Professor Macauley
Visual Group Theory, Lecture 1.3: Groups in science, art, and mathematics
Groups are always lurking where symmetry arises. In this lecture, we explore many beautiful examples of groups that arise from natural symmetries in science, art, and mathematics. This includes shapes of molecules, repeating patterns in 1, 2, and 3 dimensions, and finally, how groups arise in braids.
Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/math4120-online.html
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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.
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…
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.
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
Are there more symmetries? What about rotating a single diamond?
Closely related to wallpaper groups and crystallographic groups are quasilattices and quasicrystals, which are worth mentioning.
Why not some patterns in 3d space but only repeat in one direction? So, if some group contains T == G3, what is it?
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.
At 19:57, isn't it valid also that R=TV?
~ 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.
frieze pattern is better explained in this
https://www.youtube.com/watch?v=RcoTKka7rxw
20:22 HGT group has R as well (at some center)
Who else got strange visual artefacts around 23:16?
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??
You confused me between horizontal and vertical flip 10:34
amazing..
Liking these videos a lot. Well-paced, nice visuals and cleanly explained. Really good job. Thanks.
14:50 there is really only one operation there.
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?
23:30 — The mysterious code / notation for each wallpaper pattern is explained on p. 443 of: Schattschneider, Doris (1978) "The plane symmetry groups: their recognition and notation," American Mathematical Monthly, 85 : 439-450 — which is available here — http://www.math.fsu.edu/~quine/MB_10/schattschneider.pdf
22:00 uhh why? gh=hg
awesome!!
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?
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 😉
Around 18:00. Does the 2nd frieze group on the left also have an R symmetry?