Putting Algebraic Curves in Perspective

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  • เผยแพร่เมื่อ 25 ก.ย. 2024
  • Ever wonder what happens when you combine graphing algebraic curves with drawing in perspective? The result uncovers some beautiful relationships between seemingly different shapes, and all because of what happens when you include infinity through projective geometry.
    This video was a project for MA 721 - Projective Geometry, as part of the Master of Science program in Mathematics at Emporia State University.
    Special thanks to Kevin Turner for assisting with post-production!
    References:
    * Ash, Avner, and Robert Gross. Elliptic Tales: Curves, Counting, and Number Theory. Princeton, NJ: Princeton University Press, 2014.
    * Brady, Zarathustra Elezar. “Cross Ratios.” MIT Mathematics. Accessed November 24, 2019. math.mit.edu/~....
    * Coxeter, H. S. M. Projective Geometry. New York: Springer, 2003.
    “Cross Ratio.” from Wolfram MathWorld. Accessed November 24, 2019. mathworld.wolfr....
    * Hisel, Jordan. “Addition Law on Elliptic Curves." 2014.
    * Leykin, Anton. “Systems of Polynomial Equations.” Lecture notes from MATH 4803: Introduction to Algebraic Computation. Accessed November 24, 2019. people.math.gat....
    * “Projectively Extended Real Numbers.” from Wolfram MathWorld. Accessed November 24, 2019. mathworld.wolfr....
    Image credits:
    * Albert Durer - Public Domain
    * Charles Rex Arbogast/AP, CC BY 2.0, www.flickr.com...
    * Claudio Rocchini - Own work, CC BY-SA 3.0, commons.wikime...
    * Hans Vredeman de Vries - Public Domain
    * www.cashadvance...
    * Mikael Hvidtfeldt Christensen - Own work, CC BY 2.0, www.flickr.com...
    * new 1lluminati - Own work, CC BY 2.0, www.flickr.com...
    * Theon - Own work, CC BY-SA 3.0, commons.wikime...
    Music: DM Ashura vs. Enoch - Chaotic White

ความคิดเห็น • 429

  • @kidredglow2060
    @kidredglow2060 ปีที่แล้ว +372

    "wraps around at infinity", that BLEW MY MIND, and I think nothing in my life will ever blow my mind as much as that again.

    • @vindi167
      @vindi167 3 หลายเดือนก่อน +11

      i've accepted that the number line loops around as a fact for years now

    • @PluetoeInc.
      @PluetoeInc. 3 หลายเดือนก่อน +4

      @@vindi167 real question is , does that interfere logically with any other results ? if its not inconsistent i dont see a reason it is not to be acceptable

    • @w花b
      @w花b 3 หลายเดือนก่อน

      Chill

    • @PluetoeInc.
      @PluetoeInc. 3 หลายเดือนก่อน

      @@w花b cool*

    • @trufflefur
      @trufflefur 3 หลายเดือนก่อน +4

      @@vindi167 Same, since I was in middle school I got convinced myself that the numerical plane is actually a sphere

  • @auroraaustralis7
    @auroraaustralis7 3 หลายเดือนก่อน +140

    you can tell he did this all in one take by the breath, what a legend

    • @daveslamjam
      @daveslamjam 2 หลายเดือนก่อน +8

      or alternatively he just didn't bother to edit out the breathing like other creators do

    • @LordFishTheSecond
      @LordFishTheSecond 2 หลายเดือนก่อน +6

      aw dude now i can't stop focusing on the breathing

    • @user-fp5qd7fi6w
      @user-fp5qd7fi6w หลายเดือนก่อน +2

      I didn't even notice his breath until I read this

  • @danielsteel5251
    @danielsteel5251 4 ปีที่แล้ว +1679

    This is what the Internet is for.

    • @williamchamberlain2263
      @williamchamberlain2263 ปีที่แล้ว +60

      Well, this and sending missile launch codes between silos.

