WEBVTT
00:00:00.240 --> 00:00:05.200
Suppose you love math.
00:00:05.520 --> 00:00:09.920
And you had to choose just one proof to show someone to explain why it is that math is beautiful.
00:00:10.640 --> 00:00:17.680
Something that can be appreciated by anyone from a wide range of backgrounds while still capturing the spirit of progress and cleverness in math.
00:00:18.440 --> 00:00:19.120
What would you choose?
00:00:20.080 --> 00:00:26.720
Well, after I put out a video on Fineman’s last lecture, about why planets orbit in ellipses, published as a guest video over on MinutePhysics.
00:00:27.160 --> 00:00:37.120
Someone on Reddit asked about why the definition of an ellipse given in that video, the classic two thumbtacks in a piece of string construction, is the same as the definition involving slicing a cone.
00:00:38.080 --> 00:00:41.160
Well, my friend, you’ve asked about one of my all-time favorite proofs.
00:00:41.160 --> 00:00:47.720
A lovely bit of 3D geometry, which, despite requiring almost no background, still captures the spirit of mathematical inventiveness.
00:00:48.880 --> 00:00:54.760
For context and to make sure we’re all on the same page, there are at least three main ways that you could define an ellipse geometrically.
00:00:55.680 --> 00:00:59.120
One is to say you take a circle and you just stretch it out in one dimension.
00:00:59.880 --> 00:01:03.800
For example, maybe you consider all of the points as 𝑥, 𝑦 coordinates.
00:01:03.800 --> 00:01:07.840
And what you do is multiply just the 𝑥-coordinate by some special factor for all the points.
00:01:08.840 --> 00:01:12.000
Another is the classic two thumbtacks in a piece of string construction.
00:01:12.640 --> 00:01:18.160
Where you loop a string around two thumbtacks stuck into a piece of paper and pull it taut with a pencil.
00:01:18.560 --> 00:01:22.480
And then trace around keeping the string taut the whole time.
00:01:22.480 --> 00:01:25.840
What you’re drawing by doing this is the set of all points.
00:01:26.120 --> 00:01:31.600
So that the sum of the distances from each pencil point to the two thumbtack points stays constant.
00:01:32.760 --> 00:01:35.680
Those two thumbtack points are each called a focus of the ellipse.
00:01:36.120 --> 00:01:41.400
And what we’re saying here is that this constant-focal-sum property can be used to define what an ellipse even is.
00:01:42.240 --> 00:01:46.320
And yet another way to define an ellipse is to slice a cone with a plane at an angle.
00:01:46.880 --> 00:01:49.560
An angle that’s smaller than the slope of the cone itself.
00:01:50.400 --> 00:01:55.240
The curve of points where this plane and the cone intersect forms an ellipse.
00:01:55.480 --> 00:01:58.520
Which is why you’ll often hear ellipses referred to as a conic section.
00:01:59.840 --> 00:02:01.760
Now, of course, an ellipse is not just one curve.
00:02:02.040 --> 00:02:06.760
It’s a family of curves, ranging from a perfect circle up to something that’s infinitely stretched.
00:02:07.480 --> 00:02:12.280
The specific shape of an ellipse is typically quantified with a number called its eccentricity.
00:02:12.960 --> 00:02:15.560
Which I sometimes just read in my head as squishification.
00:02:16.320 --> 00:02:18.240
A circle has eccentricity zero.
00:02:18.680 --> 00:02:22.800
And the more squished the ellipse is, the closer its eccentricity is to the number one.
00:02:23.920 --> 00:02:29.960
For example, Earth’s orbit has an eccentricity 0.0167, very low squishification.
00:02:30.560 --> 00:02:32.720
Meaning, it’s really close to just being a circle.
00:02:33.400 --> 00:02:39.880
While Halley’s comet has an orbit with eccentricity 0.9671, very high squishification.
00:02:40.480 --> 00:02:46.360
In the thumbtack definition of an ellipse based on the constant sum of the distances from each point to the two foci.
00:02:46.880 --> 00:02:50.840
This eccentricity is determined by how far apart the two thumbtacks are.
