Invert about the 4-sphere
by BOGTRO, Sep 3, 2015, 4:50 AM
Wolstenholme also wrote up his perspective on this excellent process here
2 years ago, both me and Wolstenholme attended AMSP Cornell 2013. Since everyone knows how incredible I am at geo, I took Geo 3 (aka "Cosmin murders you"). Wolstenholme took this for dumber reasons (i.e. he actually knows geo).
Anyway, I got pretty murdered throughout that class, but that's not the point of this particular post. At some point during Geo 3, we talked about the excellent topic of inversion, which Wolstenholme knew and I did not. So I was asking him how inversion worked, he explained it to me, and I asked him whether this was limited to 2 dimensions -- i.e. can we invert about a sphere? He responded "yeah... we could invert about a 4-sphere if we wanted to". Little did we know...
Fast forward 2 years to MIT orientation, part of which involves taking ASEs (Advanced Standing Exams) to test out of classes you already know the material for. In my case, since multivariable calculus is pretty boring and I didn't really want to bother with it for an entire semester, I decided to take the 18.02 (multivariable calculus) ASE without knowing multivariable calculus.
Perhaps that wasn't a particularly great idea, but I still had a week or so to learn the subject, so it was probably going to be ok. Also I was taking the makeup version of the ASE, because at the time it was scheduled I would be taking the 8.01 (Physics I) ASE (which I didn't end up actually taking, but that's a story for another day...). On the other hand, most of the math crew here (including Wolstenholme) had already gotten credit for 8.01 through AP Physics, so they took it the day before I did.
Alright, so 2 weeks pass, and it's time for me to demonstrate my extensive knowledge of multivariable calculus. The day before, Wolstenholme mentioned how easy the ASE was, and that it didn't actually have questions on most of the harder material (e.g. surface integrals), so I was pretty chill. So I breezed through the first ~9 (of 20) questions, struggled through the next few, and finally reached #14 (ok maybe not #14 but w/e), which I was completely stumped on...
Also I should mention that spherical coordinate problems made me very sad, because there was a lot of computation, so I basically skipped them during my hardcore training.
Anyway, so first of all we have to figure out what this
thing means, which is basically saying we have a sphere of radius 
centered at 
. Ok that part's done, yay. Unfortunately, the rest isn't too easy to deal with: we can do the standard substitution of 
, but unfortunately the bounds here aren't so nice: we still have 
, but bounding 
involves some semi-annoying trig that results in a pretty ugly integral.
So I figured there was probably a better way, especially considering the answer choices (sidebar: why the hell are there answer choices???) were of the point
where 
was rational (
and 
were the only reasonable ones). Unfortunately I couldn't find said better way, burned 40 minutes, and just moved on to other problems (eventually I made up some garbage like the answer was 
where 
is the expected distance from the ball centered at the origin with radius [which comes out to 
], but it's totally wrong).
Ok, so the ASE ends, but I can't really get this problem out of my head because I'm so certain that there must be a nice way to do it. So after some unsuccessful conceptual ideas, I go ask Wolstenholme, whose first reaction was "that's 100x harder than anything we had on our test". Darn. So we briefly try this, conclude there's probably no good way, and go off to lunch.
At lunch, we meet Christopher Shao, who's also pretty good at math so I give him this problem also. Unfortunately, the three of us still can't find a particularly great way to do this problem. Finally, at some point I'm like "guys, what if we like invert about the sphere". I was actually serious (more out of desperation than anything), but Wolstenholme just started laughing before he realized I wasn't trolling. So at first we're all like "ok this is dumb", but then gradually we realize "wait........ does this actually work"
Indeed, inversion is actually quite motivated here: the ball in question is sent to a half-space (this took a while for us to work out though), and eventually we realized that this half-space doesn't actually start at
, but instead the half-space is bounded by the plane containing the intersection of the two spheres (since the inversion sends the "top" of the ball in question to that plane). So we end up with the triple integral 
from the definition of inversion, which seemed relatively promising.
So anyway, we couldn't really work this out without paper, so we went to go do other stuff and eventually Wolstenholme and I reconvened to solve this problem. After many, many, many failed computational attempts, we finally came up with the following excellent solution...
The great thing about this solution is that it works in effectively the same exact way for other dimensions, which means that -- after 2 years -- "invert about the 4-sphere" is finally a legitimate strategy for solving a problem. I'm unreasonably happy right now.
2 years ago, both me and Wolstenholme attended AMSP Cornell 2013. Since everyone knows how incredible I am at geo, I took Geo 3 (aka "Cosmin murders you"). Wolstenholme took this for dumber reasons (i.e. he actually knows geo).
Anyway, I got pretty murdered throughout that class, but that's not the point of this particular post. At some point during Geo 3, we talked about the excellent topic of inversion, which Wolstenholme knew and I did not. So I was asking him how inversion worked, he explained it to me, and I asked him whether this was limited to 2 dimensions -- i.e. can we invert about a sphere? He responded "yeah... we could invert about a 4-sphere if we wanted to". Little did we know...