    • @El_Pendejo_De_Si_Mismo
      @El_Pendejo_De_Si_Mismo ปีที่แล้ว +32

      ​@@williamchamberlain2263 the main task of the internet was to see cats from anywhere on the world

    • @andrewsemenenko8826
      @andrewsemenenko8826 ปีที่แล้ว +7

      ​@@williamchamberlain2263 oh wow, this video got recommended to everyone I guess🌟

    • @phixidelta
      @phixidelta ปีที่แล้ว +11

      What about cat memes 😔

    • @eboone
      @eboone ปีที่แล้ว +5

      @@williamchamberlain2263 boring

  • @theflaggeddragon9472
    @theflaggeddragon9472 ปีที่แล้ว +763

    As a student of arithmetic geometry, this is one of the best videos on algebraic geometry on YT, especially as an introduction. This is criminally underrated.

    • @azzteke
      @azzteke ปีที่แล้ว +2

      "criminally underrated." By whom please??

    • @luker.6967
      @luker.6967 ปีที่แล้ว +16

      @@azzteke No need to be pedantic, he clearly means this should have more views.

    • @tomdownes1g
      @tomdownes1g 8 หลายเดือนก่อน

      I hear prison food is pretty good...
      So no "Like" from me! LOL

  • @DankePrime
    @DankePrime 2 หลายเดือนก่อน +30

    Bro, math is so much cooler than people give it credit for 😢

  • @thezipcreator
    @thezipcreator ปีที่แล้ว +118

    I've heard the phrase "Point at Infinity" so many times before in math, but this video made me finally understand what exactly it meant

  • @bw0n6
    @bw0n6 4 ปีที่แล้ว +464

    I remember combing TH-cam for videos on Projective Geometry a few months back and wishing there was a good introductory single video. Now there is a great one! 10/10.

    • @inkdfist3702
      @inkdfist3702 ปีที่แล้ว +2

      Yeah the same thing happened to me!! Finally got a clear grasp on homogenous coordinates :D

    • @bw0n6
      @bw0n6 ปีที่แล้ว +1

      @AP.17 It means I was watching many videos and trying to find a good one. The use of "combing" refers to running a comb through hair to try to find something within.

  • @jojojorisjhjosef
    @jojojorisjhjosef 2 หลายเดือนก่อน +6

    This man makes me realize I had an intuition for infinity.

  • @julianlastname5730
    @julianlastname5730 ปีที่แล้ว +68

    A parabola stretched to infinity being an ellipse is so cool to me, because in my dynamics class we have been studying orbits, and they have four shapes: circle, ellipse, parabola, and hyperbola, in different perspectives, these are all ellipses!

    • @lookupverazhou8599
      @lookupverazhou8599 ปีที่แล้ว +12

      Honestly, ellipses are the key to the future of science and mathematics, just as circles and triangles have been for millenia.

    • @galacticdragon9841
      @galacticdragon9841 3 หลายเดือนก่อน +5

      there is a great reason for learning conic sections in pre-calculus, applies to way more than people would think.

    • @angrykirby_yt
      @angrykirby_yt 3 หลายเดือนก่อน

      ​@@lookupverazhou8599 Please, can you elaborate more about ?

  • @NonTwinBrothers
    @NonTwinBrothers ปีที่แล้ว +406

    17:03 I'm surprised you didn't mention the best part!
    In computer graphics when you create a function which projects 3D space down to a plane, you divide by the Z component of the camera's vision. You never want Z to be negative, however if you allow that to happen anyways (i.e. not clipping the world behind you). Everything that isn't normally visible actually shows up ABOVE the horizon, and flipped 180°. For the case of the hyperbola, this means the rest of the ellipse image actually continues perfectly as expected, which is awesome! :)
    I had created a Desmos graph last year which demonstrated exactly that, unfortunately youtube has a field day when links are posted in the comments so I cant share right now, oh well

    • @Bankosek
      @Bankosek ปีที่แล้ว +13

      Hey i'm curious to see the video, do you think you could upload it to your channel or something like that?

    • @NonTwinBrothers
      @NonTwinBrothers ปีที่แล้ว +29

      @@Bankosek I added the link to my channel description. Have fun!

    • @Bankosek
      @Bankosek ปีที่แล้ว +3

      @@NonTwinBrothers Thanks a lot!

    • @kylebowles9820
      @kylebowles9820 ปีที่แล้ว +7

      (this is the case in 3D) You usually divide by W, the homogenous coordinate, into "NDC coordinates" where Z is then used to write to the depth buffer. Depth testing wouldn't work if you were to divide by Z. Check out the full parameterization of a projection matrix for the rest of the info on how it works.

