The Calculus of Explanations

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@0megaSapphire 2 ай бұрын

I was studying this in my book on complex analysis. It was showing the relationships in terms of a circle centred at q. We have z inside the circle, and z-tilda is the geometric inversion which is outside the circle. Distance from q to z is |z - q|, and to z-tilda |z-tilda - q|. Radius of circle is R. |z - q|*|z-tilda - q| = R^2. It then proceeds to say that it follows that (z-tilda - q)*(z-bar - q-bar) = R^2. But I don't see how this follows at all. Someone help.

@0megaSapphire 2 ай бұрын

I figured it out! Let k = |z - q|. (z-tilda - q) = ((R^2)/k)*e^{i*theta} and (z-bar - q-bar) = k*e^{-i*theta), then (z-tilda - q)*(z-bar - q-bar) = ((R^2)/k)*e^{i*theta} * k*e^{-i*theta} = R^2.

@rockapedra1130 3 ай бұрын

Cool!

@erawanpencil 3 ай бұрын

@6:40, the image of those Pringle-shaped oblong circles as they undergo the inversion... is that just artistic effect or is that the shape circles actually assume 'during' a conformal transformation? I could be totally wrong but I thought getting skewed circle shapes like that happens during non-conformal skewing/translation actions... is that mathematically part of the process during the transform or just what the graphics show for non-mathematical effect? thanks

@LVCID777 4 ай бұрын

Now imagine a sphere. The inverse of a sphere may be what’s inside a black hole

@nsgoneape9899 4 ай бұрын

bro cooked and then dipped.

@thecalculusofexplanations 4 ай бұрын

🙏 more will come - just got busy with work and other projects. I’m not done with the channel yet!

@erawanpencil 5 ай бұрын

I'm kind of confused on the basics here... what type of 'space' (if that's the right term) does circle inversion take place in? Since the Pythagorean Theorem was used in your example, does that mean circle inversion is a process in regular old flat Euclidean space? I thought 'inversive geometry' (not clear if that's the same as circle inversion) had something to do with hyperbolic spaces, or projective geometry.

@thecalculusofexplanations 5 ай бұрын

That's fair, it's not intuitive I suppose - and you're right in that there are many different 'depths' to which you can understand things in maths. At the most basic level I understand inversion as a transformation taking points in R2 to other points in R2, just like rotating, scaling, skewing etc, but whereas those are linear, circle inversion in non-linear. If you watch Part 2, you'll see me talk about how circle inversion correlates precisely with the complex conformal (angle-preserving) map f(z) = 1/z, which is a special case of a "Mobius transformation" - here is the link to stereographic projections / projective geometry.

@p4rk756 5 ай бұрын

Really really good video on inversions!

@vahurpaist511 5 ай бұрын

Totally agree

@thecalculusofexplanations 5 ай бұрын

@p4rk756@@vahurpaist511 thank you both

@thecalculusofexplanations 5 ай бұрын

Thanks so much, if you enjoyed this make sure you check out part 2!

@windigo77 6 ай бұрын

i want more like this!! Amazing.

@thecalculusofexplanations 6 ай бұрын

Appreciate it!

@sirawatkhongnum5389 7 ай бұрын

I'm solving a problem that requires this theorem and need a reference book. Could you recommend a book that has the theorem in this video?

@thecalculusofexplanations 7 ай бұрын

I haven't read it, but it looks like this one would be a decent reference! en.wikipedia.org/wiki/Geometry_of_Complex_Numbers

@AryanKumar-vo1ic 7 ай бұрын

can you please explain spiral similarity?

@thecalculusofexplanations 7 ай бұрын

Possibly! I'll look into it as I'm not familiar

@rat_king- 8 ай бұрын

OK..... but what if a circle has the origin of the inversion circle with its perimeter, or even has the same origin? Isn't that just magnitude... yet what is the area? and what happens when i change the radius?

