Thanks so much for this video! I was doing a math competition problem that needed to use Burnside's Lemma, so I came here. The explanations were intuitive and made sense (unlike the other videos I've seen lol), and the animations were really engaging as well!
I really liked this video, audio was fine I somehow can't wrap my head around objects * symmetries = sum of symmetrical pictures, because it looks like a rows * columns = interior, but the interior is sparse so that shouldn't work in my mind. I must be missing something
At 9:51, every row contains all the pictures symmetrical by a certain symmetry. So the total number of pictures there is the "sum of symmetrical pictures". On the other hand, we obtained all the pictures there by multiplying "objects * symmetries". I tried to explain it a bit in more detail in the next version of the video: www.olsak.net/mirek/manim/burnside_en.mp4 (8:42)
How would it be to calculate the combinations being 9 different elements (or 9 different colors)? For each square I must put exactly one color and there can be no repetition of colors.
Then there are 9! pictures in total, and no picture symmetrical by a non-trivial symmetry. So the number of object is 9!/8 = 45360. In general, the numbers of symmetrical pictures can get a bit tricky when dealing with restrictions like that (e.g. exactly 4 blue squares) but not in this particular case as there are no symmetrical pictures.
Does this change for the case when the matrix dimensions are even (like 2x2)? What about when it is not a square matrix (2x3)? For a 2x3 matrix with 4 possible values, you get an answer of 430?
The lemma holds in general, although the set of symmetries and particular computation of symmetrical pictures may differ case by case. In particular, in the case of a non-square table, there are no 90 degree rotations, or diagonal flips. I don't understand what the number 430 is supposed to represent, or how you have calculated it.
@@procdalsinazev 430 refers to how many orbits are possible I believe. In the words of the problem I am trying to solve, the number of non-equivalent configurations of this 2x3 matrix. Anyway, thanks, that answers my question mostly and that was a very quick response you earned a subscription :).
@@procdalsinazev I'm still working through the problem so I'm not sure why that is the answer and I haven't touched upon group theory much and struggled with linear algebra. It is a 2x3 matrix with a total of 4 possible colors for any index in the matrix. So you have 4^6 possibilities with a 0-degree rotation. I agree that it doesn't make sense, I'm not sure how 430 is the answer since even with (4^6) / 8 you get 512. The only way I was able to get 430 as a final answer was by subtracting some of the flips and rotations away from the initial 512 but it doesn't make much sense to me as to the reasoning behind it or if I was just lucky. I am 99% sure the answer (430) is correct though since this is from a Google Foobar coding challenge. In the other example, a 2x2 matrix with 2 colors gives the answer of 7 distinct non-equivalent configurations. You are given a width of 2, height of 3, and 4 possible states per position in the matrix. So a few examples of some configurations of this particular 2x3 matrix would look like this 0 0 0 0 0 0 or 1 0 0 0 0 0 or 2 2 0 0 0 0 or 1 2 3 0 0 0 and any other possible combinations of these numbers, using those dimensions. If you find the problem to be interesting I can give you a StackOverflow link to look at it but I am by no means asking for help for any of this, so I appreciate the response.
I will now apply this idea to my programming problem. I have a matrix 12 by 12. Each cell can be in 20 different states. The symmetry is swapping any rows and columns. So I need to find the number of objects. Hope it works.
Sure, just keep in mind that the set of all the symmetries we are considering must be closed under composition. So in general, a symmetry is any permutation of rows combined with any permutation of columns. A similar programming problem was already discussed here, look into the comment section.
(1) Unfortunately, this is in contrary to almost any other feedback I receive -- most people appreciate graphics. (2) I am presenting a proof of a theorem, so it takes a bit of effort. Do you understand the concept of a proof? I get that compared to a kitten video, understanding a proof-video is pretty hard, I still think that I present the proof in a more understandable way than it is presented for example on Wikipedia here: en.wikipedia.org/wiki/Burnside%27s_lemma#Proof But maybe, you prefer formulae over pictures, and the Wikipedia proof is then more suitable for you...
Although, thanks for the feedback. I can understand that this topic is not the easiest. I appreciate that you could at least grasp the main idea of the calculation (although not completely). Maybe, you would appreciate explaining a bit more about how to use the lemma?
![Forced & Damped Oscillations Visuals [ Physics in minutes ][ JEE advanced ]](http://i.ytimg.com/vi/5F4bomQQlfY/mqdefault.jpg)
One of the most intutive explanation on burnside theorem!
I agree!
The animations are simply amazing. A true masterpiece
Thanks so much for this video! I was doing a math competition problem that needed to use Burnside's Lemma, so I came here. The explanations were intuitive and made sense (unlike the other videos I've seen lol), and the animations were really engaging as well!
Thanks so much!! you've saved me, I've been lost around 4 hours in wikipedia articles to understand this, thank you for this intuitive explination
Best video I've seen on Group Theory! Well done explaining it simply, instead of sticking to textbook-jargon.
You saved my Modern Algebra course, thank you!
Amazing explanation. Probably my favorite video on this topic.
Thanks for the compliment :-).