    • @fattestallenalive7148
      @fattestallenalive7148 ปีที่แล้ว +6

      You can break up the link with spaces to still post it
      I do want to look at it
      (Never mind just saw your channel but in case of future cases you can try breaking the link up
      Btw the desmos graph is AWESOME

  • @NoNTr1v1aL
    @NoNTr1v1aL 2 ปีที่แล้ว +197

    Your videos inspired me to pursue higher mathematics back in 2016. I just finished my MSc degree in math. I thank you from the bottom of my heart.

  • @zanityedpo1398
    @zanityedpo1398 2 หลายเดือนก่อน +8

    I always thought 0÷0 should be ANYTHING other than nothing, but my math teachers always told me, "You just can't divide by 0, shut up." So later in the year, when she brought in a REAL mathematician, SHE said, "You can't divide by zero because zero is invalid on the bottom of a fraction, shut up."
    Hearing you say that 0÷0 is just *not meaningful* as it can represent any number fills me with such glee.
    I don't care that I was wrong thinking 0÷0=1 now, because you treated the question seriously and gave me an answer that doesn't treat 0 ÷ 0 = "ERROR" as dogma

  • @robertmatthews6532
    @robertmatthews6532 ปีที่แล้ว +29

    As an Art teacher, I taught this to my students, except I humanized it by usi g the 60° Cone of Vision, to find the intersections in the first instance. Did it work? Surprisingly well!
    Robert.

    • @robertmatthews6532
      @robertmatthews6532 ปีที่แล้ว +7

      This was during the 1980's.

    • @elisalabarge3516
      @elisalabarge3516 ปีที่แล้ว +3

      As an art student I know what the cone of vision is but what is the first instance?

  • @reidflemingworldstoughestm1394
    @reidflemingworldstoughestm1394 3 หลายเดือนก่อน +12

    New and exciting ways to confuse flat earthers.

  • @antonseoane1127
    @antonseoane1127 ปีที่แล้ว +76

    How pretty... Done my thesis on projective geometry also and this guy has made an incredible good and easy to understand explanation of this beautiful field of mathematics

  • @Waffle_6
    @Waffle_6 9 หลายเดือนก่อน +8

    projective algebra is the coolest thing ive ever seen

  • @incommunicado_o
    @incommunicado_o 4 ปีที่แล้ว +52

    I watched your introductory series on maths for the first time 5 years ago and I recently just graduated, I can say your videos gave me the insight I needed to get here in the first place, thank you for dedication, I'm especially glad to see a new video coming out!

  • @camrouxbg
    @camrouxbg ปีที่แล้ว +62

    I took a course on projective geometry, but we never made it as far as homogeneous coordinates and there wasn't a lot of perspective (pun intended) on how to view these things or how it all comes together. It really was one of my favourite geometry courses (we covered inversive geometry as well), but this video really helped to put a neat little bow on it. Thanks so much, and please keep up the great work!

    • @MultiAndAnd
      @MultiAndAnd ปีที่แล้ว +1

      It's impossible to do projective geometry without homogenous coordinates. You cannot do computations.

    • @camrouxbg
      @camrouxbg ปีที่แล้ว +2

      @@MultiAndAnd actually there is a lot you can do with just the cross ratio. This course was proof-oriented rather than computation.

    • @MultiAndAnd
      @MultiAndAnd ปีที่แล้ว

      @@camrouxbg basically on on the prjojective line then... Not that interesting in my opinion.

    • @camrouxbg
      @camrouxbg ปีที่แล้ว +1

      @Andrea Merlo suit yourself, but you're definitely missing some of the beautiful stuff. But hey, if it's beneath you then who am I to say otherwise.

    • @MultiAndAnd
      @MultiAndAnd ปีที่แล้ว

      @@camrouxbg projective geometry is very basic stuff and not such a deep idea. Imho is not that interesting. Especially if you limit yourself to biratios. It’s mathematics for the elementary school.

  • @luker.6967
    @luker.6967 ปีที่แล้ว +7

    Last quarter of this video was legitimately mind-blowing. Thanks for inspiring me and no doubt many others!

  • @RomanNumural9
    @RomanNumural9 ปีที่แล้ว +11

    Captivating video! I'm a PhD student in math finance which is a heavily applied field, but I found this deeply interesting!