@7trdcc110 8 ай бұрын

That was actually very jntestering. Legitimately got me drawing type stuff .Hope u make more vids in the future 👍

@thecalculusofexplanations 8 ай бұрын

Thanks, I really appreciate it. A lot of my early work on this was just drawing a lot of circles and lines in a notebook. Hopefully you saw the first video in the series? I guarantee the next time you see a problem involving circles and lines you'll think about it! I will be making more videos in the future, probably on a different topic, please let me know if you have any suggestions.

@arcofficial516 Ай бұрын

@@thecalculusofexplanations how about you do a video on barycentric coordinates. Distribution of parts for a single video: Part 1: The problem (to initiate motivation.) Part 2: The theory Part 3: Tackling the Problem Patt 4: Similar problems

@sungminson3658 8 ай бұрын

Hi. Thanks for the clear explanation here. Now I know a circle touches origin will be mapped to the line geometrically, but not sure how to specifically prove this in equation form. Is there any reference I can check the proof?

@Fractured_Scholar 9 ай бұрын

Looking at this video again and trying to connect it to Linear Algebra. Three questions: 1) What textbook(s) do you recommend to learn this? 2) Do you know of any examples that directly relate this to Set Theory? 3) Has any class you've taken related this to vectors? (i.e. Consider r to be a vector \vec{r} and OA to be its projection onto another \vec{v}. Given a specific magnitude OA' for \vec{v}, then \vec{r} · \vec{v}=\vec{r}^2.)

@thecalculusofexplanations 9 ай бұрын

Hey, glad you're finding it interesting. The connection to linear algebra is a bit difficult, given linear algebra deals with linear transformations, and circle inversion is decidedly non-linear! 1. Not particularly, I think what drew me to the subject was the lack of obvious available resources. If you are wanting to learn the foundations of complex analysis, which is where all this ends up, I've heard good thing's about Needham's "Visual Complex Analysis". There may be some nice books on nonlinear geometric transformations in isolation but I'm not aware of them. 2. I'm not aware of any connection between those concepts 3. Again, the reason I found this interesting was because it's not taught in classes usually. Although it is an interesting though, it's a scaling of the vector from the origin of the circle of inversion, so its a little bit like an eigenvector, but instead of being scaled by a constant its scaled by its own length. I'm not sure if there's anything to that beyond idle speculation, though.

@thecalculusofexplanations 9 ай бұрын

P.S. Hopefully you caught Part 2 as well. kzbin.info/www/bejne/ioK3oqWmmrGLiZo&ab_channel=TheCalculusofExplanations

@comuniunecuosho-campulbudi7611 9 ай бұрын

good content

@thecalculusofexplanations 5 ай бұрын

thank you!

@spiderjerusalem4009 9 ай бұрын

Stewart's theorem does not require trigonometry at all, either you meant it in the sense of its involvement at the midst of solving the problem or its derivation. OP=3-r AP=2+r BP=1+r AO=1, BO=2 (3-r)²=[2(r+2)²+(r+1)²]/3 - 2 3(3-r)²=2(r+2)²+(r+1)²-6 0=2[(r+2)²-(3-r)²]+[(r+1)²-(3-r)²]-6 =10(2r-1)+8(r-1)-6 5(2r-1)+4(r-1)-3=0 14r = 5+4+3=3(4) r = 6/7

@spiderjerusalem4009 9 ай бұрын

6:45 that may make sense intuitionally, but if you use the inversion formula, let P=centre of the smallest circle P'=centre of the inverted circle since r=¼, then OP=R-r=1-¼=¾ 1=OP•OP'=¾OP'= OP' = 4/3 therefore, the radius of the inverted circle = OP'-1=4/3-1 = ⅓ ?????

@spiderjerusalem4009 9 ай бұрын

Ah, or perhaps because the extended straight line connecting the origin and P does not pass through intersection between the smallest circle and the largest one?

@leocrino5144 11 ай бұрын

Awesome video

@youssefdirani 11 ай бұрын

thanks

@2fifty533 11 ай бұрын

i think 2 circle inversions gives you a conformal transformation just like how 2 reflections gives a rotation

@thecalculusofexplanations 11 ай бұрын

Circle Inversion is actually a conformal map because while distances are changed, angles are preserved!

