one way a mathematician may study symmetry.
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- Опубліковано 22 сер 2024
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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)
I kinda like writing it as rs = sr^-1
Actually, it's even easier. rs = -sr (since r^3 = -r)
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.
Advanced Algebra instructor: And what is e?
Me: 2.71828...
Math is so soothing and lovely
I love your teaching style. Subscribing to your other channel is a no brainer.
Nice use of the computer graphics near the beginning.
thank you!!
-Stephanie
This was very satisfying, thank you very much!
Love this Group since my undergrad days.
I do love me some group theory! Any chance you'd be willing to go over braid groups too? 🙏
Beautiful diagram
12:20
Very neat
Cool stuff, reminds me of state machines in animating software.
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.
not me mispronouncing the title as diarrhea group.
sr = r^3 s Is that correct?
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)
@@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.
@@samwalko ye that would be easier ^^
waaaait a minute where is the weird super fun and totally nonsensical description??
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
@@MichaelPennMath Get well soon!
@@kristianwichmann9996 ditto
What about for n-Groups?
Why do you use r^2, r^3, etc. instead of 2*r, 3*r, etc.?
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.
Nice content!
Hi! Could you please share where you got your chalkboard, and what type of chalk you use? I love the sound of it.
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.