  • @MusicEngineeer
    @MusicEngineeer ปีที่แล้ว +11

    Very interesting. I like this "math for artists" stuff a lot. Thank you!

  • @UteChewb
    @UteChewb 4 ปีที่แล้ว +15

    Outstanding. I remember loving projective geometry when a student eons ago but this is several levels above and even more awesome. Well done.

  • @pascaljosiah6866
    @pascaljosiah6866 3 ปีที่แล้ว +16

    Loved your videos a few years back. I am now looking to do a PhD in pure mathematics.
    Thanks for the videos!

  • @weasel6843
    @weasel6843 ปีที่แล้ว +2

    18:33 hahahahah
    also omg u made the music at the beginning love this so much

  • @SOBIESKI_freedom
    @SOBIESKI_freedom 4 ปีที่แล้ว +7

    Beautiful.
    MORE!!

  • @pawelsubko7277
    @pawelsubko7277 2 ปีที่แล้ว +13

    This is a ridiculously good video. Just crazy.

  • @ioannismalamos8842
    @ioannismalamos8842 2 ปีที่แล้ว +14

    Great video. Most probably the best I ve watched in projective geometry!

    • @General12th
      @General12th ปีที่แล้ว +5

      It really let me... see things from a new angle!

  • @mystifiedoni377
    @mystifiedoni377 ปีที่แล้ว +2

    I can't believe it took me this long to stumble on this video. I was learning how to draw and found perspective very interesting. Projective Geometry was exactly what I was looking for.

  • @Alex-jk2qy
    @Alex-jk2qy 9 วันที่ผ่านมา

    Please do more! This was an incredible experience, what a great video and great explanation!

  • @ceremonious_houseplant
    @ceremonious_houseplant ปีที่แล้ว +3

    This video is severely underrated. Incredibly well done explanation!

  • @OndrejPopp
    @OndrejPopp 4 ปีที่แล้ว +5

    Bill is back! Welcome back mr Shillito!

  • @ts-wo6pp
    @ts-wo6pp หลายเดือนก่อน

    Great clarity, great visualizations, great everything

  • @stkhan1945
    @stkhan1945 3 หลายเดือนก่อน

    ... this is so conceptually rich.. infinitely beautiful ..the horizon, the curve wrap at infinity, the perspective/projective dimension.... just amazing ..thnq

  • @henrycoxd450
    @henrycoxd450 8 หลายเดือนก่อน +1

    This is the best video on that topic so far, your visualizations are extremely helpful

  • @eugenetheant
    @eugenetheant หลายเดือนก่อน

    Wow! Never thought of the conic sections from this *perspective*! 😮
    Thanks!

  • @inscitia
    @inscitia 16 วันที่ผ่านมา

    This presentation is a work of art.

  • @SHPfz
    @SHPfz ปีที่แล้ว +1

    I have been thinking about mathematics, specifically graphs, just like this (primarily: "There is only one unsigned infinity") all through my school life. Now, with everything put together, really hit a sweet spot...

  • @PushingFriend28
    @PushingFriend28 2 หลายเดือนก่อน

    Very fascinating. Especially the line at infinity

  • @Cuerdoylisto
    @Cuerdoylisto ปีที่แล้ว +2

    Estoy sorprendido. . .gracias por su magnífico aporte en la transición del espacio proyectivos relajado al espacio euclidiano. . .

  • @mikono2022
    @mikono2022 ปีที่แล้ว +1

    excellent production, introduction, deduction, induction, tiontion

  • @bettercalldelta
    @bettercalldelta ปีที่แล้ว +11

    imagining a point that is infinitely far away from everything is already hard but now i also gotta imagine how it wraps around to the other side

  • @Cblizy
    @Cblizy 4 ปีที่แล้ว +3

    Some of my favorite music from my child hood was created by a math teacher...Astounded.

  • @amydebuitleir
    @amydebuitleir 4 ปีที่แล้ว +7

    Oh wow oh wow oh wow! I had given up hope that you were going to do more videos. So glad to see that I'm wrong. You have such a clear way of explaining things.

    • @BillShillito
      @BillShillito  4 ปีที่แล้ว +7

      Thank you, that means a lot! I really have been wanting to get back into making these videos, but they take quite a while to produce. I need to come up with a better process...

