statement of principles.
Veritasiuk has at least stuck making soldi educational videos, as Mark Rober has let slip away his past effort to educate in addition to demonstrate his cool toys.
Re: Beautiful Abelian Sandpiles
#24> “…an abelian group is both associative and commutative…” If something is not associative it is not a group. An abelian group is a group which is commutative.
So...an abelian group is both associative (because it's a group) and commutative (because it's abelian), which is exactly what the OP said? It sounds like you're disagreeing about something, but I'm not clear what your objection is.
Re: Beautiful Abelian Sandpiles
#25> “…an abelian group is both associative and commutative…” If something is not associative it is not a group. An abelian group is a group which is commutative.
So...an abelian group is both associative (because it's a group) and commutative (because it's abelian), which is exactly what the OP said? It sounds like you're disagreeing about something, but I'm not clear what your objection is.
I’m not disagreeing. I’m pointing out that in TFA it sounds as associativity is a property of abelian groups specifically whereas it as a property of all groups in general. In that sense it’s not wrong, just the emphasis is a bit misleading.
If you look in an abstract algebra textbook they all basically say the same definition for abelian groups (eg in Hien)
> “A group G is called abelian if its operation is commutative ie for all g, h in G, we have gh = hg”.
Re: Beautiful Abelian Sandpiles
#26It seems the sand only spills up and to the left.
It seems like it spills to 4 directions on Chrome, but only up and left on Firefox.
The really weird part is that when I fetch https://eavan.blog/sandpile.js in Chrome, I see a "toppleAll" function near the top, but that same function is not defined when the script is fetched with Firefox.
Re: Beautiful Abelian Sandpiles
#27Earlier quoted context omitted.
So...an abelian group is both associative (because it's a group) and commutative (because it's abelian), which is exactly what the OP said? It sounds like you're disagreeing about something, but I'm not clear what your objection is.
I’m not disagreeing. I’m pointing out that in TFA it sounds as associativity is a property of abelian groups specifically whereas it as a property of all groups in general. In that sense it’s not wrong, just the emphasis is a bit misleading. If you look in an abstract algebra textbook they all basically say the same definition for abelian groups (eg in Hien) > “A group G is called abelian if its operation is commutat…
In an abstract algebra textbook, they define groups first and then abelian as a property that some groups have. Here, the author is defining abelian groups "from scratch" and doesn't have an earlier definition of groups to lean on.
In more advanced texts, they could simply say that a group is a moniod with inverses and could (by your reasoning, should) avoid specifying that groups are associative since this is a property of all monoids.
Re: Beautiful Abelian Sandpiles
#28Earlier quoted context omitted.
I don't believe that Game of Life is Abelian.
I don't think you could even define an associative binary operator on states in the Game of Life because of its computational irreducibility.
CGOL is is turing complete. If you can make a NOR gate, you can make anything.
Re: Beautiful Abelian Sandpiles
#29Earlier quoted context omitted.
I’m not disagreeing. I’m pointing out that in TFA it sounds as associativity is a property of abelian groups specifically whereas it as a property of all groups in general. In that sense it’s not wrong, just the emphasis is a bit misleading. If you look in an abstract algebra textbook they all basically say the same definition for abelian groups (eg in Hien) > “A group G is called abelian if its operation is commutat…
In an abstract algebra textbook, they define groups first and then abelian as a property that some groups have. Here, the author is defining abelian groups "from scratch" and doesn't have an earlier definition of groups to lean on. In more advanced texts, they could simply say that a group is a moniod with inverses and could (by your reasoning, should) avoid specifying that groups are associative since this is a prop…
Well if I check such a book that takes a category-theoretic approach to teaching abstract algebra (Aluffi “Algebra Chapter 0”), he says the following:
> “ A semigroup is a set endowed with an associative operation; a monoid is a semigroup with an identity element. Thus a group is a monoid in which every element has an inverse”.
Re: Beautiful Abelian Sandpiles
#30Earlier quoted context omitted.
In an abstract algebra textbook, they define groups first and then abelian as a property that some groups have. Here, the author is defining abelian groups "from scratch" and doesn't have an earlier definition of groups to lean on. In more advanced texts, they could simply say that a group is a moniod with inverses and could (by your reasoning, should) avoid specifying that groups are associative since this is a prop…
Well if I check such a book that takes a category-theoretic approach to teaching abstract algebra (Aluffi “Algebra Chapter 0”), he says the following: > “ A semigroup is a set endowed with an associative operation; a monoid is a semigroup with an identity element. Thus a group is a monoid in which every element has an inverse”. So according to Aluffi at least, the operation of a monoid is also associative. As you can…
Right. And so is the article. When you are introducing an object you need to specify its properties, _including_those_it_inherits from objects you haven't defined.
If I haven't defined mammals, I say that bats are warm blooded animals that produce milk for their young, etc., but if I have (or expect my readers to know what a mammal is) I can just say they are mammals.