Here is a short quiz for you: What group is encoded in the undirected Cayley graph created at the end of the video? In general, what family of groups corresponds to the animated sequence of the Cayley graphs? Here is an extra hint for you: pay attention to how the colors are used throughout the video.
They should be the other dihedral groups, right? I forgot about the colours part initially and just saw a cyclic transformation of arbitrary length along with an additional self-inverse transformation. So f^2 = 1 and r^n = 1 for some n (for the triangle, n = 3). I'm not certain what the general form of the commutator looks like though.
Right, and the relation with the inverse can be generalized to the case when the r^n = 1 relation is dropped. I will probably talk about this in one of the future videos.
You finished it! Very cool. Loved this. I had made a few simple animations for my abstract algebra class but was impressed to see yours here. Very well Done!
The animations very clearly showed the different symmetries of the triangle and creation of the Cayley table (I liked how the vertices of the triangle were different colors to visualize the manipulations), and the presentation made me realize that there is similarity between listing tables of symmetries for groups and listing probability tables in statistics. I have yet to explore abstract algebra yet, but this video was a clean and intriguing introduction to it!
Thank you for the constructive feedback! A lot more will be coming in this series. And good luck with your entry. I think it should definitely make it to the short list, and possibly end up as one of the winners.
I love how you can define the group through just a few small axioms like f² = 1, r³ = 1, fr = r²f. All you have is how to return to the identity and how the different transformations interact with each other. The complex numbers have a similar definition of i⁴ = 1, so you only have 1 loop (and thus no need for a commutator), but it takes 4 rotations to reach the identity. (One minor caveat here is that i² isn't completely independent from 1 when adding scalar multiples, but those don't exist yet in this series.) Clifford algebras are a personal favourite of mine and have eᵢ² = ±1 and eᵢeⱼ = eⱼeᵢ for any unequal i and j. You need to specify which values of i give positive or negative squares, but that's all you need in order to make highly complex algebras capable of describing so much of geometry and physics. Go ahead, try and build a Cayley table for your choice of base elements!
What is your recommendation for a 16 year profoundly interested in mathematics and not satisfied intellectually by the government’s education system I’ve attempted Single variable calculus but due to lack of ares where I can express I’m abilities I’ve been slowly losing my knowledge I wish to understand more solve more not for my pleasure but in aid of others in someway
This is all very well, but what is the point of it all? Is there anything more than the flipping and rotation of geometric objects? Does Group Theory reveal a property or pattern in nature? Probably not, as they are constructed by humans. Do they provide a useful tool for the solution of problems? If so, i would dearly like to know.Just a hint would be nice. Preferably without being told anything more about symetries of objects, which strike me as kind of obvious. Good video though.
Thank you for the feedback, Stafford. A lot more is coming in this series, and these and other questions will be answered in the future videos. For now, we showed a historic motivation for the notion of a group and played a bit with some more advanced notions such as generators and relations.
Isnt it like saying what is the point of logarithm when you literally defined something for equations that you cant solve, of course in math you start with an idea that seems like have no application but you can expand the idea itself for more application in the future
@Stafford It is not exaggerating to say that groups are one of the main tools in mathematics. When there are groups, you have a way to harness structure to understand your object, and conversely, when studying an object with useful structure, at some point you will find a group which encodes the information. Almost all of mathematics is studying very nice groups, albeit not always with the very general perspective one associates with abstract algebra. Of course, the applications are more advanced than what is covered in lecture one (or here, the introductory video). Three conceptually simple classes of examples for useful groups are 1) groups which directly correspond to other objects. There are a few areas where you can define groups whose subgroups have a nice bijection to the substructures you want to understand (examples: Galois group of field extension, fundamental group of space) 2) groups are rather nice to study, so it is sometimes helpful to assign an object you want to understand a group which captures some information about it - this is the idea of using groups as invariants: If the groups are different, then the object are different too, which is a powerful tool to distinguish them. The prime „example“ for this is the idea of (co)homology. 3) when you have *really* nice groups, you can use them as a framework for further work. What I have in mind here are vector spaces, which are fundamental for huge parts of mathematics in their own right, but still primarily special groups. And of course, mathematics has applications to describe nature! What do you mean, manmade ideas cannot describe the physical world!? ;) Physicists are quite good at using mathematics (read here: groups) to make predictions and enable technologies.
Lots of amazing responses in this thread! To add to Stafford's remark that groups are constructed by humans. It is an open philosophical question whether mathematics is created or discovered. But when it comes to groups, I believe they are discovered because there are too many patterns and structures in nature that fit the notion of a group. I plan to address this exact question in one of the future videos in this series.
The fundamentals of particle physics actually stems from certain kinds of group theories. So it's actually types of symmetries from which fundamental interactions that make all matter and energy as we know it exist. For example, Electromagnetic theory comes from the U(1) group which is basically the circle rotation group, and it explains why stuff has charge. More abstract groups like SU(2) and SU(3) explain the weak and strong forces, and the generators of these groups correspond with force particles like photons, W bosons, Z bosons, gluons, etc.
Here is a short quiz for you: What group is encoded in the undirected Cayley graph created at the end of the video? In general, what family of groups corresponds to the animated sequence of the Cayley graphs? Here is an extra hint for you: pay attention to how the colors are used throughout the video.