    • @amydebuitleir
      @amydebuitleir 4 ปีที่แล้ว +3

      @@BillShillito The polished graphics and animations are lovely, but in my opinion what's special about your videos is that you present things so clearly. I'd happily watch videos of you lecturing at a whiteboard, especially if it would mean we could get more videos. (N J Wildberger has been doing exactly that for years, and his videos are quite popular.) But that's just my suggestion. If having a high production quality makes the process more rewarding for *you*, I'll be patient. At least, I'll try to be patient. ;^)

    • @OndrejPopp
      @OndrejPopp 4 ปีที่แล้ว

      @@amydebuitleir Now, how did you like the infinity talk of mr Norman Wildberger? Personally I think mr Wildberger is spreading false teachings confusing especially the minds of children. So it would be cool if there would be a field of mathematics to prove him wrong, especially for the sake of these children. The problem with mr Wildberger is in my opinion that he appears to me like a modern mr Pythagoras who did not believe in irrational numbers like the square root of 2, or infinity all by itself.. He also claims that the rules of arithmetic break down with big numbers so you can not be certain about anything over there, yet he is clever enough to hide just beyond the reach of the modern calculator, so the kids have no way to prove him wrong with their calculators. Yet what's even more freaky that there is a whole bunch of people that agree with him and they call themselves "purists", just like with the Pythagorean sect... And so you can discuss infinity or the sqare root of 2 endlessly over there... While no concensus whatsoever ever comes out of this. At one time I thought I was going to write him a letter, but I never did, because I was getting this impression that mr Wildberger is not confused nor mentally sick at all, but that he is doing all this on purpose to have a false scientific island for himself where he can enjoy infinite glory while he is hiding with his examples just out of reach of a hand calculator, confusing children. In my opinion. I filed a complaint about him at the University in Australia that he was affiliated with, but no one responded. And so I have not listened to mr Wildberger in years, but now we have seen infinity in this lecture, maybe we can use this to make mr Wildberger stop confusing children, if that is what he is still doing. Still, for me, the most freaky part is all these followers of mr Wildberger, who actually appear to agree with him, maybe for their own glory, but as you may have guessed this already, I am not one of those. So Ok, putting infinity into perspective that would really be something that needs to be done over there in Australia, that is, if this in my opinion, craziness of mr Wildberger is still going on, especially when it confuses children with examples just beyond the reach of their calculators....

    • @amydebuitleir
      @amydebuitleir 4 ปีที่แล้ว +1

      @@OndrejPopp I have only watched a few dozen of his videos, so I may not have seen the same ones you did. I was initially a bit concerned by his opinions on infinity and irrational numbers. However, in the videos I watched he made it very clear whenever his view differed from the consensus, and why, so I personally didn't feel misled. I only watched the videos on more advanced topics such as algebraic topology, and I felt that students at that level can benefit from considering alternative views as long as they are labelled as such. His absolute insistence on rigor and clarity leads him to reject real numbers, but those same qualities make him good at explaining complex ideas. I didn't watch any of his videos for children, so I can't comment on how he presents his ideas to them.

    • @OndrejPopp
      @OndrejPopp 4 ปีที่แล้ว

      @@amydebuitleir Hi Amy. There is no problem expressing alternative views, as long as they are valid ofcourse... Because mathematics is not a religion or is it? And that's kind of an issue here. My concern about the children is that some of mr Norman Wildberger's followers are school teachers, so teachers who admire his ideas, and so mr Wildberger's alternative views may creep into the heads of children in this way. Anyway it is an endless discussion, but the best indication I got is that mr Wildberger's examples are just a little bit out of reach for a calculator... I don't know if you ever saw that one, this pyramid number 10 to the power of 10 to the power of 10 ... to the power of 10 and so on, and apparently so claims mr Wildberger, these pyramid numbers are so big that normal rules of arithmetic do not apply.. And you can not calculate them either because they do not fit into a calculator and some school teachers love this.... and are discussing the possibilities how to introduce this in schools and to the kids...

  • @vaevfunc
    @vaevfunc ปีที่แล้ว +1

    I rarely write comments but it hit me well!
    Back in school I finished art classes and perspective was always something intuitive but I tried to describe it mathematically. I ended up with massive formulas for even simple things and now it turns out mathematicians created a more convenient language for that.
    It would be great if you reveal RP3 (which as I understand represents how we see the world in 3D). For example, imagine you have a cube in 3D perspective. How to find the coordinates of an inscribed sphere? In usual geometry we just say that the sphere intersects the cube at middlepoints of cube surfaces, however in perspective it is not the case.
    Anyway, thank you!

