Chapter 1: Symmetries, Groups and Actions | Essence of Group Theory
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- Опубліковано 5 лют 2020
- Start of a video series on intuitions of group theory. Groups are often introduced as a kind of abstract algebraic object right from the start, which is not good for developing intuitions for first-time learners. This video series hopes to help you develop intuitions, which are useful in understanding group theory.
In particular, this video is going to be about thinking groups as symmetries (or isometries to be precise) because they are much more visualisable, and that symmetries of an object do form a group using the abstract definition of the group that is usually given.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
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If you want to know more interesting Mathematics, stay tuned for the next video!
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If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #grouptheory #groups #intuition
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Sure sir
found your channel through the dream controversy and gotta say your videos are very pogchamp
3b1b is now a genre.
Haha :)
@@mathemaniac also I filled out your survey! Your channel is actually amazing and underrated, hopefully the yt algorithm will make another vid famous like the dream analysis one :)
Was hungry for more group theory stuff after 3b1b did his monster group video, and found this. Thanks!
(Though I understand we probably aren't going to get to things like the sporadic groups for a intuition level overview)
Hope it helps your understanding of group theory!
exactly the same
Amazing series, shared with everyone that wants to know more about abstract algebra outside a uni course. Thank you for you work !
great explanation, I like the 3blue1brown style
That's because he uses the program made by 3blue1brown and tbh it's hard not to
@@telnobynoyator_6183 that's not true
he explicitly states (in the description) that although his style does resemble that of Grant's he doesn't use his animation engine manim
@@machineman8920 Didn't see it ! Thanks for correcting me
Beautifully clearly visualized video. Thank you so much for the hard great work!!
It really helps me to have a little sense of group theory instead of just memorizing a lot math symbols and proofs! Thank you!
Probably the best description I've seen of Group actions. Thank you!
Thanks so much for the appreciation!
I really enjoyed this. Thank you!
I usually try to refrain from looking at a subject in a known way if the teacher uses a different one - especially if they explicitly say that they do so.
I think it often holds back from a better understanding because one tries to fit known (but not fully understood) with totally new knowledge and is even more confused in the end.
But I really like seeing the group axioms represented as properties of symmetry! It makes the symmetry explanation of groups feel very intuitive. (Having understood the definition of a group but not much more.)
This is a great series!
How excellent and understanding video it is!! Thanks a lot, Sir!!
Mathemaniac "I'm not going to use the group axioms"
also Mathemanic: describes the group axioms in terms of symetries.
Clever
please make a video series on functional analysis and real analysis too, your series is awesome.
Superbe vidéo. Les explications sont très claires.
Dode, thanks for visualizing it. That's really helpful, I wasn't imagining the symmetry of a group , I hoped u added the symmetry of more things like galwa group just in the same minor, and again thank u for the video ❤
Thank You So Much💐
Thank you so much!!!
you are better than most of the maths and science channels I am subbed to, definitely deserve more recognition. happy to be here before you blow up w subs ;)
Thanks so much for your appreciation!
How come I have never seen this series before? I was looking for an explanation of polya's theorem and I found this. Looking forward to it
Thank you !
thank you very much
starting from today, next is chapter 2, i hope it gave me motivation to learn since i need math elegance and application for my motivation rose up.
BRILLIANT explanation- THANK YOU
Thanks for the appreciation!
Thanks a lot
You are deserving of way more subscribers sir!
Thank you so much!
I think I do have an idea which UA-camr served as an inspiration :D. By the way, true story, I was like the 10,000th subscriber of his :).
AWESOME WORK SERIES!!!
Glad that you enjoyed it!
This was very helpful. Thank you!
Glad that you enjoyed it!
This video is amazing, thanks for making it
Glad you liked it!
Symmetries, groups and actions? More like “Sounds like your channel is the main math attraction!” Thanks for putting together so many high-quality videos.
i like how you use the 3 blue 1 brown design
it's a great explanation
Thanks so much! Glad that you enjoyed the video!
This channel is gold.
Aww thanks so much for the appreciation!
Great video explaining group theory
Thanks so much for the appreciation!
Outstanding video lecture.
Many thanks!
Excelent presentation on group actions
Thanks!
10/10...👍🤗
Just watched this first video but u definitely deserve more subs
Thanks so much for the appreciation and subscription!
The fundamental level of reality is in the language of group theory
Nice video series and nice animations. Just as in description of this video, it's actually a higher level of understanding or to understand the topic from a view that is different from traditional textbooks, those videos are not for any beginner, it needs considerable mathematical maturity to fully understand those videos.
There's no way that this fantastic group theory vid only got 30k views:)
Aww thanks for the compliment!
Hope I see you with 1 mil subscribers
That's a compliment too high for me to take, but thanks for enjoying the videos!
@@mathemaniac ya....you deserve more subscribers....keep up the great content....
