Visual Group Theory, Lecture 1.3: Groups in science, art, and mathematics

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.

[alanhere]

Who else got strange visual artefacts around 23:16?

[sajateacher]

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.

[ChonGeeSan]

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 ;)

[RooftopDuvet]

Liking these videos a lot. Well-paced, nice visuals and cleanly explained. Really good job. Thanks.

[JacobFeldman]

20:22 HGT group has R as well (at some center)

[dcterr]

Closely related to wallpaper groups and crystallographic groups are quasilattices and quasicrystals, which are worth mentioning.

[oumplzo]

I think m in wallpaper patterns stands for mirror so you can see reflection symmetries there.

[anyad]

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?

[phyein]

~ 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.

[briankrebs]

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?

[Ludwig]

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.

[troller]

You confused me between horizontal and vertical flip 10:34

[kusy]

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?

[alexanderbecker]

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?

[evariste]

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??

[hericklenin]

Are there more symmetries? What about rotating a single diamond?

[davidefanchini]

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.

[ahmedalbyati]

Please if you can help me
I need an algorithm for a program design and evaluate cryptosystem based on braid groups

[nicolasmpgutierrez]

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...

[ub]

Around 18:00. Does the 2nd frieze group on the left also have an R symmetry?