  • @savonliquide7677
    @savonliquide7677 ปีที่แล้ว +2

    Very nice !! especially the choice of the order on witch concepts are intruduced! The principles notion and subtilities are presented with clarity , pedagogy in a rigourous way. The beginer might have to use the "pause" bouton quite a lot, in order to get the worth of this lesson, but this will be a great benefit because this is not vulagarisation but real maths😍

  • @lyricalcarpenter
    @lyricalcarpenter 3 หลายเดือนก่อน +1

    The Fermat’s Last Theorem cameo hit me like a ton of bricks

  • @ardenspiro1436
    @ardenspiro1436 2 หลายเดือนก่อน

    I spent an entire week puzzled imagining the perspective and how it applies but my teacher just legit wouldnt grasp what i was saying. It's all just a round (circular for 2d) curve.
    Amazing video I feel less alone in my head.

  • @violetsweet1660
    @violetsweet1660 4 หลายเดือนก่อน

    This video has made so many things so much clearer to me.

  • @Galva94a
    @Galva94a ปีที่แล้ว

    This video is gold! Having studied this stuff some years ago at uni, I was able to recover all the lost knowledge in just 20 minutes!

  • @shjescaresme
    @shjescaresme ปีที่แล้ว +1

    What a perfect introduction for this subject! Thank you.

  • @choty7066
    @choty7066 ปีที่แล้ว +1

    Its 3 am and my brain is truning into a fine mushy paste as the cycle of trying to comperhend the contents of the video and failing repeats itself every second

  • @ninjanolan6328
    @ninjanolan6328 ปีที่แล้ว +2

    This is amazing. I had no idea such interesting mathematics existed

  • @henrycgs
    @henrycgs ปีที่แล้ว

    wow! that's so cool! this is the most intuitive and "clean" way to deal with infinity as a number I've ever seen!

  • @R2242V
    @R2242V ปีที่แล้ว +2

    Wow, is this some kind of apocryphal and forbidden part of mathematics? It somehow reminds me of what our teacher told us about x and y axes that those are not lines but circles with an infinity radius. No one from 150 attending students cared about that fact except me with my friend who just laughed about it like it was joke. Teacher didn't explain it further maybe because we were mere future engineers.

  • @the54thburgemeister63
    @the54thburgemeister63 ปีที่แล้ว +5

    i’m so high this is literally so real

  • @goki6548
    @goki6548 3 หลายเดือนก่อน

    I always thought about how to properly prove the foci of a parabola (for orbital mechanic purposes). I know the equation, it is simple, but I just couldn't get my head around the "other foci being at the infinite point". This video has showed me how that works and I feel GREAT now. Thanks!

  • @wyboo2019
    @wyboo2019 11 หลายเดือนก่อน +1

    my favorite way of doing projective geometry is with projective geometric algebra

  • @joyboricua3721
    @joyboricua3721 ปีที่แล้ว

    I find Bill's explanations quite soothing. Great jpb!

  • @infinitelyexplosive4131
    @infinitelyexplosive4131 ปีที่แล้ว +1

    Wonderful animations, thank you! Helpful for understanding elliptic curves

  • @Andre-Linoge
    @Andre-Linoge หลายเดือนก่อน

    Outstanding! Great teacher !

  • @siddiquiikram5673
    @siddiquiikram5673 4 ปีที่แล้ว

    Please continue this you are real mathemacian and math teacher

  • @phoriiak2835
    @phoriiak2835 ปีที่แล้ว

    dayum im not even in high school but you explained this in such a way that it piques my interest AND is able to make me understand most of it

  • @boas_
    @boas_ 3 หลายเดือนก่อน +1

    Great video!

  • @zucc4764
    @zucc4764 ปีที่แล้ว

    this is so good, I've always wondered how logarithmic scales could be constructed, and now I have my answer

  • @iustinraznic5811
    @iustinraznic5811 ปีที่แล้ว +1

    One of the best explanations I've ever seen for Algebraic Geometry and the other branches touched in this video. Congrats!