Yay!! From 3B1B.. you got a shout out.. awesome
I really liked this video, audio was fine
I somehow can't wrap my head around objects * symmetries = sum of symmetrical pictures, because it looks like a rows * columns = interior, but the interior is sparse so that shouldn't work in my mind. I must be missing something
At 9:51, every row contains all the pictures symmetrical by a certain symmetry. So the total number of pictures there is the "sum of symmetrical pictures". On the other hand, we obtained all the pictures there by multiplying "objects * symmetries".
I tried to explain it a bit in more detail in the next version of the video: www.olsak.net/mirek/manim/burnside_en.mp4 (8:42)
thank you kind sir this was beautiful i understood it so easily it's kinda suspicious
I hope it is indeed just a suspicion :-)
How would it be to calculate the combinations being 9 different elements (or 9 different colors)? For each square I must put exactly one color and there can be no repetition of colors.
Then there are 9! pictures in total, and no picture symmetrical by a non-trivial symmetry. So the number of object is 9!/8 = 45360.
In general, the numbers of symmetrical pictures can get a bit tricky when dealing with restrictions like that (e.g. exactly 4 blue squares) but not in this particular case as there are no symmetrical pictures.
Does this change for the case when the matrix dimensions are even (like 2x2)? What about when it is not a square matrix (2x3)?
For a 2x3 matrix with 4 possible values, you get an answer of 430?
The lemma holds in general, although the set of symmetries and particular computation of symmetrical pictures may differ case by case. In particular, in the case of a non-square table, there are no 90 degree rotations, or diagonal flips.
I don't understand what the number 430 is supposed to represent, or how you have calculated it.
@@procdalsinazev 430 refers to how many orbits are possible I believe. In the words of the problem I am trying to solve, the number of non-equivalent configurations of this 2x3 matrix. Anyway, thanks, that answers my question mostly and that was a very quick response you earned a subscription :).
@@retagainez With how many colors? However, to me, 430 doesn't make sense to me with any number of colors. What numbers did you take average of?
@@procdalsinazev I'm still working through the problem so I'm not sure why that is the answer and I haven't touched upon group theory much and struggled with linear algebra. It is a 2x3 matrix with a total of 4 possible colors for any index in the matrix. So you have 4^6 possibilities with a 0-degree rotation. I agree that it doesn't make sense, I'm not sure how 430 is the answer since even with (4^6) / 8 you get 512. The only way I was able to get 430 as a final answer was by subtracting some of the flips and rotations away from the initial 512 but it doesn't make much sense to me as to the reasoning behind it or if I was just lucky. I am 99% sure the answer (430) is correct though since this is from a Google Foobar coding challenge. In the other example, a 2x2 matrix with 2 colors gives the answer of 7 distinct non-equivalent configurations.
You are given a width of 2, height of 3, and 4 possible states per position in the matrix.
So a few examples of some configurations of this particular 2x3 matrix would look like this
0 0
0 0
0 0 or
1 0
0 0
0 0 or
2 2
0 0
0 0 or
1 2
3 0
0 0 and any other possible combinations of these numbers, using those dimensions.
If you find the problem to be interesting I can give you a StackOverflow link to look at it but I am by no means asking for help for any of this, so I appreciate the response.
@@retagainez Give me the precise (quoted, possibly a link) formulation of the problem with answer 430.
Thank you so much
Your animation is similar to 3 blue 1 brown anyway great video best on this topic
Can you provide the code for this video?
Have you checked the video description?
@@procdalsinazev How on the earth you know what to write in code?
@@sniper6233 ¯\_(ツ)_/¯
Really good video!
Thanks great video
I will now apply this idea to my programming problem. I have a matrix 12 by 12. Each cell can be in 20 different states. The symmetry is swapping any rows and columns. So I need to find the number of objects. Hope it works.
Sure, just keep in mind that the set of all the symmetries we are considering must be closed under composition. So in general, a symmetry is any permutation of rows combined with any permutation of columns. A similar programming problem was already discussed here, look into the comment section.
Thank you 😊
Very good!
please, if you don't have a good mic, add subtitles, not everyone has good-hearing and is perfectly fluent in english to understand bad audio :/
added
@@procdalsinazev thank you ^^
great video !
Such a good video! Thank you!
Excellent!
3B1B style!!! Yay...
1) Reduction of Graphics would be great
2) From around 8:57 The video went to intolerable state where everything was just made complicated
(1) Unfortunately, this is in contrary to almost any other feedback I receive -- most people appreciate graphics.
(2) I am presenting a proof of a theorem, so it takes a bit of effort. Do you understand the concept of a proof? I get that compared to a kitten video, understanding a proof-video is pretty hard, I still think that I present the proof in a more understandable way than it is presented for example on Wikipedia here: en.wikipedia.org/wiki/Burnside%27s_lemma#Proof But maybe, you prefer formulae over pictures, and the Wikipedia proof is then more suitable for you...
Although, thanks for the feedback. I can understand that this topic is not the easiest. I appreciate that you could at least grasp the main idea of the calculation (although not completely). Maybe, you would appreciate explaining a bit more about how to use the lemma?
I think it's hard to understand