New video! All possible pythagorean triples, visualized
Added 2017-05-26 17:58:22 +0000 UTCHey everyone,
This is kind of a fun one. The first time I ever learned about the formula for generating pythagorean triples, it was in the context of stereographically projecting rational points onto a circle. The visual intuition of that is nice, but the actual algebra to work out a formula is quite a mess, and (at least for me) the actual result is easy to forget.
However, you can land on that same general formula just by squaring complex numbers with integer coordinates, which lends itself to a different visualization entirely. I still talk about about the projections onto the unit circle, as it makes for a very nice little proof at the end, but I hope this offers a different perspective even to those familiar with the question of finding pythagorean triples.
-Grant
Comments
Yes, definitely. Thank you Grant.
Shel Hammer
2017-06-04 07:24:07 +0000 UTCHi Shel, that's kind of you to say. It's something I've thought a lot about, and I'm still not sure I get it right. When I write, I try to do the planning slowly, and the actual sentence construction/typing quickly, as if I was speaking to a student and transcribing it. Then when I read, I try to speak with 30-50% more energy than seems natural, because quite a bit is lost through the mic, and when things are not in person. Hope that helps!
3blue1brown
2017-06-03 21:17:09 +0000 UTCHi Grant, I work as a part time mathematician and I have to make a lot of videos for my job. Every time I try and make a video I either script it out way too much, and I end up sounding like a robot reading a boring book, or I fly off the handle and go off task. Do you have any advice for how to write your dialog? It always flows so nicely and I'd hope to be able to replicate some of that. Shel
Shel Hammer
2017-06-03 19:26:15 +0000 UTCGreat point Burt! I think I may have just gotten a bit lazy when inserting that one :)
3blue1brown
2017-05-30 05:09:40 +0000 UTCHi Bill, I did make a few videos on the Jacobian Matrix for Khan Academy. They are khan-style, not 3b1b style, but you may be interested in looking at them nevertheless. I'll keep your other suggestions in mind as I plan future content.
3blue1brown
2017-05-30 05:09:21 +0000 UTCDear Grant, I can't express the depth of my gratitude to you for the educational videos you have produced. Thank you so very much! If I may suggest 3 areas of mathmatics that are of interest to me and would appreciate your description and explanation of how they can be applied to fields and possibly multiple dimensionality analysis. They are, The Jacobian matrix The Kronker delta The Mandelbrot equation I have brain damage due to MS, however I'm still very capable if grasping visual geometric shapes. Tensors, vectors, scalars, and gradients. Your application of the visuals is out standing for me when watching your lectures. If you could display a 3d blended into a 4d visualization it would be fantastic to see. I know it's asking a lot. I'm studying Mathmatics and physics both classical and quantum to fight the MS. I'm hopping there's truth to Neuro plasticity if the brain. Again thank you and cheers, William G Russell wgr@lightspeed.net PS I look forward to watching your future lectures and re-watching all if the multiple times to fully grasp the concepts your teaching!
Bill Russell
2017-05-27 19:02:23 +0000 UTCTowards the end, when you're talking about circle geometry and how if a given angle is theta, then the angle between the points at any other point on the circle is 2 theta, would you animate that idea instead of leaving it static? Mebbe track theta as the point varies along the track of the circle? I think the concept would be more clear. (I think I get what you're saying but I'm not sure.)
Burt Humburg
2017-05-27 18:19:48 +0000 UTCGreat thought! There is indeed some great fodder for a video around these ideas. Often these sorts of visuals touch math in completely surprising ways, as how represnetations of the j-function include circle diagrams akin to this.
3blue1brown
2017-05-26 21:58:01 +0000 UTC<a href="https://www.wolframalpha.com/input/?i=(3+*+sum(10%5Ex,x,0,2n%2B1))%5E2+%2B+(4*10%5E(n%2B2)+*+sum(10%5Ei,i,0,n-1)+%2B+500*sum(10%5Ei,i,0,n-1)%2B56)+%5E+2+-+(5*10%5E(n%2B2)+*+sum(10%5Ei,i,0,n-1)+%2B+6+*+10%5E(n%2B1)+%2B+40+*+sum(10%5Ei,i,0,n-1)+%2B+5)%5E2" rel="nofollow noopener" target="_blank">https://www.wolframalpha.com/input/?i=(3+*+sum(10%5Ex,x,0,2n%2B1))%5E2+%2B+(4*10%5E(n%2B2)+*+sum(10%5Ei,i,0,n-1)+%2B+500*sum(10%5Ei,i,0,n-1)%2B56)+%5E+2+-+(5*10%5E(n%2B2)+*+sum(10%5Ei,i,0,n-1)+%2B+6+*+10%5E(n%2B1)+%2B+40+*+sum(10%5Ei,i,0,n-1)+%2B+5)%5E2</a>
john kraemer
2017-05-26 21:06:04 +0000 UTC1:19 -- "What's you're favorite proof?" ruh-roh
faisal
2017-05-26 21:03:57 +0000 UTCPlaying around with this I noticed a strange pattern that I hadn't encountered before. In general '3'*(2n+2), '4'*n+'5'*n+'56', '5'*n+'6'+'4'*n+'5' is a Pythagorean triple, so for instance, (333333, 445556, 556445), or (333333333333, 444445555556, 555556444445), or (33, 56, 65). Has anyone else encountered this before?
john kraemer
2017-05-26 20:15:18 +0000 UTCOne other fun idea: Grant, do you know about Ford circles and the Apollonian gasket? Well, if you take the Ford circles along a vertical line x = 1, i.e. with unit-diameter circles centered at x=1.5, y = n for any integer n, and then other rational points along the line y=1 represented by smaller circles, then you can invert all of those circles across the unit circle, and you’ll end up filling the circle x^2 + y^2 = x with circles that touch every rational point on it, and also all have rational diameters. It’s fun to see how the stereographic projection between a circle C and a line L can be represented either as a straight-line projection through a point on the circle, or equivalently can be represented as a circle inversion through a new circle touching circle C at one point. More generally you can find that every Apollonian gasket is a Möbius transformation of every other Apollonian gasket, and the Ford circles represent part of the gasket between two lines (the part touching one of the lines).
Jacob Rus
2017-05-26 18:43:49 +0000 UTC