Fast forward 2 years to MIT orientation, part of which involves taking ASEs (Advanced Standing Exams) to test out of classes you already know the material for. In my case, since multivariable calculus is pretty boring and I didn't really want to bother with it for an entire semester, I decided to take the 18.02 (multivariable calculus) ASE without knowing multivariable calculus.
Perhaps that wasn't a particularly great idea, but I still had a week or so to learn the subject, so it was probably going to be ok. Also I was taking the makeup version of the ASE, because at the time it was scheduled I would be taking the 8.01 (Physics I) ASE (which I didn't end up actually taking, but that's a story for another day...). On the other hand, most of the math crew here (including Wolstenholme) had already gotten credit for 8.01 through AP Physics, so they took it the day before I did.
Alright, so 2 weeks pass, and it's time for me to demonstrate my extensive knowledge of multivariable calculus. The day before, Wolstenholme mentioned how easy the ASE was, and that it didn't actually have questions on most of the harder material (e.g. surface integrals), so I was pretty chill. So I breezed through the first ~9 (of 20) questions, struggled through the next few, and finally reached #14 (ok maybe not #14 but w/e), which I was completely stumped on...
MIT wrote:
The sphere is tangent to the 
-plane. Compute the average distance to the origin over the set of all points inside this sphere.
is tangent to the 
-plane. Compute the average distance to the origin over the set of all points inside this sphere.Also I should mention that spherical coordinate problems made me very sad, because there was a lot of computation, so I basically skipped them during my hardcore training.
Anyway, so first of all we have to figure out what this
thing means, which is basically saying we have a sphere of radius 
centered at 
. Ok that part's done, yay. Unfortunately, the rest isn't too easy to deal with: we can do the standard substitution of 
, but unfortunately the bounds here aren't so nice: we still have 
, but bounding 
involves some semi-annoying trig that results in a pretty ugly integral.So I figured there was probably a better way, especially considering the answer choices (sidebar: why the hell are there answer choices???) were of the point
where 
was rational (
and 
were the only reasonable ones). Unfortunately I couldn't find said better way, burned 40 minutes, and just moved on to other problems (eventually I made up some garbage like the answer was 
where 
is the expected distance from the ball centered at the origin with radius [which comes out to 
], but it's totally wrong).Ok, so the ASE ends, but I can't really get this problem out of my head because I'm so certain that there must be a nice way to do it. So after some unsuccessful conceptual ideas, I go ask Wolstenholme, whose first reaction was "that's 100x harder than anything we had on our test". Darn. So we briefly try this, conclude there's probably no good way, and go off to lunch.
At lunch, we meet Christopher Shao, who's also pretty good at math so I give him this problem also. Unfortunately, the three of us still can't find a particularly great way to do this problem. Finally, at some point I'm like "guys, what if we like invert about the sphere". I was actually serious (more out of desperation than anything), but Wolstenholme just started laughing before he realized I wasn't trolling. So at first we're all like "ok this is dumb", but then gradually we realize "wait........ does this actually work"
Indeed, inversion is actually quite motivated here: the ball in question is sent to a half-space (this took a while for us to work out though), and eventually we realized that this half-space doesn't actually start at
, but instead the half-space is bounded by the plane containing the intersection of the two spheres (since the inversion sends the "top" of the ball in question to that plane). So we end up with the triple integral 
from the definition of inversion, which seemed relatively promising.So anyway, we couldn't really work this out without paper, so we went to go do other stuff and eventually Wolstenholme and I reconvened to solve this problem. After many, many, many failed computational attempts, we finally came up with the following excellent solution...
(we could have easily used 
instead of 
because if 
we have issues with the inversion not sending anything to inside the ball, but whatever). The ball (which we shall henceforth denote by 
)
above the 
is sent to the point 
where![\[ u = \frac{r^2x}{x^2 + y^2 + z^2} \: \: v = \frac{r^2y}{x^2 + y^2 + z^2} \: \: w = \frac{r^2z}{x^2 + y^2 + z^2} \]](http://latex.artofproblemsolving.com/7/1/d/71d826c5c024b80f3986498a7c8587f703f3a50b.png)
![\[ \frac{d(u, v, w)}{d(x, y, z)} = \frac{r^6}{(x^2 + y^2 + z^2)^6}\begin{vmatrix}y^2 + z^2 - x^2 && -2xy && -2xz\\-2yx && z^2 + x^2 - y^2 && -2yz\\-2xz && -2yz && x^2 + y^2 - z^2\end{vmatrix} = -\frac{r^6}{(x^2 + y^2 + z^2)^3} \]](http://latex.artofproblemsolving.com/9/1/2/912accce53ee6d3d6a3ffac6905973f2fda584f0.png)
so 
and also 
.
The great thing about this solution is that it works in effectively the same exact way for other dimensions, which means that -- after 2 years -- "invert about the 4-sphere" is finally a legitimate strategy for solving a problem. I'm unreasonably happy right now.
This post has been edited 1 time. Last edited by BOGTRO, Sep 3, 2015, 4:52 AM
Reason: Added link
Reason: Added link
August 2018
December 2015