one way a mathematician may study symmetry.

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[@kkanden]

even though i absolutely loved my abstract algebra course, dihedral groups were something i struggled with the most due to its geometric nature involving a lot of good "artistic" imagination (which i don't have)

[@TheEternalVortex42]

I kinda like writing it as rs = sr^-1

[@flowingafterglow629]

Actually, it's even easier. rs = -sr (since r^3 = -r)

[@Ttarler]

One of my favorite parts of your videos is your chalkboard skills - they remind me of a favorite math professor in undergrad. But that rotations animation really did a good job of showing the symmetries. We’ll produced and thank you.

[@JayTemple]

Advanced Algebra instructor: And what is e?
Me: 2.71828...

[@JustNow42]

Math is so soothing and lovely

[@zacmilne9423]

I love your teaching style. Subscribing to your other channel is a no brainer.

[@GandalfTheWise0002]

Nice use of the computer graphics near the beginning.

[@MichaelPennMath]

thank you!!
-Stephanie

[@Ny0s]

This was very satisfying, thank you very much!

[@doctorb9264]

Love this Group since my undergrad days.

[@jneal4154]

I do love me some group theory! Any chance you'd be willing to go over braid groups too? 🙏

[@howwitty]

Beautiful diagram

[@goodplacetostop2973]

12:20

[@quantumskull2045]

Very neat

[@seedmole]

Cool stuff, reminds me of state machines in animating software.

[@flowingafterglow629]

Yeah, I think I am bothered by the fact that a square is actually D4h (in 3D), so you are missing all the horizontal reflections and there are rotation axes that go through all those reflection planes.

[@rain2001]

not me mispronouncing the title as diarrhea group.

[@bobh6728]

sr = r^3 s Is that correct?

[@reeeeeplease1178]

Yes
From rs = sr^3 apply r^3 to the left of both sides to get
s = r^3 s r^3 and then r to the right of both sides to get
sr = r^3 s
(r^4 vanishes as it is the identity)

[@samwalko]

​@@reeeeeplease1178 Alternatively, you could also apply s to the left and right of both sides of the equation, and now it's the s^2 that cancels.

[@reeeeeplease1178]

@@samwalko ye that would be easier ^^

[@paokaraforlife]

waaaait a minute where is the weird super fun and totally nonsensical description??

[@MichaelPennMath]

Unfortunately, my mental health hasn't been super great lately so those have been a casualty. I'll try to get back to them for ya :)
-Stephanie

[@kristianwichmann9996]

@@MichaelPennMath Get well soon!

[@JayTemple]

@@kristianwichmann9996 ditto

[@dec1aim_flow432]

What about for n-Groups?

[@charlessmith1931]

Why do you use r^2, r^3, etc. instead of 2*r, 3*r, etc.?

[@GoldenAgeMath]

You can use either, the first would be called "multiplicative notation" and the second would be called "additive notation". If you consider the cyclic group of size 4 as an example, we can describe this in multiplicative notation as = {1, a, a^2, a^3} (here the group operation is multiplication, so a^2 * a^3 = a), or in additive notation as {0, 1, 2, 3} where 4 = 0 and 2 + 3 = 1. Notice how adding/multiplying the generator (a or 1, I could've used the same symbol for both if I wanted to) to itself 4 times produces the identity in both cases, 1 is the multiplicative identity and 0 is the additive identity.

[@JBMJaworski]

Nice content!

[@nicholaslear7002]

Hi! Could you please share where you got your chalkboard, and what type of chalk you use? I love the sound of it.

[@user-ys3ev5sh3w]

r is a rotation around 0D point in 2D
s is a rotation around 1D line in 3D
r2 is a rotation around 2D plane in 4D
And so on
We can construct Pascal-like triangle
for 1-chain 1
for square 4 4
for cube 16 32 16
for 4-cube 64 192 192 64
for 5-cube 256 1024 1536 1024 256
Wich is face-vectors of donut4.
For example cube has 16 rotation aroun points,
32 rotation around lines, 16 rotations around planes.
( 1-donut4 is 4-ring (point shifted 4 times first and last points glued, i.e rotated 4 times) with face-vector 4 4
2-donut4 is 1-donut4 rotated 4 times with face-vector 16 32 16
3-donut4 is 2-donut4 rotated 4 times with face-vector 64 192 192 64 )
So we see that d-D_n is:
1) d-Donut_n polytope.
2) d-nested n-rotation(n-symmetry,n-cycle,n-orbit,n-reflection ...) where n is amount of rotation.
3) 2*n-ary (d-1)-digit number system.