@2fifty533 11 ай бұрын

@@thecalculusofexplanations i mean, if you do 2 successive circle inversions you can get different looking conformal maps that arent the same as a single circle inversion

@2fifty533 11 ай бұрын

@@thecalculusofexplanations slightly unrelated, but I've heard of a thing called conformal geometric algebra, and it's like linear algebra except vectors are circles rather than line segments you can get a conformal transformation by taking the geometric product of 2 vectors

@thecalculusofexplanations 11 ай бұрын

@2fifty533 ah yes, apologies. It’s an interesting thought! It also ties in with the brief mention of iterated inversions I made at the end of the sequel to this video.

@thecalculusofexplanations 11 ай бұрын

@@2fifty533 Sounds fascinating!

@gonzaaa_t Жыл бұрын

ayo loved this video, super clear and concise, great job

@thecalculusofexplanations Жыл бұрын

Thank you so much! Make sure you check out Part 2

@viktorbergman517 Жыл бұрын

hey, in that last problem, instead of using the diamenter of the inverted circle plus the radius of the reference circle, i used the radius of both to get the distance of the center of the circle from the origin, so (with C as the center of the circle whos radius we want to find) OC * OC' = 1 (=) OC * (1 + 0.5) = 1 (=) OC = 1/1.5 = 2/3. since the both circles are tangent the radius should be the radius of the big circle minus the distance between both origins, so 1 - 2/3 = 1/3, which isnt what i get when doing it with the whole diameter, did i miss something or am i doing something wrong?

@viktorbergman517 Жыл бұрын

ah wait i got it, the center of a circle gets distorted when you do the inversion, its just that the outer edge still forms a circle anyway

@thecalculusofexplanations Жыл бұрын

@@viktorbergman517Well done, this is something that has tripped me up as well, and I animated an explanation of why the inverted center is not the center of the inverted circle (the non-linearity of the tranformation ensures this) - but it ruined the flow of the video, and made it a bit too long so I chose eventually to cut it out. Good work thinking it through.

@Xianleft Жыл бұрын

Hey there, would you be interested in how this applies to golf and putting?

@thecalculusofexplanations Жыл бұрын

It does?? I'd be very interested.

@Xianleft Жыл бұрын

@@thecalculusofexplanations consider the tangent of the arc you stand within is relative to the sec and cos of your angles of the follow through as you consider speed and pi. If you could help with the math I could provide more details

@victorhmingthanmawia4773 Жыл бұрын

Could be another system.

@AndrewBrownK Жыл бұрын

What about projecting a circle through a concentric circle?

@thecalculusofexplanations 11 ай бұрын

That should give you another concentric circle on the opposite side of the inversion circle!

@40watt53 Жыл бұрын

Oh my god you don't even have 1000 subscribers and this is incredible dude.

@thecalculusofexplanations Жыл бұрын

Wow thank you so much for the compliment - I feel the same way about many of the newer maths creators, consider watching some #SoME3 videos (including the second part in this series) But I'm definitely looking forward to hitting the 1000 mark, I'd love if you can share these videos with your friends :)

@fulla1 Жыл бұрын

Dude, clear your throat and speak up! The topic was pretty interesting, though.

@thecalculusofexplanations Жыл бұрын

Apologies, I was a bit sick before recording this but I couldn't wait any longer!

@themathguy3149 Жыл бұрын

Please sir unban trigonometry is kinda hard outhere without it I want my mom 🥲

@thecalculusofexplanations Жыл бұрын

Haha, you have my permission.

@marioeraso3674 Жыл бұрын

Awesome! I'm interested in the road coloring throrem. Would love to see your approach to enliving it!

