I read a book about particle physics, "Deep Down Things" and they talk a little about Group Theory and then Lie Groups so I thought I would try to learn the basics of Group Theory since it sounds interesting.
At 26:54 I think you missed -1 in the answer exponent. S1 S3^-1 S1^-1 = S3^-1 Just saying, but I like your videos and I will keep watching them. Thank you ;)
On 12:30 Do you mean that doing a glide reflection after doing the gh (glide to the right and then horizontal flip), is a glide reflection that glides to the left?
In regards to the Frieze Pattern exercise: would we not be able to rotate the pattern 180 degrees around any point exactly between any two nearest diamonds? I am struggling to see how this could be equivalent to any combination of translations and glide reflections. Is it not necessary to include a vertical reflection in our generators so that we may construct a rotation by composition of vertical and horizontal reflection?
Performing a glide reflection is just a translation horizontally followed by a vertical flip over the horizontal axis, so the vertical reflection you're talking about is contained inside the glide reflection
How can a frieze group flip/reflect horizontally along an "infinite" axis? The midpoint of infinity that allows you do the reflection is not even defined. What kind of fundamental theorem to allow this is omitted here?
At 15:48, when constructing the Translation from G. Should it not read G squared = T squared? It seems to me, that T here is in fact a double Translation. J At least, it is twice the length of the previous example.
~ 18:45 I don't understand why the 2nd one on the left requires a horizontal (over a vertical axis) reflection... wouldn't a translation+glide or even just 2x glide reflection be enough? Same for any, I don't see the need for the horizontal flip as a generator for any of them.
James, I had almost the same question. Seems to me that a guide reflection (which comes with an inverse, of course) would be the only needed generator. But translation and glide are the same things, no? And I'm not sure what you mean by 2x glide reflection.
I also have the same question but I guess probably because we started with only the left half or the right half of the diamond so we need horizontal flip to obtain a whole diamond.
Hi, If only actions are allowed that preserve the footprint, the horizontal flip at 12:30 should not be a permitted acition. Also if it is indistinguishable, does that not represent the Identity action?
I think when he says horizontal flip he means flipping w.r.t. a vertical axis (going through the centre of some diamond). In this case, the horizontal flip is permitted and it is not equal to e, since if you label any three consecutive diamonds 1-2-3, then horizontal flip yields 3-2-1.
In the 2/7 Frieze, shouldn't there be also a "negative" glide reflection? Otherwise we would not be able to "undo" any action - i.e. we won't have a reverse for each action. Same question for T in the 1/7 and 6/7 Frieze.
I’m late to the party but for anyone else who may come across this, yeah it does but it can be generated with G and H. G involves a vertical flip and we know that a vertical flip times a horizontal flip is a rotation
@@ProfessorMacauley I mean, at 14:30, patterns for 1d space is shown in 2d. I feel it somehow limited the possibilities if it's shown in 2d. If a helix(something like the graph to show DNA) is sliced along its axis, the result is in 3d space, but only repeats in 1 direction. Its translation equals to (glide+rotation 120 degrees)^3 or t == ( glide+rotation 90 deg)^4. Is this situation anything different from all the patterns you shown in ppt at 14:30. According to the wiki page, What I'm curious about is called screw axes. So the question should be explained like, is screw axes of 2 different from screw axes of 3? Or are they somehow in a series? Thank you.
Has anyone here done the homework asking to draw cayley diagrams for the frieze patterns? I'm not sure about the one pattern that has three generators...
Yeah I remember in my class the instructor defined horizontal and vertical reflections the other way and there were a bunch of students that were confused by the definition.
That wouldn't change the symmetry group. There are infinitely many different patterns, but they fall into one of 17 "types", characterized by the group.
@@ROForeverMan i believe the author meant, that it will not generate a pattern, different from described 17. But surely, the particular wallpaper will change it's symmetry group if you change the color of its part.
How math should've been created: Dude: bro, think sbout this. Bro: makes no sense, dude. Dude: bro, wants me to draw? Bro: I doubt u can, dude. Dude - *invents math*
Who else got strange visual artefacts around 23:16?
I read a book about particle physics, "Deep Down Things" and they talk a little about Group Theory and then Lie Groups so I thought I would try to learn the basics of Group Theory since it sounds interesting.
At 26:54 I think you missed -1 in the answer exponent. S1 S3^-1 S1^-1 = S3^-1
Just saying, but I like your videos and I will keep watching them. Thank you ;)
great videos!🥰
On 12:30 Do you mean that doing a glide reflection after doing the gh (glide to the right and then horizontal flip), is a glide reflection that glides to the left?
20:22 HGT group has R as well (at some center)
In regards to the Frieze Pattern exercise: would we not be able to rotate the pattern 180 degrees around any point exactly between any two nearest diamonds? I am struggling to see how this could be equivalent to any combination of translations and glide reflections. Is it not necessary to include a vertical reflection in our generators so that we may construct a rotation by composition of vertical and horizontal reflection?
