_By the way..._ By 5 minutes in, something you said made me write a tiny little proof (my first, actually) for something I didn't realize needed one. Thank you.
Great! You're diving in deep. I'm curious: which statement needed a proof? Feel free to share your proof, or some other details about your train of thought. I'm really interested.
@@AllAnglesMath Whoops, phrasing. I was indeed listening very carefully and when you said: _"[applying a transformation then its inverse is equivalent to] not doing anything at all"_ I realized I have an inversion in my own research that I'd taken as self-evident (when it needs to be shown). It took you spelling it out for me to realize I was overlooking it.
Hold on a moment! The blocks that you divided the square's group into at the end; if you treat each sub-block as a single element, the result looks suspiciously like the diagram between the 4 basis quaternions. Considering the individual elements, it looks a lot like the Geometric Algebra Cl(2), with r2 as -1, which is why ignoring it and looking just at the blocks is just the transition between basis elements (which is equivalent to the quaternions not including sign). Also worth mentioning is that the upper left 2x2 block-of-sub-blocks would then be the basis ℂomplex numbers, which forms the even-subalgebra of Cl(2) (aka Cl(0,1)), meanwhile joining each of the other two sub-blocks with upper left sub-block gives two copies of Cl(1) (aka the split-complex numbers). It's always fun to find when two finite groups are actually the same. I think the more general instance of these blocks of sub-blocks, and Geometric Algebra's subalgebras, would be factoring into the "simple" groups.
First off, great video; looking forward to the rest of the series. Though, I'm a little confused by how you use the word symmetry here. You showed that a plain square with the same color on both sides can be rotated 0, 90, 180, or 270 degrees or flipped accross four axes and it will look exactly the same as before. But if the square is plain then shouldn't all of these transformations be equivalent to the identity? It seems that we were only able to differentiate between these transformations when you added your logo onto the square and colored the two sides differently; let's call that the colored square. The only way I can wrap my head around this is that the set of symmetries of the plain square form a group on the colored square, but these transformations are NOT symmetries of the colored square. Did I understand that correctly?
You've basically got it right. There's a couple subtle things going on here. First, the symmetries of the plane square are better thought of actions you do to space, or the paper/transparency on which the square is drawn. Doing those actions leaves the plane square looking the same, but would change how the colored square looks. The other part is that in modern mathematical language, the "group" is basically just the table. And then you can define ways that table can act on things (numbers in the case of addition modulo 4, or a square that is plain or colored). In the right context, someone might say that the group is acting trivially on the plain square (since those symmetries leave it looking the same) but faithfully on the colored square (since they each leave the logo looking differently).
I think it has to do with how you define the equivalence of any 2 squares. Does the edges' color count? Does the faces' color count? What if I had a weird square that has to be reflected 4 times on itself in order to get back the original? Can I even have such a square, or is reflection defined to be the transformation that returns the original shape after exactly 2 applications?
@@alegian7934 "Reflection" typically *means* a certain type of operation on regular Euclidean space, like reflecting the plane over a line or reflecting 3D space through the origin. And all instances of what would be called a "reflection" undo themselves when applied again to a space. And since the group table says that one of the reflections in the video becomes the identity when it's applied twice, no action of that group element could need to be applied four times. The closest thing that comes to mind to what you're getting at would be a transformation on, say, "the plane and a clock" which reflects the plane in some way and also moves the a clock hand a quarter turn in one direction. Then after four applications of this "reflection and clock move", the "plane and clock" would be back to it's original state.
Hi, i'm Claudio from Italy: first of all compliment for your fantastic video!!! All very clear. I'd like to ask you a shareware/freeware software to manage groups and do exercises. thank's a lot and i hope you continue to make great videoes.
You said in your previous video that there must only be one neutral element, but doesn't rotating by 0, 360, 720... degrees count as having more than one neutral element? To me they all do the same (which is nothing), but they are still different.
Very good point. The solution is to say that all of those rotations are one and the same: They're all identical to a zero degree rotation. So there is still really only one neutral element. I'm sure you could also solve this by somehow creating an equivalence class of all multiples of 360. But that would go too far for an introductory video.
Almost. There is an extra condition: an isomorphism is a homomorphism which is invertible. The inverse itself is then also an isomorphism. For example, when you map from a group with 12 elements to a group with only 4, you might have a homoomrphism, but it cannot be an isomorphism.
At 14:00 I found this diagram unintuitive, because the arrows themselves depict a geometrical rotation that is NOT what is depicted by each colored square. I believe the arrows are just intended to show that there is a cyclic sequence of rotation operations, but they fact that they are drawn as 90 degree rotations (and which are not the same as the 90 degree rotation represented by the group operation) was confusing.
There is nothing really unintuitive about that diagram, by the simple fact that the direction of arrows does NOT have to be related with what the arrows describe. And that's it.
Exceptional graphiics!!
Man I just found your channel as a self-studier and I can tell it is a gold mine!
Welcome. I hope you will enjoy the videos and learn a lot from them. Feel free to ask questions in the comments!
Love the clear explanations!
Great video, intuitively informative, nice graphics and animation. Thanks a lot!
Thank you so much!
Absolute gem of a video!
Thank you!
_By the way..._ By 5 minutes in, something you said made me write a tiny little proof (my first, actually) for something I didn't realize needed one. Thank you.
Great! You're diving in deep. I'm curious: which statement needed a proof? Feel free to share your proof, or some other details about your train of thought. I'm really interested.