  • @prod.winterxphool6227
    @prod.winterxphool6227 ปีที่แล้ว +1

    Wow that was crazy to learn about, so excited to become a mathematician

  • @runakovacs4759
    @runakovacs4759 ปีที่แล้ว

    Your enunciation is very clear and easy to focus on, keep my attention. Subscribed!

  • @alphonsobutlakiv789
    @alphonsobutlakiv789 ปีที่แล้ว +1

    This is the first time I've see someone use the set up I use in my work. Though I manly used it for art, it was really to understand the nature of distortion. Have also considered the horizon as infinity, but I also used localized infinity in light projections, and if you use this to generate a copy of the whole picture, in the picture in perspective in perspective, you can find infinity with the new finite infinity perseved in the images image of its self in it'd self. Also, was looking into how to transfer the governing lines of infinity to localized light infinity without disturbing an object, hard to explain what I mean by that part.

  • @saulijamsa6165
    @saulijamsa6165 4 ปีที่แล้ว +2

    Greatly interesting! But: The parabola looks like an ellipse when you look at it from a point y0?

  • @mk-nw4si
    @mk-nw4si 4 ปีที่แล้ว

    the great return after few years

  • @atlas_of_prescottia
    @atlas_of_prescottia ปีที่แล้ว

    I'll admit, I struggled to follow a lot of this, but the reveal at 17:12 was so satisfying!

  • @lucasvicentim3650
    @lucasvicentim3650 หลายเดือนก่อน

    This is a beautiful video.

  • @nicholasstrauss
    @nicholasstrauss ปีที่แล้ว +1

    thank you 🙂

  • @AscendedOnOsu
    @AscendedOnOsu ปีที่แล้ว

    this came up on my recommended a while ago, and i didn't even know you were the musician that made deltaMAX; love your works

  • @petrouchkita0000
    @petrouchkita0000 5 วันที่ผ่านมา

    Fascinating

  • @Rawi888
    @Rawi888 ปีที่แล้ว +1

    Just being blazed going “wtffff” downloaded some data into the brain stem. I’m not to sure what I learned, but I sure learned it.

  • @DanielVCOliveira
    @DanielVCOliveira ปีที่แล้ว +3

    Does the wrapping around at infinity explain why Riemann's zeta function with x = -1 (the sum of all positive integers) results in a negative number?

    • @theflaggeddragon9472
      @theflaggeddragon9472 ปีที่แล้ว

      It does not (afaik). The analytic continuation of zeta functions is quite mysterious and is related to modularity theorems in the Langlands program.

  • @Bottle_O_Glue
    @Bottle_O_Glue ปีที่แล้ว

    I have absolutely no clue what is happening, and I am here for it

  • @danielbriggs991
    @danielbriggs991 3 หลายเดือนก่อน

    This was strange. Just 20 minutes ago, I was imagining making a video where a curve that looks like a parabola up close is actually an ellipse when you zoom out. And then I get recommended this!

  • @pichass9337
    @pichass9337 4 ปีที่แล้ว +6

    WAIT ARE YOU DM ASHURA?

    • @BillShillito
      @BillShillito  4 ปีที่แล้ว +5

      Yes. :P But I'm mostly a math teacher!

    • @pichass9337
      @pichass9337 4 ปีที่แล้ว +4

      @@BillShillito OH MY GOD. YOUR LECTURES AND MUSIC ARE AMAZING

  • @physira7551
    @physira7551 2 ปีที่แล้ว

    This is so so satisfying to generalize FTA even futher!

  • @justinting1422
    @justinting1422 4 ปีที่แล้ว +2

    This is awesome!

  • @minecrafting_il
    @minecrafting_il ปีที่แล้ว +6

    The inclusion of complex numbers is like someone losing at an argument:
    "We can clearly see that the circle doesn't intersect infinity"
    "Great argument! However, *6-dimesional space* "

    • @metachirality
      @metachirality 6 หลายเดือนก่อน

      It's more like concluding that grass doesn't exist because there is none in your room, not considering that it is more useful and enlightening to consider the outside world.

    • @tone618
      @tone618 4 หลายเดือนก่อน

      Throws punch. Great effort however expands spacetime.

  • @markupnone733
    @markupnone733 3 หลายเดือนก่อน

    Excellent.