Performing a glide reflection is just a translation horizontally followed by a vertical flip over the horizontal axis, so the vertical reflection you're talking about is contained inside the glide reflection
I think m in wallpaper patterns stands for mirror so you can see reflection symmetries there.
Please if you can help me
I need an algorithm for a program design and evaluate cryptosystem based on braid groups
How can a frieze group flip/reflect horizontally along an "infinite" axis? The midpoint of infinity that allows you do the reflection is not even defined. What kind of fundamental theorem to allow this is omitted here?
Closely related to wallpaper groups and crystallographic groups are quasilattices and quasicrystals, which are worth mentioning.
Thank you for mentioning it! Now the interested viewer can Google those things. :-)
At 15:48, when constructing the Translation from G. Should it not read G squared = T squared? It seems to me, that T here is in fact a double Translation. J
At least, it is twice the length of the previous example.
~ 18:45 I don't understand why the 2nd one on the left requires a horizontal (over a vertical axis) reflection... wouldn't a translation+glide or even just 2x glide reflection be enough? Same for any, I don't see the need for the horizontal flip as a generator for any of them.
James, I had almost the same question. Seems to me that a guide reflection (which comes with an inverse, of course) would be the only needed generator. But translation and glide are the same things, no? And I'm not sure what you mean by 2x glide reflection.
I also have the same question but I guess probably because we started with only the left half or the right half of the diamond so we need horizontal flip to obtain a whole diamond.
does the Cayley diagram for the frieze work? in the frieze pattern gr^2 = f but this doesn't seem to be the case in the Cayley diagram??
Are there more symmetries? What about rotating a single diamond?
amazing..
Hi,
If only actions are allowed that preserve the footprint, the horizontal flip at 12:30 should not be a permitted acition. Also if it is indistinguishable, does that not represent the Identity action?
I think when he says horizontal flip he means flipping w.r.t. a vertical axis (going through the centre of some diamond). In this case, the horizontal flip is permitted and it is not equal to e, since if you label any three consecutive diamonds 1-2-3, then horizontal flip yields 3-2-1.
Is the braid group a free group?
In the 2/7 Frieze, shouldn't there be also a "negative" glide reflection? Otherwise we would not be able to "undo" any action - i.e. we won't have a reverse for each action. Same question for T in the 1/7 and 6/7 Frieze.
I think there is by default the inverse action in the group which accomplish what you asked
Around 18:00. Does the 2nd frieze group on the left also have an R symmetry?
Yes, I think it does. It is at every midpoint between the figures in the horizontal axe.
No, it doesn't. Rotation would change the orientation of the objects. It's to the left, not to the right.
I’m late to the party but for anyone else who may come across this, yeah it does but it can be generated with G and H. G involves a vertical flip and we know that a vertical flip times a horizontal flip is a rotation
Liking these videos a lot. Well-paced, nice visuals and cleanly explained. Really good job. Thanks.
awesome!!
Why not some patterns in 3d space but only repeat in one direction? So, if some group contains T == G3, what is it?
I'm not sure I understand your question. There are definitely patters in 3D space that only repeat in one direction. en.wikipedia.org/wiki/Space_group
@@ProfessorMacauley I mean, at 14:30, patterns for 1d space is shown in 2d. I feel it somehow limited the possibilities if it's shown in 2d. If a helix(something like the graph to show DNA) is sliced along its axis, the result is in 3d space, but only repeats in 1 direction. Its translation equals to (glide+rotation 120 degrees)^3 or t == ( glide+rotation 90 deg)^4. Is this situation anything different from all the patterns you shown in ppt at 14:30.
According to the wiki page, What I'm curious about is called screw axes. So the question should be explained like, is screw axes of 2 different from screw axes of 3? Or are they somehow in a series?
Thank you.
Has anyone here done the homework asking to draw cayley diagrams for the frieze patterns? I'm not sure about the one pattern that has three generators...
You confused me between horizontal and vertical flip 10:34
Yeah I remember in my class the instructor defined horizontal and vertical reflections the other way and there were a bunch of students that were confused by the definition.
Agree, his definition of vertical/horizontal flip is very confusing. It is like playing a video game with inverted controls.
I find the use of the notion 'isomorphism' strange here since you have yet to define what that means in the course.
Watch previous video
What if you add different colors to the wallpapers ?
That wouldn't change the symmetry group. There are infinitely many different patterns, but they fall into one of 17 "types", characterized by the group.
@@matthewmacauley5441 Why wouldn't it change the symmetry group if by moving a color into another colour you would get a different looking wallpaper ?
@@ROForeverMan i believe the author meant, that it will not generate a pattern, different from described 17. But surely, the particular wallpaper will change it's symmetry group if you change the color of its part.
At 19:57, isn't it valid also that R=TV?
That sounds more like G than R
14:50 there is really only one operation there.
How math should've been created:
Dude: bro, think sbout this.
Bro: makes no sense, dude.
Dude: bro, wants me to draw?
Bro: I doubt u can, dude.
Dude - *invents math*
22:00 uhh why? gh=hg
Wow❤😂😢😮😅😊