@@AllAnglesMath Whoops, phrasing. I was indeed listening very carefully and when you said: _"[applying a transformation then its inverse is equivalent to] not doing anything at all"_ I realized I have an inversion in my own research that I'd taken as self-evident (when it needs to be shown). It took you spelling it out for me to realize I was overlooking it.
@@oddlyspecificmath Thanks for clarifying. Glad to be of use, even if very indirectly. Good luck with your research!
Hold on a moment! The blocks that you divided the square's group into at the end; if you treat each sub-block as a single element, the result looks suspiciously like the diagram between the 4 basis quaternions. Considering the individual elements, it looks a lot like the Geometric Algebra Cl(2), with r2 as -1, which is why ignoring it and looking just at the blocks is just the transition between basis elements (which is equivalent to the quaternions not including sign). Also worth mentioning is that the upper left 2x2 block-of-sub-blocks would then be the basis ℂomplex numbers, which forms the even-subalgebra of Cl(2) (aka Cl(0,1)), meanwhile joining each of the other two sub-blocks with upper left sub-block gives two copies of Cl(1) (aka the split-complex numbers).
It's always fun to find when two finite groups are actually the same. I think the more general instance of these blocks of sub-blocks, and Geometric Algebra's subalgebras, would be factoring into the "simple" groups.
Great video. I'm excited to see how this series evolves!
Great video, i gained a much better intuitive understanding. Thanks!
I really enjoyed the video, thank you
Thanks for the positive feedback.
Amazing content! Simply amazing!
Thank you!
This video is wonderful and the animations are beautiful! What tool are you using for making the animations?
We use a custom Python library built around OpenCV.
I _really_ like this series.
Also, I find it very funny that you "missed" the L while talking about four digit clocks at 12:40. :'D
😄
You're the first to pay such detailed attention.
First off, great video; looking forward to the rest of the series. Though, I'm a little confused by how you use the word symmetry here. You showed that a plain square with the same color on both sides can be rotated 0, 90, 180, or 270 degrees or flipped accross four axes and it will look exactly the same as before. But if the square is plain then shouldn't all of these transformations be equivalent to the identity? It seems that we were only able to differentiate between these transformations when you added your logo onto the square and colored the two sides differently; let's call that the colored square.
The only way I can wrap my head around this is that the set of symmetries of the plain square form a group on the colored square, but these transformations are NOT symmetries of the colored square. Did I understand that correctly?
You've basically got it right. There's a couple subtle things going on here. First, the symmetries of the plane square are better thought of actions you do to space, or the paper/transparency on which the square is drawn. Doing those actions leaves the plane square looking the same, but would change how the colored square looks. The other part is that in modern mathematical language, the "group" is basically just the table. And then you can define ways that table can act on things (numbers in the case of addition modulo 4, or a square that is plain or colored). In the right context, someone might say that the group is acting trivially on the plain square (since those symmetries leave it looking the same) but faithfully on the colored square (since they each leave the logo looking differently).
What a great and very subtle question by Entangled kittens. I think diribigal's answer hits the nail on the head.
I think it has to do with how you define the equivalence of any 2 squares. Does the edges' color count? Does the faces' color count?
What if I had a weird square that has to be reflected 4 times on itself in order to get back the original? Can I even have such a square, or is reflection defined to be the transformation that returns the original shape after exactly 2 applications?
@@alegian7934 "Reflection" typically *means* a certain type of operation on regular Euclidean space, like reflecting the plane over a line or reflecting 3D space through the origin. And all instances of what would be called a "reflection" undo themselves when applied again to a space.
And since the group table says that one of the reflections in the video becomes the identity when it's applied twice, no action of that group element could need to be applied four times.
The closest thing that comes to mind to what you're getting at would be a transformation on, say, "the plane and a clock" which reflects the plane in some way and also moves the a clock hand a quarter turn in one direction. Then after four applications of this "reflection and clock move", the "plane and clock" would be back to it's original state.
Thank you!
Hi, i'm Claudio from Italy: first of all compliment for your fantastic video!!! All very clear. I'd like to ask you a shareware/freeware software to manage groups and do exercises. thank's a lot and i hope you continue to make great videoes.
Hi Claudio. Sorry, but I'm not aware of any such tools. Good luck with your further studies!
You said in your previous video that there must only be one neutral element, but doesn't rotating by 0, 360, 720... degrees count as having more than one neutral element? To me they all do the same (which is nothing), but they are still different.
Very good point. The solution is to say that all of those rotations are one and the same: They're all identical to a zero degree rotation. So there is still really only one neutral element.
I'm sure you could also solve this by somehow creating an equivalence class of all multiples of 360. But that would go too far for an introductory video.
Hey, isn't isomorphic the same as saying an homomorphism exists?
Almost. There is an extra condition: an isomorphism is a homomorphism which is invertible. The inverse itself is then also an isomorphism.
For example, when you map from a group with 12 elements to a group with only 4, you might have a homoomrphism, but it cannot be an isomorphism.
First one to see that
Congratulations ;-) Great to see you.
At 14:00 I found this diagram unintuitive, because the arrows themselves depict a geometrical rotation that is NOT what is depicted by each colored square.
I believe the arrows are just intended to show that there is a cyclic sequence of rotation operations, but they fact that they are drawn as 90 degree rotations (and which are not the same as the 90 degree rotation represented by the group operation) was confusing.
There is nothing really unintuitive about that diagram, by the simple fact that the direction of arrows does NOT have to be related with what the arrows describe. And that's it.