  • @KenNguyen-xl9bj
    @KenNguyen-xl9bj 4 ปีที่แล้ว

    It is nice to see you are posting videos again. I’m an alumni from your GHP game theory class. i always enjoyed your teachings. :)

  • @vindi167
    @vindi167 3 หลายเดือนก่อน

    11:11 YES. finally someone said it
    -∞ and +∞ are the *same*. everything works the same at that. one graph that shows this very well is y=1/x, where both go to ∞. it looks like different directions, but really the number line just loops at infinity.

    • @Ms-xq6jx
      @Ms-xq6jx 3 หลายเดือนก่อน +1

      Not really

  • @PoorDog69
    @PoorDog69 3 ปีที่แล้ว +3

    Hey, are you DM Ashura? From making electronic music? I heard your remix MaxX ResurrexXion and I wish Konami would've put it as their boss song in DDR.

  • @toddn1128
    @toddn1128 2 หลายเดือนก่อน

    How is this free? This was an incredible watch!

  • @kruksog
    @kruksog 3 หลายเดือนก่อน

    Fantastic video.

  • @themfu
    @themfu ปีที่แล้ว

    What a great exposition! Well done!

  • @Gunslinger-us1ek
    @Gunslinger-us1ek 2 วันที่ผ่านมา

    that was amazing

  • @Sr.Estroncio38
    @Sr.Estroncio38 หลายเดือนก่อน

    What an amazing video, you are awesome

  • @assassinosoldato92
    @assassinosoldato92 ปีที่แล้ว

    hats off to this marvellous video

  • @korigamik
    @korigamik ปีที่แล้ว +3

    This is amazing! Masi I ask what do you use for there equation morphing animations and graphics? Is the source code for video open?

    • @BillShillito
      @BillShillito  ปีที่แล้ว +5

      I used PowerPoint for all the animations in this video. 😅 It took some work to get it to do the things it did, but I've sorta learned to push PPT to its limits. (I DO wish it would update its animations though - a few changes could make it radically better for animation.)
      By the way, I have to ask, did somebody post this video somewhere or something? All of a sudden I'm getting a bunch of comments all at once.

    • @korigamik
      @korigamik ปีที่แล้ว +4

      @@BillShillito haha thank you for answering. But no, I guess TH-cam just decided to recommend this to everyone

  • @didan470
    @didan470 3 หลายเดือนก่อน

    astonishing

  • @kh4y3m
    @kh4y3m ปีที่แล้ว

    Awesome presentation! The link with art, love this part. Thanks

  • @NoisqueVoaProduction
    @NoisqueVoaProduction 3 หลายเดือนก่อน

    The fact that Parabolas are just ellipses stretched to infinity may not be as a surprise if we remember about the conic sections.
    Circles are when the plane cutting it is paralel to the base. Ellipses are when they are oblique, but not parallel to the side, and parabolas are when the plane is parallel to the side of the cone, so it is the first ellipse that "couldn't find" the other plane to close on itself.
    Hyperbolas are when the plane is orthogonal to the base.
    I didn't know hyperbolas were parabolas in projective geommetry, but it makes sense, since they are orthogonal, they would only be a parabola in the extreme case where the angle of the cone is orthogonal, but in that case it wouldn't be a cone, but a cylinder(in which, in projective geommetry, it would be a cone at infinity). Pretty cool!

  • @kylebowles9820
    @kylebowles9820 ปีที่แล้ว

    Nice presentation, clear and well done. Love to see what other people do with math. I use these for computer vision and graphics

  • @angeldude101
    @angeldude101 2 ปีที่แล้ว

    Lines in 3D space have their own representation similar to homogenous coordinates capable of representing arbitrary lines in space as well as lines at infinity, which are called Plücker coordinates. They can be represented as a pair of 3D vectors with a direction and a "moment," though lines at infinity would have (0,0,0) for the moment if you were just using vectors. With the right product, it's actually possible to join any two points or meet any two planes two get the Plücker coordinates for their intersection, even if the planes are parallel or if one of the points is at infinity. This can actually lead to finding algebraic representations for intersections of practically arbitrary curves if you have enough components, including representing imaginary roots or non-intersecting curves.

  • @mahdiasad1353
    @mahdiasad1353 2 ปีที่แล้ว +2

    I was having a problem drawing curved roads in perspective and figureing out the scales of objects....but......................................................seems like it's not just me, math got problems too🙂🙂🙂🙂🙂