They should be the other dihedral groups, right? I forgot about the colours part initially and just saw a cyclic transformation of arbitrary length along with an additional self-inverse transformation. So f^2 = 1 and r^n = 1 for some n (for the triangle, n = 3). I'm not certain what the general form of the commutator looks like though.
Yes, they are dihedral groups. The general relation is rf = fr^{-1}.
@@mathflipped That makes sense. Since r^n = 1, r^(n-1)r = 1, so r^(n-1) = r^-1.
Right, and the relation with the inverse can be generalized to the case when the r^n = 1 relation is dropped. I will probably talk about this in one of the future videos.
@@mathflipped I'm looking forward to the rest of your series on visual algebra.
You finished it! Very cool. Loved this. I had made a few simple animations for my abstract algebra class but was impressed to see yours here. Very well
Done!
Thank you, Tom! We should collaborate sometime in the future.
The animations very clearly showed the different symmetries of the triangle and creation of the Cayley table (I liked how the vertices of the triangle were different colors to visualize the manipulations), and the presentation made me realize that there is similarity between listing tables of symmetries for groups and listing probability tables in statistics. I have yet to explore abstract algebra yet, but this video was a clean and intriguing introduction to it!
Thank you for the constructive feedback! A lot more will be coming in this series. And good luck with your entry. I think it should definitely make it to the short list, and possibly end up as one of the winners.
@@mathflipped Thank you, and good luck with your series!
I love how you can define the group through just a few small axioms like f² = 1, r³ = 1, fr = r²f. All you have is how to return to the identity and how the different transformations interact with each other. The complex numbers have a similar definition of i⁴ = 1, so you only have 1 loop (and thus no need for a commutator), but it takes 4 rotations to reach the identity. (One minor caveat here is that i² isn't completely independent from 1 when adding scalar multiples, but those don't exist yet in this series.) Clifford algebras are a personal favourite of mine and have eᵢ² = ±1 and eᵢeⱼ = eⱼeᵢ for any unequal i and j. You need to specify which values of i give positive or negative squares, but that's all you need in order to make highly complex algebras capable of describing so much of geometry and physics. Go ahead, try and build a Cayley table for your choice of base elements!
Great lecture. Thanks master.
You are welcome, Jorge. Glad it was useful.
Beautifully done video
Thank you, Whitney.
What is your recommendation for a 16 year profoundly interested in mathematics and not satisfied intellectually by the government’s education system I’ve attempted Single variable calculus but due to lack of ares where I can express I’m abilities I’ve been slowly losing my knowledge I wish to understand more solve more not for my pleasure but in aid of others in someway
Is this voice that again?
The voice is that of my collaborator for this series, Matthew Macauley. He is credited in the video description.
Just send ya an email...hit me back to confirm you got it Igor
Thanks, Josh! Got your email and sent a reply.
This is all very well, but what is the point of it all? Is there anything more than the flipping and rotation of geometric objects? Does Group Theory reveal a property or pattern in nature? Probably not, as they are constructed by humans. Do they provide a useful tool for the solution of problems? If so, i would dearly like to know.Just a hint would be nice. Preferably without being told anything more about symetries of objects, which strike me as kind of obvious. Good video though.
Thank you for the feedback, Stafford. A lot more is coming in this series, and these and other questions will be answered in the future videos. For now, we showed a historic motivation for the notion of a group and played a bit with some more advanced notions such as generators and relations.
Isnt it like saying what is the point of logarithm when you literally defined something for equations that you cant solve, of course in math you start with an idea that seems like have no application but you can expand the idea itself for more application in the future
@Stafford It is not exaggerating to say that groups are one of the main tools in mathematics. When there are groups, you have a way to harness structure to understand your object, and conversely, when studying an object with useful structure, at some point you will find a group which encodes the information. Almost all of mathematics is studying very nice groups, albeit not always with the very general perspective one associates with abstract algebra. Of course, the applications are more advanced than what is covered in lecture one (or here, the introductory video).
Three conceptually simple classes of examples for useful groups are
1) groups which directly correspond to other objects. There are a few areas where you can define groups whose subgroups have a nice bijection to the substructures you want to understand (examples: Galois group of field extension, fundamental group of space)
2) groups are rather nice to study, so it is sometimes helpful to assign an object you want to understand a group which captures some information about it - this is the idea of using groups as invariants: If the groups are different, then the object are different too, which is a powerful tool to distinguish them. The prime „example“ for this is the idea of (co)homology.
3) when you have *really* nice groups, you can use them as a framework for further work. What I have in mind here are vector spaces, which are fundamental for huge parts of mathematics in their own right, but still primarily special groups.
And of course, mathematics has applications to describe nature! What do you mean, manmade ideas cannot describe the physical world!? ;) Physicists are quite good at using mathematics (read here: groups) to make predictions and enable technologies.
Lots of amazing responses in this thread! To add to Stafford's remark that groups are constructed by humans. It is an open philosophical question whether mathematics is created or discovered. But when it comes to groups, I believe they are discovered because there are too many patterns and structures in nature that fit the notion of a group. I plan to address this exact question in one of the future videos in this series.
The fundamentals of particle physics actually stems from certain kinds of group theories. So it's actually types of symmetries from which fundamental interactions that make all matter and energy as we know it exist. For example, Electromagnetic theory comes from the U(1) group which is basically the circle rotation group, and it explains why stuff has charge. More abstract groups like SU(2) and SU(3) explain the weak and strong forces, and the generators of these groups correspond with force particles like photons, W bosons, Z bosons, gluons, etc.