Thanks a lot ! Please could you help with one thing which makes me struggle : If I understand clearly a symmetry is like a function whose domain is the set of vertices, or the set of edges, or the set of faces, and the codomain is the set of vertices, or the set of edges, or the set of faces (respectively). This function needs to follow two rules :
· distances between elements needs to be preserved
· I think that the fact that the whole object is preserved follows the fact that distances between elements needs to be preserved
So here is the question :
If the set of symmetries form a group over the set G of symmetries : do we consider the whole functions that symmetries are when we compose them or do we consider the output of those functions when composing symmetries? What does it means when we compose symmetries : for example if a and b are symmetries a·b = c which is also a symmetry, but symmetries act on elements of sets so does a·b ∀x,y,z∈X , b*x=y, a*y=z but z here is ∈X not in G, so clearly "·" is not equivalent to "*"....so a·b is not equivalent to a*(b*x)....
NIce new emerging channel
Welcome aboard!
0:06 just having donne the vector spaces last term, what does the arrows here represent??
Damm your channel name is original and your content too!!
Not sure if you are being sarcastic... if not then sorry that I am mistaken and thanks for your compliment!
After 17 Sec. .....i love it
Can you please do a video on the under-appreciated work on symmetries by Emmy Noether? I know the theorem, but how was it proven?
It's not easy to get there from what the channel is usually about - but could try. This video idea is up on the list!
please reply,which tool are u using for animations?
I don’t think this video is written well enough to help beginners with group theory and symmetry like myself. There’s is so many point where your assertions would cause a cascade of confusion for an absolute beginner, but it is well made for those who already are familiar with some of the moving parts
I just love your videos. Could you permit me to take some peice of informations from your content to put in mine content. I mean to say that i would not do copy paste of your video and audio, i will only take up the info. I am your hearted subscriber and i love your videos. Keep up the great content.
It's okay to use my content *if you credit me clearly*.
@@mathemaniac sure ! And thanks
You can check my videos to check whether i give credit to you or not. I am your biggest fan and student.
Good video. Thanks. Suggestion: When you say "distances" you should specify distances between some elements. E.g., symmetries preserve distances between points, between elements, between figures, between points and line segments, etc. Transformations that preserve distances between .... what?
What I mean is all distances, so it means everything you just said.
@@mathemaniac I think your example is a little obscure, because the objects acted on are two marked triangles (not just triangles) and as such they are different objects (although they have parts that are the same, i.e., the triangle parts). You are right, of course, that the markings break the translational symmetry of the triangles, but the markings make them completely different objects to begin with.
If you have only one object to start with, then won’t an object preserving transformation also preserve distances? In the example at 1:20, two objects are exchanged so distance is not preserved.
Sigma Ball
-Sigma Grindset
Excellent video! I just find that the music is a little bit distracting.
Thanks for the comment! The background music has hopefully been better in recent videos, and I will keep this in mind for the future ones.
at 0:45, isn't the group associated with symmetries of a square called D_4?
The convention is probably different across the world. When I learnt it, the subscript denote the order of the group, not the number of vertices. But probably your convention is D_4. There is nothing wrong about these conventions, but just that I like to stick with the convention I learnt.
😍😍😍 3blue 1brown style tnx
Love from india
Thanks so much!
@@mathemaniac hey will you make videos on some topology
Is there a name if the object is preserved but not the lengths? Like an eigenvalue eq Av=lambda v with lambda =/=1 ?
It is simply a permutation of the points within an object, if that's what you are looking for. I have talked about permutations in Chapter 7 of the video series.
Permutation involves much more than linear transformations though, like they can be not continuous.
Isn't A4 the associated group of symmetries of a tetrahedron? Please lmk if I am missing something. Awesome video/channel thanks!!!
S4 is the associated group; A4 is simply the rotational symmetries, and if we count reflections as well, that would be S4.
@@mathemaniac So for the symmetries that are not A4 you swap vertices that are opposite a plane bisecting the tetrahedron but you don't get these symmetries from moving the entire tetrahedron in space?
@@samwisegamski No you can't get those symmetries by simple rotation: a reflection flips orientation in space, and a rotation preserves orientations. If you would prefer the matrix lingo, a reflection has determinant -1, and a rotation has determinant +1. You can try to see if rotating the tetrahedron could actually reproduce the symmetries that you described!
As a somewhat related note, the fact that you can't rotate the tetrahedron to get its reflected image is called chirality in chemistry.
Why do methematicians call a group a "group"? Why not the closym, for instance (it is closed and preserves symetries)
Thanks, nice content, but why don't you go your own way? It is definitely possible for you to build another paradigm of visualizing mathematics.
there exists e such that
xe=ex=x
I guess
Hi :D can you make a discord server?
Can you persuade me why I would want to do that? I don't use Discord, so I don't know why or how people use it.
16th comment
thats in the bottom 20% of comment newness, m8
@@douglaspantz at the time yes not after 20 years hopefully
@@hzkzg1614 i highly doubt an early youtube comment will be at the top of your bucket list after 20 years, but ok. I guess its nice to have a marking
@@douglaspantzmake no doubt it will be.
@@hzkzg1614 understandable, have a nice day
Don't mind bruh but could you tell me truth that how many times you have done Ph.D in maths?
I don't have a PhD...
@@mathemaniac you are genius
god
theory of least action is not mean doing nothing. it mean use the least waste called efficiency. just like if take a division to hunt for single terrorist is inefficient but if use a million of terrorist can overturn the world that is efficient. just kidding don't do it at 🏡 me unleast got certificate