00:02:51.520 --> 00:02:57.840
Specifically, it’s the distance between the foci divided by the length of the longest axis of the ellipse.
00:03:00.840 --> 00:03:05.920
For slicing a cone, the eccentricity is determined by the slope of the plane that you used for the slicing.
00:03:07.200 --> 00:03:14.280
And you might justifiably ask, especially if you’re a certain Reddit user, why on earth should these three definitions have anything to do with each other?
00:03:14.960 --> 00:03:20.520
I mean, sure, it kind of makes sense that each should produce some vaguely oval-looking, stretched-out loop.
00:03:21.160 --> 00:03:27.560
But why should the family of curves produced by these three totally different methods be precisely the same shapes?
00:03:28.440 --> 00:03:34.920
In particular, when I was younger, I remember feeling really surprised that slicing a cone would produce such a symmetric shape.
00:03:35.560 --> 00:03:42.120
You might think that the part of the intersection farther down would kind of bulge out and produce a more lopsided-egg shape.
00:03:43.000 --> 00:03:51.080
But nope, the intersection curve is an ellipse, the same evidently symmetric curve you’d get by just stretching a circle or tracing around two thumbtacks.
00:03:52.120 --> 00:03:54.080
But of course, math is all about proofs.
00:03:54.480 --> 00:03:59.640
So how do you give an airtight demonstration that these three families of curves are actually the same?
00:04:00.840 --> 00:04:04.840
For example, let’s focus our attention on just one of these equivalences.
00:04:04.840 --> 00:04:09.920
Namely that slicing a cone will give us a curve that could also be drawn using the thumbtack construction.
00:04:10.920 --> 00:04:16.720
What you need to show here is that there exist two thumbtack points somewhere inside that slicing plane.
00:04:17.480 --> 00:04:24.720
Such that the sum of the distances from any point of the intersection curve to those two points remains constant.
00:04:25.160 --> 00:04:27.200
No matter where you are on that intersection curve.
00:04:28.280 --> 00:04:32.760
I first saw the trick to showing why this is true in Paul Lockhart’s magnificent book, Measurement.
00:04:33.320 --> 00:04:38.560
Which I would highly recommend to anyone young or old who needs a reminder of the fact that math is a form of art.
00:04:39.480 --> 00:04:47.320
The stroke of genius comes in the very first step, which is to introduce two spheres into this picture, one above the plain and one below it.
00:04:48.000 --> 00:04:49.680
Each one of them sized just right.
00:04:50.080 --> 00:04:55.920
So as to be tangent to the cone along a circle of points and tangent to the plane at just a single point.
00:04:59.280 --> 00:05:04.440
Why you would think to do this, of all things, is a tricky question to answer, and one that we’ll turn back to.
00:05:05.000 --> 00:05:11.480
Right now, let’s just say that you have a particularly playful mind that loves engaging with how different geometric objects all fit together.
00:05:12.400 --> 00:05:17.640
But once these fears air sitting here, I actually bet that you could prove the target result yourself.
00:05:18.520 --> 00:05:20.120
Here, I’ll help you step through it.
00:05:20.120 --> 00:05:24.360
But at any point, if you feel inspired, please do pause and just try to carry on without me.
00:05:25.440 --> 00:05:31.640
First of, these spheres have introduced two special points inside the curve, the points where they’re tangent to the plane.
00:05:32.640 --> 00:05:37.280
So reasonable guess might be that these two tangency points are the focus points.
00:05:38.160 --> 00:05:43.360
That means that you’re gonna wanna draw lines from these foci to some point along the ellipse.
00:05:43.880 --> 00:05:49.360
And ultimately, the goal is to understand what the sum of the distances of those two lines is.
00:05:50.080 --> 00:05:55.600
Or, at the very least, to understand why that sum doesn’t depend on where you are along the ellipse.
00:05:58.400 --> 00:06:03.440
Keep in mind, what makes these lines special is that each one does not simply touch one of the spheres.
00:06:03.840 --> 00:06:07.240
It’s actually tangent to that sphere at the point where it touches.
00:06:07.960 --> 00:06:13.360
And in general, for any math problem, you want to use the defining features of all of the objects involved.
00:06:14.320 --> 00:06:17.440
Another example here is what even defines the spheres.
