Inversion (extra) - Numberphile

Main video: • A Miraculous Proof (Pt...
See also: • Pentagons and the Gold...
Zvezda's Numberphile playlist: bit.ly/zvezda_videos
Zvezda's webpage: math.berkeley.edu/~stankova/
Epic Circles: • Epic Circles - Numberp...
Triangle Magic Highway: • Triangles have a Magic...
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[@bobcunningham6953]

This sequence of inversion videos is the first in quite a while that makes me want to work through all of it on my own. Many topics covered on Numberphile are "interesting to see and become aware of, but not something I wish to learn to do". This is somehow different for me, in that simple math and apparently simple geometry combine to do "wonderful things": - Inverting an arbitrary inscribed quadrilateral gives is Ptolemy. - Inverting an inscribed rectangle gives us Pythagoras. - Inverting an inscribed equilateral triangle with an arbitrary extra point give us a surprising linear correspondence. - Inverting 4/5 of a pentagon gives us the Golden Ratio. This is just way too wild.

[@cheaterman49]

Also, it's not been mentioned, but from looking at how Pythagora's is a special case of Ptolemy's, and the way it was represented on paper, I think that applies to Thales' too?

[@pahom2]

This whole thing can be a subject of a cool animation of a circle moving around the circle touching and crossing center and its sides, showing how its inversion shrinks and grows and turns into a line here and there.

[@MrQwefty]

Paging 3Blue1Brown!

[@ygalel]

One of my pass time is to solve the problem of Apollonius using a ruler, compass and a pencil using circular inversion. It is so satisfying.

[@TamaraTkacova]

The circle at 1:35 is the most perfect circle I‘ve ever seen

[@revenevan11]

The bit with the circle being squished onto the line between the two points where it intersected the circle of inversion reminded me of a 3blue1brown video about transformations and matrices and stuff. I'd *love* to see some of Grant Sanderson's visualization animations done on this inversion type of transformation, it'd be cool to see points on a grid of different coordinate systems relative to the circle of inversion move in different ways for the whole plane overall, to get an overall sense of it. It also strikes me as a process that may have some relation to chaos, since two points just slightly separated can end up super far away after inversion, or end up arbitrarily close, which intuitively feels like extreme dependence on initial conditions, and like the center is an attractor to far away points!

[@jursamaj]

Agreed, I'd like to see Grant explain inversion. But it's not really like chaos, any more than 1/X is. Everything in an arbitrarily small neighborhood will still be in an arbitrarily small (altho much bigger, if moving outward) neighborhood after inversion.

[@ca-ke9493]

It would be interesting to see how this relates to gravitational fields, as we see r^2 and undefined centers popping up in the math

[@totaltotalmonkey]

Great comment. It's interesting to think of the different ways of collapsing two dimensions into one.

[@numberphile2]

A little extra bit about inversion from this video: kzbin.info/www/bejne/mHuypq2nqpiAi7M

[@cheaterman49]

Zvesda is soo cool :-) she always explains like we're third graders, and that feels good, plane inversion is something I personally never came across!

[@JavSusLar]

2:16 Actually, the arc in the inversion circle goes outside it, and the arc out of the inversion circle goes in it. So in this case we would obtain a circle containing the inner arc. Another special case was not treated here: if you draw a line through the center of inversion and the center of the circle being inverted, that line intersects the circle being inverted at two points. If (and only if) that two points are each other's inversion, then the inversion of that circle is itself (the outer arc inverts to the inner arc, and vice-versa).

[@ca-ke9493]

If you think about it this way, a circle passing through the center of inversion just maps onto a circle with an infinitely large radius, such that it becomes a line? (Aka too much of the circle is inside the circle of inversion/too close to the center - causing the inner arc to balloon drastically when mapped outside.

[@boudicawasnotreallyallthat1020]

I have only one question: what's a circle?

[@pierreabbat6157]

If the circle being inverted is perpendicular to the circle of inversion, it inverts to itself.

[@OrangeC7]

@@ca-ke9493 Exactly so! I love these kinds of things because you get to play around with infinity and infinite values in a way that you can actually control and experiment on

[@RedBar3D]

Maybe you could do a podcast with Zvezda, like the one you did with Holly Krieger? I at least would love to listen to it.

[@dougpowers]

Done! Newest episode on Numberphile2.

[@EdoTimmermans]

Another special case not mentioned in the video is there: a circle going through two (different) points that are each others inversion points. That circle remains identical when inverted, also when the center of that circle is not on the line between these points!

[@Zeldaretter]

Your videos about inversion have been quite beneficial to me. I never truly understood the significance of inversions when I was studying them at university ten years ago, but I now see their full potential. I never enjoyed geometry in university, but after seeing your video, I can see how important it is in many areas of mathematics. Now I'm able to lecture with a lot more enthusiasm about this subject. Thank you very much.