@thecalculusofexplanations Жыл бұрын

Thanks! Looks interesting! I’ll put it on the list of potential topics for a Graph Theory series :)

@mykolanikolayev1714 Жыл бұрын

Please share source code for this animation

@thecalculusofexplanations Жыл бұрын

Hey, thanks for your interest. Unfortunately the code for these is impossibly messy and unreadable, but I will be sharing Github access to code for future videos on my Patreon. I was also thinking about making a video for Patreon supporters about how I make these (the tools and process I use) www.patreon.com/TheCalculusofExplanations/membership

@angeldude101 Жыл бұрын

One of the neat things about circle inversion is that you can get any 2D conformal (angle-preserving) transformation just by preforming a sequence of circle inversions. In addition, lines are really just circles with infinite radius, and doing a circle inversion across them is the same as a traditional reflection across a mirror. So you can compose these inversions to get things like uniform scaling, translations, rotations around the pair of points where two circles (or lines) intersect, hyperbolic boosts, and much more. And all of this generalizes to arbitrary dimensions. (Technically even 1D where you can generate scalings and boosts by inverting across a dipole.) Of course this also includes 3D inversion across spheres, 4D inversion across hyperspheres, or whatever dimension you want.

@thecalculusofexplanations Жыл бұрын

Really? Wow that’s awesome, I didn’t know that. Do you have any good sources, might include that in a follow up video one day.

@angeldude101 Жыл бұрын

@@thecalculusofexplanations It's mainly from Conformal Geometric Algebra, which as the name suggests can be used to represent arbitrary conformal transformations in a manner similar to ℂomplex numbers and quaternions. These transformations are formed by multiplying some number of vectors together. The vectors in question are often used as mirrors/reflections, and CGA vectors specifically represent circle/sphere inversions. Since the product is associative, the inversions can be composed rather than applied one after the other. Depending on what inversions you composed and in what order, you can form any transformation representable by the algebra, which is any conformal transformation. A more rigorous source? I honestly don't have one. There are some good sources on CGA, but usually in specific contexts, and the interpretation I gave is actually not the one found in most sources. Probably the best source I can think of for CGA would be GA4CS (Geometric Algebra for Computer Science).

@thecalculusofexplanations Жыл бұрын

@@angeldude101 Fascinating stuff, I won't pretend to understand completely but I'm not surprised to hear about another deep connection to another field.

@Xianleft Жыл бұрын

You could apply the Schrödinger equation to that for directional velocity right?

@Fractured_Scholar Жыл бұрын

Brilliant! Great work. Did not expect you to reupload the whole video. That said, this is MUCH easier to follow. In response to your request for future content, have you considered connecting what you're showing back to early math? This aids viewers at all levels. Viewers early in their studies get to see things they've recently learned or are learning, while your most advanced viewers get to see how they are simply applying the same concepts learned decades ago and connect them to each other. For example, you casually stated Curvature=Inverse Radius at 2:40. That's a pretty important and succinct statement -- it deserves some weight! 😂 If you were to expand on that section alone, you could talk about how it is geometrically representing the Reciprocal Function, which grade schoolers use to learn Fractions. A mid-skill viewer like myself gets to say: That looks suspiciously similar to a derivative. Is there any relationship there? Then you and your peers (the advanced "viewers"/creators) get to generate more content. Thank you kindly for the update. I look forward to your next video.

@thecalculusofexplanations Жыл бұрын

Thank you so much, I was hoping you would see the improved version, as your feedback was invaluable. I tried to take it all into account and streamline some things, and include the formula for inversion at the start so everyone was on the same page, this necessitated re-recording the audio as well. I'm glad you like it. In terms of explaining things at different levels, that's another great point and something I want to keep in mind. One thing I'm finding is that trying to follow every thought and explain every concept is simply impossible, as the videos become too long, too unfocused and too time consuming to produce. The concept of curvature, for example, deserves its own video (at least) - I can't possibly do it justice within this one, and it wasn't central to explanation. There are also people who've covered it far better than I have on KZbin already. I want to focus on topics I can cover to a decent level in 10 minutes or so, while keeping quality high in both animation and explanation, and if possible highlight areas of maths that I haven't seen a lot of visual explanations for on KZbin. Appreciate the interest, and I'll see you in the next video!

