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| | |.---.-..----.| |--..-----..----. | | |.-----..--.--.--..-----.
| || _ || __|| < | -__|| _| | || -__|| | | ||__ --|
|___|___||___._||____||__|__||_____||__| |__|____||_____||________||_____|
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COMMENT PAGE FOR:
(HTM) Beautiful Abelian Sandpiles
lupire wrote 22 hours 43 min ago:
Wikipedia has a picture/animation of the Identity for rectangular and
square grids
(HTM) [1]: https://en.wikipedia.org/wiki/Abelian_sandpile_model
LegionMammal978 wrote 1 day ago:
It looks like the author has a pretty simple procedure for computing
the 'identity' sandpile (which they unfortunately don't describe at
all):
1. Fill a grid with all 6s, then topple it.
2. Subtract the result from a fresh grid with all 6s, then topple it.
So effectively it's computing 'all 6s' - 'all 6s' to get an additive
identity. But I'm not entirely sure how to show this always leads to a
'recurrent' sandpile.
EDIT: One possible route: The 'all 3s' sandpile is reachable from any
sandpile via a sequence of 'add 1' operations, including from its own
successors. Thus (a) it is a 'recurrent' sandpile, (b) adding any
sandpile to the 'all 3s' sandpile will create another 'recurrent'
sandpile, and (c) all 'recurrent' sandpiles must be reachable in this
way. Since by construction, our 'identity' sandpile has a value ⥠3
in each cell before toppling, it will be a 'recurrent' sandpile.
OgsyedIE wrote 1 day ago:
In the case of piling sand exactly in the centre, the intermediate
states between the initial state and reaching the final equilibrium
seem to get closer to having a circular boundary as the grid size
increases, instead of the diamond-shaped boundary you might expect for
a symmetrical object in a planar grid. Take a look at the largest
resettable grid doing this within a couple seconds of being reset.
pmcarlton wrote 1 day ago:
I found 'xsand.c' (X11) in 1995 by Michael Creutz, that simulated these
sandpiles; I had fun with the sand but also learned C from it.
mcphage wrote 1 day ago:
> The rules of abelian groups guarantee that these identity sandpiles
must exist, but they tell us nothing about how beautiful they are.
This has causality backwardsâbeing a group requires an identity
element. You can't show something is a group without knowing that the
identity element exists in the first place.
In fact, a good chunk of how this article talks about the math is
just... slightly off.
seanhunter wrote 1 day ago:
> ââ¦an abelian group is both associative and commutativeâ¦â
If something is not associative it is not a group. An abelian group is
a group which is commutative.
MarkusQ wrote 21 hours 35 min ago:
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.
seanhunter wrote 21 hours 21 min ago:
Iâm not disagreeing. Iâm 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 itâs 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)
> âA group G is called abelian if its operation is commutative ie
for all g, h in G, we have gh = hgâ.
MarkusQ wrote 16 hours 45 min ago:
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.
seanhunter wrote 54 min ago:
Well if I check such a book that takes a category-theoretic
approach to teaching abstract algebra (Aluffi âAlgebra
Chapter 0â), he says the following:
> â 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
inverseâ.
So according to Aluffi at least, the operation of a monoid is
also associative. As you can see he does in fact also remove
the associativity criterion from the description of a group by
defining it in terms of a monoid. So heâs consistent with me
at least.
FredrikMeyer wrote 1 day ago:
I implemented this in Rust some years back. It is connected to some
serious research mathematics (see f.ex [1] )
(HTM) [1]: https://www.ams.org/notices/201008/rtx100800976p.pdf
(HTM) [2]: https://github.com/FredrikMeyer/abeliansandpile
haritha-j wrote 1 day ago:
Very related (yet idiotically titled, as always) veritasium video
(HTM) [1]: https://youtu.be/HBluLfX2F_k?si=6lVPLvJNc2YH_4go
JimmyBuckets wrote 1 day ago:
It's like reverse clickbait with him
lupire wrote 22 hours 40 min ago:
[1] "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.
(HTM) [1]: https://m.youtube.com/watch?v=S2xHZPH5Sng
SiempreViernes wrote 1 day ago:
Yeah, I wish he'd do a second channel that is just reposts with
normal titles.
recursive wrote 1 day ago:
It seems the sand only spills up and to the left.
omoikane wrote 20 hours 41 min ago:
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 [1] in Chrome, I see a
"toppleAll" function near the top, but that same function is not
defined when the script is fetched with Firefox.
(HTM) [1]: https://eavan.blog/sandpile.js
ggm wrote 1 day ago:
Isn't this single frame state of a classic cellular automata? Note, not
"just" because I mean no disrespect. I don't understand how this
differs from Conway's life other than nuances of the live or die rule.
Sharlin wrote 1 day ago:
I don't believe that Game of Life is Abelian.
tripplyons wrote 1 day ago:
I don't think you could even define an associative binary operator
on states in the Game of Life because of its computational
irreducibility.
ggm wrote 13 hours 5 min ago:
CGOL is is turing complete. If you can make a NOR gate, you can
make anything.
gsf_emergency_6 wrote 1 day ago:
CGL doesn't have the scale invariance ("fractality") of ASM. ASM
criticality is stable and persistent. "fractal life on edge"?
what that looks like
(HTM) [1]: https://youtu.be/rKD51IUNK3A?t=40s
ggm wrote 1 day ago:
So that gets to how it differs, but it doesn't say its not a
cellular automata. It could say "it's a cellular automata with
different rules"
gsf_emergency_6 wrote 1 day ago:
It is a cellular automata distinguished by commutativity. You
used CGL as the basis for comparison, that's highly nonAbelian.
According to Wolfram (& I agree :), everything is a cellular
automaton, so comparing to CGL made more sense to me.
skeltoac wrote 1 day ago:
Now I want to redo a bathroom. Good job, writer!
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