[@unclem7816]

That voice & accent is simply spectacular.

[@xyz.ijk.]

Just as beautiful and informative and educational. Thank you.

[@valentincorman1578]

Thank you Brady, I had the same questions!

[@ClaskoTheKnight]

More uses for inversion please!

[@pourveegupta2783]

0:55 What an accent! I replayed it many times.

[@davidwilkie9551]

Abstract relationships and hidden cause-effect connection construction inversion of point-line-circle time-timing sequence => effectcause reflection inside-outside holographic Inflation process +/- (?). Interesting

[@jannegrey593]

Super-Professor.

[@parvezyunas6622]

sir, very informative lecture. I wonder what would be the case if we use an eclipse ..... instead of a circle...

[@jungleprophet1408]

I'm so sad and shocked that I never learned about circle inversion in my youth. I assume there's a similarly useful 3D "sphere" inversion as well. And what about defining inversion with some other shapes in the plane? Such as across an equilateral triangle? Other regular polygon approximations to a circle...

[@daddymuggle]

The circle of inversion is a special case of a mirror. The simplest case of reflection in the plane is a straight line mirror. Other smooth curves would produce reflections, but the mathematics would be more difficult, and linearity of transformation would be lost. For some curves, the transformation would produce non-unique images of some points. Non-smooth mirrors (ie mirrors with an angle on them) would produce non-unique images of some points, and perhaps fail to act on certain other points. Thus defining those transformations would require some decision-making about what results are wanted. In the case of a mirror which loops back on itself (such as the circle of inversion), the same considerations would apply. For example, I would expect an ellipse to produce results in the same general vein as a circle. (A circle is, after all, a special case of an ellipse). However, some curves would certainly present interesting problems - any curve with a focal point, for example, would behave in interesting ways. A non-smooth inversion figure such as a triangle would present some interesting problems and require some care in defining the inversion. Likewise any inversion figure with any straight sides would have peculiar properties which would be determined by how you define the inversion. (What point or points would you choose as the centre?)

[@EebstertheGreat]

The special case of a circle that passes through the center can be removed if the center of inversion maps to a point at infinity, since any circle passing through the center must invert to a circle that passes through the point at infinity, making it a circle of infinite radius (i.e. a line).

[@oscargr_]

I think you would be right, but you would have to define infinity not just as a distance away from the centre of inversion. The problem is that the centre of inversion would be the only point that does not map onto a single other point.

[@EebstertheGreat]

@@oscargr_ The center of inversion maps to the point at infinity.

[@oscargr_]

@@EebstertheGreat which point at infinity? In what direction from the center?

[@EebstertheGreat]

@@oscargr_ No, like, you extend the real plane to add a single point at infinity. That makes the plane an infinitely large sphere. I guess this is called the real projective plane. A neighborhood of infinity contains all points outside of some disk centered at the origin.

[@DanielFrance81]

@@EebstertheGreat Not the real projective plane (where you add a point at infinity for each direction), rather the complex projective line (or Riemann sphere)

[@PuerinTheHunter]

What if our universe was contained inside such circle and we had only an illusion of being outside the inversion circle? As we approach the center of the circle (but never reaching it), our orientation senses give the appearance that we're going infinitely in the direction of outer space. Would the physical laws still apply the same way?

[@IDoNotLikeHandlesOnYT]

They should, I think. Inversion seems equivalent to reflection, just with a curved reflecting line and the space on one side distorted so an infinite amount of space can be fit there.

[@debjitkhaskel7879]

Nice

[@cheaterman49]

Quick question - is Thales' theorem also a special case of Ptolemy's?

[@vsm1456]

I don't think so. you can probably use Ptolemy's theorem, but you can also prove Thales' theorem with simple geometry: en.wikipedia.org/wiki/Thales%27s_theorem#Third_proof

[@user-ej2jg1wl8r]

I want 3b1b version of this!

[@culwin]

Circles, man.

[@JavSusLar]

uoᴉsɹǝʌuᴉ ۵ I

[@JavSusLar]

6 thumbs up and I sincerely read nine. I'm ill!!! 🙈

[@gerdkah6064]

my mind inverses ^v^

[@jeansavard4990]

And what does an elephant look like after inversion?

[@MrPointness]

Aladdin's shoe

[@jeansavard4990]

@@MrPointness LOL!

[@AlabasterJazz]

I think it would strongly depend on how much of the elephant was inside or outside of the circle of inversion :)

[@davidcampos1463]

I am unmoved. Because, I still ask which came first, the circle or the sphere? Is this just mechanics?

[@xyz.ijk.]

Freehand circles ...

[@d.SAiNi.]

😶!!