Duality: magic in simple geometry

[@elijahberegovsky8957]

Wow! Why have I never heard of this??? When I saw the thumbnail I thought you’d talk about inversion, which is amazing, but have been explained to the ground on KZbin. I was not prepared to see a different transform, completely unfamiliar to me. Kudos!

[@otta3680]

I really loved this video. So beautiful and perfect with the music. I can’t believe that this video is not more recognized. Common algorithm do your thing

[@radek2483]

Hi, thanks for showing duality using animations. I have few things that I disliked and few that I liked a lot. I disliked the introduction to duality, as I believe it's better to introduce it without any circle being involved, as then there is nothing left to prove. But what I really loved was the animation of "degrees of freedom". Like when we know the polar is a line passing through some point, you first make it move and then when we know some other properties, they fix it in place :-) Radek (I am the author of the handout you have in description)

[@andpoincaresaid9011]

Wow, hi! Your handout is super well organized and was a pleasure to see! It's a pity I only found it when the video was already finished. I was attracted by the purely geometric definition of the polar and by the beauty of the fact itself, though in retrospective it would've been indeed easier to introduce it using the dot product

[@thebees955]

What a beautiful re-imagining of Thales' theorem!

[@eugeneprimak2572]

Wow, I really became twice as wiser!

[@jabbahatt8082]

bro this is amazing

[@LumaSloth]

Damn, that's incredible. I didn't know there was a name for it!

[@SlavaTsarev]

i love the narration and animations - well done guys, keep it up (right up!)

[@SlavaTsarev]

kzbin.infoUgkxVSgbIB25jOABljvhl2dj9xrBfKztCUh1

[@birdbeakbeardneck3617]

i remever the first time watching a video about line point duality i was like wow

[@CognitiveOffense]

Syns and taxes, this is a great lesson! Thanks you for new understanding.

[@camrouxbg]

Thank you for this! I really enjoyed my projective geometry class, and this was a nice revisitation to it. I do love how you depicted and motivated desargue's theorem. It makes total sense thinking of it thay way.

[@spiderjerusalem4009]

any book recommendation?

[@Mathematica702]

You definitely need to make more videos. I imagine it was a lot of work, but this was an amazing contribution.

[@MarvinLoterte-qk3zu]

Thank youu, this is a big help for constructing duality in my finals.

[@thomasthrane4199]

Very nice video. Duality is definitely one of the reasons I love projective geometry. My profile picture is in fact a theorem about poles and polars: If you take a triangle ABC, and the triangle formed from the polars of the points A, B, C, then these triangles are perspectively related like in Desargue's theorem!

[@tedsmith9726]

Great video, and I really enjoyed some of the problems in it - tell me I'm not the only person who couldn't resist taking a crack at the 'you can draw a line connecting exactly two points etc.' problem.

[@alphalunamare]

Duality isn't simply two ways of looking at geometric problems. It is fundamental to scientific thought. Different formulsations and mathematical techniques allow problems to be attacked from different directions. Each to his own and perhaps success. Matrix mechanics or Schrodinger sort of thing. This beautiful presentation exposes the reality of different view points and drives home the benefit of being open minded and taking different Paths to a solution. Duality is nothing special, it is just looking at the same thing from different aspects. Something which modern day Physics abhors. I forget his name but someone showed that all expressions of String Theory are essentially the same i.e. Duals of each other. 50 Shades of bolluix :-)

[@discreet_boson]

Beautiful video!

[@algorithmgalactic6143]

Absolutely mind-blowing!

[@johnchessant3012]

Very nice video!

[@diegosuarez5331]

Maybe I am late to the party, but just wanted to say this is one of the best videos I have seen ever. The animations are flawless and intuitive (the use of colors made it 1000000 times easier), the examples are super interesting and their solutions stunning, and the topic itself is something I have never seen and is absolutely amazing! I cannot believe somethimg this beautiful is so unpopular in the geometry scene... had I known of this theorem, one of the problems in the IMO I went to would have deginetly been easier. Please keep on making content... it was fantastic!

[@fhlbadenhorst]

Thank you so much for taking the time to make this inspiring video

[@JohnDlugosz]

When I was doing geometry proofs in high school, it never occurred to me to bring a set of colored markers.

[@andrewmartin2321]

beautiful

[@WeiPan_article]

Dear Sir, thanks for the video. May I say that the dual relation bewteen two objetcs (eg. point-line) can be viewd in the way that these two objects belong to the same entity in a higher space?

[@gabitheancient7664]

:0 that's beautiful

[@skmaths-help]

Lovely video

[@lostinthebluecity]

No .. this can't be true.. understanding polar can never be this simpler.. .. 😄 a great video .. 👏

[@keishakwok4333]

Thnks for your insightful video! How do I use this directly to proofs? It's not like I could say "by duality the other diagram is true so this one is true"

[@andpoincaresaid9011]

Actually, that's pretty much what you do! The key here is that applying the polar transform twice brings you back to the original diagram. So, once you've proven that the dual statement is true, you simply apply the transform again and conclude that the primal statement was true as well

[@tgwehahsjdjdow9829]

Hey guys, do you have any tips for being good at geometry? Cuz I struggled against them but I am alright in other maths topics

[@damyankorena]

1:07 proof is trivial by pascal's theorem

[@ojas3464]

👍

[@jazzjohn2]

This is a nice video, but unfortunately the thin lines' colors are hard to see for people with even mild color blindness.

[@lordchickenhawk]

G'Day mate. Thanks for explaining about the polar aspect. I accidentally discovered something in my trade (fitting and turning) a few years back that is related to this. I was trying to solve a practical problem I had at the time when I discovered how to invert Thales theorem to my advantage. kzbin.info/www/bejne/l2a5p59nntllj7M I named this method "finding a circumcentre by Thales theorem", it's useful because a circumcentre or a PCD can be found with just a boilermaker's square on the work site

[@pragyanpranay3681]

wait, this is way too fast

[@shruggzdastr8-facedclown]

The diagrams seen in this video (employed as visual aids) are not helpful for any viewer who is colorblind.