@arcofficial516 Ай бұрын

@@thecalculusofexplanations Bruv? Why does the distance of a point (a,b) from a line(Ax+By+C=0) in a cartesian plane have formulae |Aa+Bb+C|/√(A²+B²). I do know how to prove the formulae using algebra. But at the same time i can sense the beauty lying in the formulae. Algebra does no good in representing that beauty. Btw awesome video.❤

@invictus327 Жыл бұрын

I was watching Norman Wildberger's playlist on algebraic topology and this video perfectly illustrated inverse geometry on a circle for me. Thank you.

@thecalculusofexplanations Жыл бұрын

I'm so glad to hear it! I will have a second video up with a slightly harder problem soon

@MilanStojanovic9 Жыл бұрын

well done, 3b1b would be proud Also very useful for me

@thecalculusofexplanations Жыл бұрын

That's a huge compliment, thank you.

@thecalculusofexplanations Жыл бұрын

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

@tiong2351 Жыл бұрын

Great complement to this topic in the visual complex analysis textbook :)

@thecalculusofexplanations Жыл бұрын

Do mean Needham? I've heard good things about it but I haven't actually read it, you'd recommend it?

@kenkiarie Жыл бұрын

Very interesting. Can't wait for those interesting problems in part 2!

@thecalculusofexplanations Жыл бұрын

Thanks, I appreciate it!

@thecalculusofexplanations Жыл бұрын

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

@aik21899 Жыл бұрын

That was very interesting and I am excited to watch the next one. Maybe you could talk about where this method has been used in proofs in a later video? A "real life application" is always interesting.

@thecalculusofexplanations Жыл бұрын

Thanks for your interest - I'll try to mention some applications in the next video

@thecalculusofexplanations Жыл бұрын

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

@guidosalescalvano9862 Жыл бұрын

I would love to see how points on circle pairs map to each other. Let's define three circles, the mapping circle, the inner circle and the outer circle. Using the mapping circle the inner is circle inverted to the outer. The inner is inside the mapping circle. Now let's assuming a set of n inner points, spaced at equal inner circle arc distances from each other on the inner circle. The inner points are mapped to outer points. Am I correct that the outer arc distance between the outer points is inversely proportional to the distance of the corresponding inner points to the mapped circle's center?

@thecalculusofexplanations Жыл бұрын

Possibly, if I understand the question correctly. Might be interesting to visualise, I'll consider it if I do a part 3.

@guidosalescalvano9862 Жыл бұрын

This video was so awesome that I wanted to watch the whole series. I then saw the recent date and figured; better subscribe. Then I thought, maybe there are a lot more videos, and realized this is seriously your first post!? Wow...

@thecalculusofexplanations Жыл бұрын

Thanks, I really appreciate it - I actually made this one in 2021 with really poor audio (almost inaudible) and re-uploaded it with better sound quality, I am really glad everyone has enjoyed it so much, part 2 is coming soon. Within the next 10 days I would say.

@guidosalescalvano9862 Жыл бұрын

@@thecalculusofexplanations Looking forward to it. This is becoming a year of math ^^

@thecalculusofexplanations Жыл бұрын

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

@sayantanchatterjee2195 Жыл бұрын

Upload part 2

@thecalculusofexplanations Жыл бұрын

It's almost ready!

@thecalculusofexplanations Жыл бұрын

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

@richarddizaji7848 Жыл бұрын

This video’s thumbnail gives a really strange visual effect while scrolling

@thecalculusofexplanations Жыл бұрын

That was somewhat intentional!

@pierre7770 Жыл бұрын

Oh no way this is your first video that’s awesome !! Was getting ready to watch part 2 haha. Really beautiful animations, perfect pace, crystal clear explications and fascinating concept. Thank you so much for this video !

