New video! Euler's formula with introductory group theory

I've been watching Bartosz Milewski's series on category theory on youtube hoping for a simple intro. I'm at about 10 hours into his series and I see the world differently; it applies to much more than math or programming. Highly recommended! Category theory definitely lends itself to visual representation with objects and arrows; unfortunately it's not obvious how to make it simpler without loosing something.

Thank you for watching, Albert. Any path in life motivated by a love of math seems like a good one to me :)

3blue1brown

Thank you for all the videos you are publishing, the way you explain, and the passion you display for mathematics just helped me find what I want to do in life. For that, I am thankful.

Albert Villeneuve Nguyen

Looks really cool, I'll check it out.

3blue1brown

Wow, this is relevant to my interests (as usual, but still, it's uncanny how your videos track with things I'm interested in). I have long gotten used to complex exponentiation being rotation, but hadn't grokked the yoking of it to group theory. I had a catastrophically bad abstract algebra professor, so group theory has been a bit of a sore point with me. If you ever do a longer series on group theory you'd be a hero. Have you played with the Escher "Print Gallery" effect at all? It can be thought of as an UNCOMMONLY beautiful combination of complex functions. I tried to do it justice in blog form: <a href="http://roy.red/droste-.html" rel="nofollow noopener" target="_blank">http://roy.red/droste-.html</a> and on the way, I drew something similar to your final animation. It has the nice property of being a very simple combination of elementary functions so it's quite easy to understand why it behaves like it does, unlike most conformal maps.

Thank you for explaining the group theory connection. I had seen the earlier video which enlightened me, but I never realise the group theory aspect. And though you already showed these concepts once, I still find it gorgeous

Magnasium

Real power is pure rotation. Imaginary power (with positive coefficient on i) is squishing. A real number corresponds to stretching/squishing by multiplication. Exponentiation of a real number by a real number (eg. 4.3^2.5) means the degree of stretching changes with your exponent. If the exponent is the input and the exponentiation is the output, the sliding the input complex plane along the real axis causes the output complex plane to stretch along the radial directions. Sliding the input complex plane in a direction perpendicular to real axis (i.e. imaginary axis) causes the output complex plane to get "stretched" in a direction perpendicular to radial direction; this effectively becomes movement along the tangential direction of a circle with centre at origin, and so rotation. Since an imaginary number corresponds to rotation by multiplication, exponentiation by a real number just varies the angle of rotation. Similar to above, exponentiation by an imaginary number causes movement in a direction perpendicular to rotation; i.e. stretching/squishing. The movement due to exponentiation of a real number by an imaginary number is 90deg anti-clockwise to movement due to exponentiation of a real number by a real number. So exponentiation of an imaginary number by an imaginary number will be 90deg anti-clockwise to normal rotation; which would correspond to squishing.

Magnasium

Same here. I can visualize the multiplication squishing the infinite line to a point for 0, and then stretching it back out in a flipped orientation for negative; but I feel that may be abstract. Then I saw the complex plane and realized it's much easier to demonstrate negative in the complex plane through rotation; and I was hoping he would harp on that.

Magnasium

And what does taking imaginary number to a power mean?

Kuba Okrzesa

Thank you so much for sharing this beauty at so many levels!

I think of this channel as art explaining maths ... inspiring and informative. Thanks.

Christopher Burke

You definitely should! I'd love to see what you produce.

3blue1brown

Excellent. My only quibble is that in discussing multiplication on the number line, you only referred to the "multiplicative group of POSITIVE numbers." And while it's pretty obvious what multiplication by negative numbers should do to the line, you didn't spell it out. And yet, when talking about i*i = -1, you referred to -1 as the "unique action" without having earlier made clear that negative numbers are also a valid part of the multiplicative group.

jason black

Perfect timing. I've just been learning some group theory. You couldn't do a similar simple intro to category theory could you? 😁

Hey Grant. Nice video. Wanted to share with you something cool in Group Theory if you'll be animating it ever. Suppose you have a group with some simple representation. For the sake of example, think of the dihedral group D8 for the square with eight elements, let's say, and take for the representation that it's generated by the 90 degree rotation \rho and the, say, horizontal flip \tau. Now draw the Cayley Digraph for that group, so that there are 8 nodes, with each node having two arrows emanating from it, one labeled \rho, the other labeled \tau. Already this is a nice visualization. But now, suppose you take a normal subgroup N. For D8 as above, a normal subgroup is N = { Id, \rho^2 }. (identity and 180 degree rotation). In your Cayley di-graph animation, you can now put the 8 nodes into four sub-sets - the four left-cosets of N. Draw faint blue circles around them to show that they're associated. But as you do this, you keep the di-graph arrows between all the nodes. Viewers will see all \rho arrows and all \tau arrows can be viewed not as arrows between nodes, but as arrows between these cosets. The fact that this is possible can be seen as a direct consequence of - in fact equivalent to - the fact that N is a normal subgroup. Thus the quotient group D8 / N is born with four elements. With animation, it could be vividly illustrated that the elements of the quotient group are subsets of elements of D8, and that the structure of the quotient group "comes from" the structure of D8. Viewers will easily recognize the final structure as that of Z_2 x Z_2. Damn, you know, I'd think I'd like to take a crack at it myself first, Grant. I'll try animating the frames in Mathematica and see if I can get anything nice.

Jacob Mirra

Loved the video!

Luop90