
Show HN: Interactive Wallpaper animated art tool (symmetry groups) - alexberke
http://coloring-book.co/wallpaper
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alevskaya
Kudos! This is a beautiful intro to the planar symmetries. If anyone here
wants to play around with doodling their own planar symmetry patterns, I made
a weird little tool to do that at
[https://eschersket.ch/](https://eschersket.ch/)

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westoncb
I've been wondering for a long time how it is that symmetry groups are used in
science, and I was thinking about it some more while looking at this.

Is the idea that when analyzing some natural phenomenon there's a bunch of
complexity, but in certain cases (where we can use symmetry groups) the
complexity follows a pattern. If we recognize the pattern, then in order to
get at the heart of the matter we only need to look at _one element_ of the
pattern for the next stages of our analysis (rather than considering all of
its repetitions).

And then maybe the next thing is that often times the patterns occur in
abstract mathematical spaces rather than being literal physical patterns.

Can anyone comment on the accuracy of that interpretation?

~~~
foxes
So in science, symmetry groups appear in physics in various forms. The ones
more commonly found in physics are not exactly like the finite symmetry groups
in the OP (I cant immediately think of a good example). Briefly, they are used
to describe transformations of your physical system that leaves "something"
(this can be abstract) unchanged.

So in physics, the best examples take the form of a continuous symmetry. A
very simple example is the special orthogonal group SO(3) - the group that
describes all length preserving rotations.

Another example in relativity are Lorentz and Poincaré groups. The Poincaré
group effectively describes affine transformations, a transformation of space
that preserves parallel lines and ratios of distances.

More abstract examples occur in field theory. The unitary group of one
dimension U(1), the special unitary groups SU(2) and SU(3) can be used to
"describe" electrodynamics, the weak and strong force respectively.

In these cases the solutions to the equations that describe how particles
interact under these forces have these symmetries. These theories all have
some conserved quantity that is unchanged by the action of these groups
(charge, color etc).

Perhaps the one of the important thing to take away is; if the equations that
describes your system has some symmetry, then there is a conserved quantity
(something that doesnt change in time). This is Noether's Theorem.

Another take away is, if your equations have some symmetry, you can use it to
describe all the possible solutions. Once you have one solution you can find
others by applying the transformation. So perhaps these are a bit more
abstract to think about initially, but there is something physical there.

~~~
westoncb
> Once you have one solution you can find others by applying the
> transformation.

Interesting. That sounds like roughly the inverse of what I was trying to
describe: using the transformations to generate rather than analyze.

That's pretty cool though. If I understand correctly, the process would be
something like: prove that an equation has some symmetries[0]; then find one
solution; then, since we know the equation's symmetries, we can use them to
gain access to a formal expression of all the solutions, which could then be
used (for example) to assert that such and such must be in the solution set or
not in the solution set, etc. (actually, bad example there since our original
equation already allows us to do that...)

If that's right, then there's probably some way of making reference to the
symmetry groups algebraically—maybe that's part of what the U(1) etc.
notations are for? And then some way of getting a symbol meaning essentially,
"group X applied to entity Y", where entity Y is something like a solution to
an equation with the appropriate symmetries, and group X is a symbol for the
symmetry group like U(1)? Or maybe "entity Y" is the symmetric equation rather
than a solution.

My initial thinking was more like that you'd start with something symmetric
like the full set of solutions, then use knowledge of the symmetry to
eliminate repetitions, so that you're only left with what's unique and can
analyze that in isolation. But I guess that would make more sense to apply to
observational data rather than to a set of solutions.

Using the electrodynamics example: maybe you'd start with a bunch of recorded
data related to electric and magnetic phenomena interacting, which in some
sense is like a sampling of the solutions to (ideally not yet discovered)
equations describing electrodynamics. So if you could determine that that data
has some symmetries, you can use it to eliminate everything that's repeated,
and what's left (a single solution I suppose...) might given you some insight
about an equation which would have it as a solution.

[0] When it's said that "an equation has a symmetry" is it just shorthand for
"the solutions to an equation have a symmetry"?

~~~
foxes
[0] Yes a solution will transform under the symmetry to be another solution of
the same equations. Perhaps for a bit of history Galois originally used finite
groups to look for solutions to algebraic equations. You might find one
solution and then be able to describe all the other solutions. A bit later on
Sophus Lie (where the name Lie groups comes from) went on to study the same
problem for differential equations. Maybe to say it briefly, if you understand
that your solutions have a symmetry you can use it to simplify your original
equation and make things easier to solve.

In the field theory example, you have something called a Lagrangian which you
can use to derive the equations which describe the dynamics of your system. In
those cases it will have a symmetry of one of those groups, you can transform
the Lagrangian by one of those groups.

Knowledge of the symmetry does help you simplify your problem - its basically
used to remove redundancy.

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tenryuu
Can you fix the infuriating history state that just disallows me from using
the back button on my browser? It's way too fast that I can't click it a
second time to navigate back to HN

~~~
alexberke
Yes -- Fixed! Thanks for pointing that out!

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chmullig
It's strangely beautiful, especially for something educational. A slow moving
version of the animations as a wallpaper would be cool.

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o0void0o
doesn't seem to work. Just a white screen.

~~~
gus_massa
I tried it IE11 and it doesn't work. But it work in Chrome and Firefox. Do you
have javascript enabled?