00:06:18.000 --> 00:06:20.080
It’s not just the fact that they’re tangent to the plane.
00:06:20.640 --> 00:06:25.440
But that they’re also tangent to the cone, each one at some circle of tangency points.
00:06:26.080 --> 00:06:30.320
So you’re gonna need to use those two circles of tangency points in some way.
00:06:31.000 --> 00:06:33.640
But how exactly?
00:06:34.160 --> 00:06:40.000
One thing you might do is just draw a line straight from the top circle down to the bottom one along the cone.
00:06:41.040 --> 00:06:47.160
And there’s something about doing this that feels vaguely reminiscent of the constant-sum thumbtack property and hence promising.
00:06:48.120 --> 00:06:50.160
You see, it passes through the ellipse.
00:06:50.600 --> 00:06:56.480
And so, by snipping that line at the point where it crosses the ellipse, you can think of it as the sum of two line segments.
00:06:56.880 --> 00:06:59.080
Each one hitting the same point on the ellipse.
00:06:59.880 --> 00:07:04.160
And you can do this through various different points of the ellipse, depending on where you are around the cone.
00:07:04.680 --> 00:07:08.000
Always getting two line segments with a constant sum.
00:07:08.640 --> 00:07:12.480
Namely, whatever the straight line distance from the top circle to the bottom circle is.
00:07:13.440 --> 00:07:17.080
So you see what I mean about it being vaguely analogous to the thumbtack property.
00:07:17.800 --> 00:07:22.040
And that every point of the ellipse gives us two distances whose sum is a constant.
00:07:22.960 --> 00:07:25.320
Granted these lengths are not to the focal points.
00:07:25.320 --> 00:07:26.760
They’re to the big and the little circle.
00:07:27.320 --> 00:07:29.880
But maybe that leads you to making the following conjecture.
00:07:30.640 --> 00:07:44.760
The distance from a given point on this ellipse, this intersection curve, straight down to the big circle is, you conjecture, equal to the distance to the point where that big sphere is tangent to the plane, our first proposed focus point.
00:07:45.760 --> 00:07:55.800
Likewise, perhaps the distance from that point on the ellipse to the small circle is equal to the distance from that point to the second proposed focus point where the small sphere touches the plane.
00:07:57.080 --> 00:07:58.880
So is that true?
00:07:59.760 --> 00:08:00.680
Well, yes.
00:08:01.120 --> 00:08:03.960
Here, let’s give a name to that point that we have on the ellipse 𝑄.
00:08:04.800 --> 00:08:09.760
The key is that the line from 𝑄 to the first proposed focus is tangent to the big sphere.
00:08:10.320 --> 00:08:14.440
And the line from 𝑄 straight down along the cone is also tangent to the big sphere.
00:08:16.280 --> 00:08:18.360
Here, let’s look at a different picture for some clarity.
00:08:18.920 --> 00:08:24.640
If you have multiple lines drawn from a common point to a sphere, all of which are tangent to that sphere.
00:08:25.280 --> 00:08:30.640
You can probably see just from the symmetry of the setup that all of these lines have to have the same length.
00:08:31.640 --> 00:08:36.960
And in fact, I encourage you to try proving this yourself or to, otherwise, pause and ponder on the proof that I’ve left on the screen.
00:08:39.400 --> 00:08:43.520
But looking back at our cone slicing set up, your conjecture would be correct.
00:08:43.960 --> 00:08:49.680
The two lines extending from the point 𝑄 on the ellipse tangent to the big sphere have the same length.
00:08:50.560 --> 00:08:58.920
Similarly, the line from 𝑄 to the second proposed focus point is tangent to the little sphere, as is the line from 𝑄 straight up along the cone.
00:08:59.640 --> 00:09:01.400
So those two also have the same length.
00:09:02.280 --> 00:09:13.480
And so, the sum of the distances from 𝑄 to the two proposed focus points is the same as the straight line distance from the little circle down to the big circle along the cone, passing through 𝑄.
00:09:14.280 --> 00:09:18.680
And clearly, that does not depend on which point of the ellipse you chose for 𝑄.