@thecalculusofexplanations Жыл бұрын

Thank *you* so much for the kind words :) It's really motivating to hear people are looking forward to part 2, I'm aiming to release it in the next few weeks!

@thecalculusofexplanations Жыл бұрын

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

@dackid2831 Жыл бұрын

@@thecalculusofexplanations You have this video on private.

@thecalculusofexplanations Жыл бұрын

@@dackid2831 I'm pretty sure this is public?

@dackid2831 Жыл бұрын

@@thecalculusofexplanations My bad, I am referring to part 2

@zach5935 Жыл бұрын

thats so cool. it seems quite abstract and hard to visualise, but your explanation is rather understandable 👍

@thecalculusofexplanations Жыл бұрын

Thank you! It does get easier with practise and solving more problems to visualise where the inverted objects end up

@gdclemo Жыл бұрын

Nice video. So if a circle is inside the inversion circle, but goes around the origin, this must map to a circle that entirely surrounds the inversion circle, right?

@thecalculusofexplanations Жыл бұрын

Thanks! Yes, that's correct, I neglected to show that case but the problem in the next video will show an example of exactly that.

@TheJara123 Жыл бұрын

brilliant effort man, thanks for the nice video, please keep posting

@thecalculusofexplanations Жыл бұрын

Thanks, that's highly motivating :)

@adrienrebollo5879 Жыл бұрын

Excellent video, thanks! Maybe the music is a little bit repetitive, but the math is finely explained.

@thecalculusofexplanations Жыл бұрын

Thanks for the feedback, I'll think about how to adjust the background music!

@guidosalescalvano9862 Жыл бұрын

@@thecalculusofexplanations I thought the music was fine. Nice and focussed.

@Fractured_Scholar Жыл бұрын

Great basic explanation -- Subscribed and looking forward to part 2! Is it possible to do a real world application of where this is useful?

@thecalculusofexplanations Жыл бұрын

Thanks, it depends what exactly you mean by real world, but I can think about including some remarks on that in Part 2.

@Fractured_Scholar Жыл бұрын

@@thecalculusofexplanations - What I mean is that working equations and graphs can be interesting, but it doesn't necessarily map onto (heh) a viewer's brain or life. Where do we see this behavior in the real world? To me, circle/sphere inversion looks like turning something inside out. But that doesn't fully explain the concept for examples beyond physics or three dimensions. Perhaps the Social Sciences offer practical examples???

@thecalculusofexplanations Жыл бұрын

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

@Fractured_Scholar Жыл бұрын

@@thecalculusofexplanations - Thank you, kindly - I'll give it a watch soon!

@user-po5mv2qi9h Жыл бұрын

I love this technique! Thank you for the video! I will wait for the second part.

@thecalculusofexplanations Жыл бұрын

Isn't it awesome? Working on Part 2 as we speak.

@Aequorin628 Жыл бұрын

I was hoping you were a #SoME2 channel and would have some other videos I could watch, but it's impressive that your first video is this good!

@thecalculusofexplanations Жыл бұрын

Thank you! I actually made this for the first SoME, but I'm getting back into it - look out for Part 2 soon.

@Hi-6969 Жыл бұрын

wait there's a part 2? These videos on inversion really do help. I find inversion one of the funkier oly geo topics to learn, and you're video makes it very clear what it is, and how to to use. Using manim makes these videos even better.

@thecalculusofexplanations Жыл бұрын

Thank you! I appreciate it - look out for Part 2 coming soon.

@thecalculusofexplanations Жыл бұрын

Actually, what do you mean by oly geo? Olympiad geometry type problem? I'm not overly familiar with competition maths, does it appear often as a problem solving technique?

@thecalculusofexplanations Жыл бұрын

Part 2 is out now! kzbin.info/www/bejne/Z6eyd31raJt-ers

@brendawilliams8062 Жыл бұрын

Maybe the geo topics are the three great spheres? The man who lived kaleidoscopes and saved geometry?

@Hi-6969 Жыл бұрын

It appears quite often on harder probs, yes.@@thecalculusofexplanations

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