Projective Plane and Homogeneous Coordinates - FLT Proof #4.1.2.5

[@evariste1686]

so many presentations of the projective plane only know the formal/abstract construction. Your explanation of the geometry underlying that abstraction makes the formalities much easier to reason about. Just excellent.

[@Hanniballectraa]

I have been going mad trying to understand homogenous coordinates/lines and why they are represented in a certain way. You video has made everything crystal clear! I now understand the "Why" part to the way the coordinates are represented! Thank you very much for posting this video!

[@andytucker9991]

Hey just want to thank you so much, out of all the videos i found about homogeneous coordinates, only yours gave me good intuition of what it really is!

[@mehdia5176]

Your explanation is grounded in common sense. You made the abstract concrete and for that you have my sincere appreciation. Thank you sir.

[@greg55666]

You are very welcome!

[@QQpapababy]

The explanation is so clear that makes me understand the reason why we introduce such homogeneous coordinates. The points on 2D space are no longer points. They are lines that run through the plane. This makes me to think that what we might be that run through this 4D spacetime.

[@greg55666]

Thank you for the kind words, QQ. That's a really interesting idea!

[@QQpapababy]

@@greg55666 I still have questions. We have introduce the homogeneous coordinates. What will be the next?

[@greg55666]

@@QQpapababy These videos are an introduction to elliptic curves and modular forms. We started by looking at elliptic curves over real numbers, and rational numbers, and complex numbers. These videos about the dot product and the projective plane are needed because the group operations over elliptic curves require homogeneous coordinates. This video is the end of that branch of this tree--we've looked at elliptic curves over real numbers (#4.1), and rational numbers (#4.2). Now we're working on the last foundation for elliptic curves, complex numbers (which is where we are now, #4.3). Unfortunately, I realized complex numbers is where the real action is in Fermat's Last Theorem, and it's way more complex (ha) or involved than the other two topics. In order to understand how elliptic curves relate to complex numbers (and hence modular forms) we really need to be very familiar with complex numbers. And that requires us to be familiar with multivariable calculus. And that requires topology. So we've kind of gone down this rabbit hole of foundations before we have a solid enough foundation to move on to a book about elliptic curves. The issue, of course, is that we can't know what we *need* to know until we already know everything, then we can look back and say, oh, we could have done without that. So we're going to be working on complex numbers (and multivariable calculus for a while). Always any discussion about math has to assume some level of knowledge in the audience. I have arbitrarily decided to assume a knowledge of everything up through freshman calculus--derivation and integration of one variable. I think multivariable calculus is not as common a topic, and it's certainly something I'm not as comfortable with as single-variable. It is is absolutely vital to have a good familiarity with multivariable calculus in order to understand complex integration. Also, we're not in any hurry. This current series on topology probably isn't necessary, but it's really interesting and fun, and that's the real goal.

[@mercedes9474]

Thank you so much! This video really helped me understand the idea of projective plane. Excellent! 👌

[@khadijakhalid9777]

So informative....i really enjoyed it...👍👍👍👍

[@greg55666]

Thank you! I'm happy to hear that.

[@yapayzeka]

best explanation out there

[@monoman4083]

good explanation. thx..

[@walts555]

👍Excellent Intro, well done on motivations and interpretation. Just getting into this subject, and I'm amazed at how many authors skip the motivations and how to interpret the word "projective" in their intro treatments. These same authors seem well aware of these aspects but don't include in their instruction. Are they even TRYING to explain this material? Like trying to explain cars without ever mentioning roads or wheels. Very puzzling.

[@davidandcookie7648]

Dude where have you been? You were chugging along with posts, but there hasn't been one.

[@greg55666]

Thank you for writing! Sorry about the delay--I was originally going to just make one video discussing elliptic curves over complex numbers, but I realized/decided it is, first, a really interesting topic, and second, way more important than I realized at first for FLT. So I've decided to make a whole series exploring the highlights of complex analysis leading up to the Weierstrass equation and how that relates to elliptic curves (amazing). Spending time to explore complex analysis will also be useful/necessary to be able to get even a basic grasp of what modular forms are before we move on to exploring elliptic curves in detail. I should have another video up shortly, within the next few days, a week at the most. Please come back!

[@reginaldanderson7896]

I always forget: Real projective plane has a line at infinity, and 1d-complex projective space has just a point at infinity?

[@greg55666]

Can you ask your question again, please, being a little more clear what it is that you want to know? I can't tell from the question what it is that you think you are forgetting. Have you watched the video?

[@reginaldanderson7896]

@@greg55666 \CP^1 minus an affine chart is a point (0 real dimensional and 0-complex dimensional), but \RP^2 minus an affine chart is one real dimensional (a line). It's fine thanks

[@greg55666]

@@reginaldanderson7896 I had a feeling the question was rhetorical. If you have any further questions, please let me know. Thanks for watching!

[@reginaldanderson7896]

@@greg55666 Ah well your feeling was incorrect, it was not rhetorical, but I reviewed it and got it