00:09:18.680 --> 00:09:22.320
Bada boom, bada bing, slicing the cone is the same as the thumbtack construction.
00:09:22.600 --> 00:09:25.280
Since the resulting curve has the constant-focal-sum property.
00:09:25.920 --> 00:09:39.160
Now, this proof was first found by Germinal-G-Germinal-Germa-, who cares, Dandelin, a guy named Dandelin in 1822.
00:09:39.800 --> 00:09:42.760
So these two spheres are sometimes called Dandelin spheres.
00:09:43.520 --> 00:09:48.320
You can also use the same trick to show why slicing a cylinder at an angle will give you an ellipse.
00:09:48.880 --> 00:09:56.880
And if you’re comfortable with the claim that projecting a shape from one plane onto another tilted plane has the effect of simply stretching out that shape.
00:09:57.560 --> 00:10:02.520
This also shows why the definition of an ellipse as a stretched circle is the same as the other two.
00:10:03.400 --> 00:10:03.720
More homework.
00:10:05.680 --> 00:10:09.800
So why do I think that this proof is such a good representative for math itself?
00:10:10.480 --> 00:10:16.960
That if you had to show just one thing to explain to a non-math enthusiast why you love the subject, why this would be a good candidate?
00:10:18.040 --> 00:10:22.320
The obvious reason is that it’s substantive and beautiful without requiring too much background.
00:10:22.840 --> 00:10:29.600
But more than that, it reflects a common feature of math that sometimes there is no single, most fundamental way of defining something.
00:10:30.440 --> 00:10:33.000
That what matters more is showing equivalences.
00:10:33.640 --> 00:10:40.240
And even more than that, the proof itself involves one key moment of creative construction, adding the two spheres.
00:10:40.720 --> 00:10:44.280
While most of it leaves room for a nice, systematic and principled approach.
00:10:44.640 --> 00:10:50.920
And this kind of creative construction is, I think, one of the most thought-provoking aspects of mathematical discovery.
00:10:51.560 --> 00:10:54.280
And you might understandably ask where such an idea comes from.
00:10:55.000 --> 00:10:59.200
In fact, talking about this particular proof, here’s what Paul Lockhart says in Measurement.
00:11:00.160 --> 00:11:02.320
How do people come up with such ingenious arguments?
00:11:02.720 --> 00:11:05.920
It’s the same as the way people come up with Madame Bovary or Mona Lisa.
00:11:06.360 --> 00:11:07.720
I have no idea how it happens.
00:11:08.240 --> 00:11:11.080
I only know that when it happens to me, I feel very fortunate.
00:11:12.240 --> 00:11:16.240
I agree, but I do think we can say at least a little something more about this.
00:11:16.600 --> 00:11:27.400
While it is ingenious, we can perhaps decompose how someone who has immersed themselves in a number of other geometry problems might be particularly primed to think of adding these specific spheres.
00:11:28.240 --> 00:11:31.960
First, a common tactic in geometry is to relate one length to another.
00:11:32.640 --> 00:11:43.280
And in this problem, you know from the outset that being able to relate these two lengths to the foci to some other two lengths, especially ones that lineup, would be a useful thing.
00:11:43.800 --> 00:11:46.800
Even though, at the start, you don’t even know where the focus points are.
00:11:48.400 --> 00:11:53.920
And even if it’s not clear exactly how you do that, throwing spheres into the picture isn’t all that crazy.
00:11:54.680 --> 00:11:58.120
Again, if you’ve built up a relationship with geometry through practice.
00:11:58.640 --> 00:12:05.080
You would be well-acquainted with how relating one length to another happens all the time when circles and spheres are in the picture.
00:12:05.680 --> 00:12:10.000
Because it cuts straight to the defining feature of what it even means to be a circle or a sphere.
00:12:10.640 --> 00:12:12.760
And this is obviously a very specific example.
00:12:13.160 --> 00:12:16.920
But the point I wanna make is that you can often view glimpses of ingeniousness.
00:12:17.120 --> 00:12:21.040
Not as inexplicable miracles but as the residue of experience.
00:12:22.240 --> 00:12:28.160
And when you do, the idea of genius goes from being mesmerizing to instead being actively inspirational.