Suppose you love math, and you had to choose just one proof to show someone to explain
why math is beautiful.
Something that can be appreciated by anyone from a wide range of backgrounds while still
capturing the spirit of progress and cleverness in math.
What would you choose?
After I put out a video on Feynman’s Lost Lecture about why planets orbit in ellipses,
published as a guest video on minutephysics, someone on Reddit asked about why the definition
of an ellipse given in that video, the classic two thumbtacks and a piece of string construction,
is the same as the definition involving slicing a cone.
Well, my friend, you’ve asked about one of my all-time favorite proofs, a lovely bit
of 3d geometry which, despite requiring almost no background, still captures the spirit of
mathematical inventiveness.
For context, there are least three main ways you could define an ellipse.
One is to say you take a circle, and stretch it in one dimension.
For example, if you consider all these points as (x, y) coordinates, maybe you multiply
the x-coordinate of each point by some factor.
Another is the classic two-thumbtacks-and-a-piece-of-string construction.
Loop a string around two thumbtacks in a piece of paper, pull it taut with a pencil, then
trace around, keeping the string taut the whole time.
What you are drawing by doing this is a set of all points so that the sum of the distances
from each point to the two thumbtacks stays constant.
The two thumbtack points each called a “focus” of the ellipse.
And what we're saying here is that this constant focal sum property can be used to define what an ellipse even is.
And yet another way to define an ellipse is to slice a cone with a plane at an angle,
an angle smaller than the slope of the cone itself.
The curve of points where the plane and cone intersection forms an ellipse, which is why
you’ll hear ellipses described as a “conic section”.
Of course, an ellipse is not just one curve, it’s a family of curves, ranging from a
perfect circle to something infinitely stretched.
The specific shape of an ellipse is typically quantified in a number called its “eccentricity”,
which I sometimes read in my head as “squishification”.
A circle has eccentricity 0, and something more squished has an eccentricity closer to
1.
For example, Earth’s orbit has eccentricity 0.0167, low squishification, meaning it’s
really close to a circle, while Halley’s comet has an orbit with eccentricity 0.9671,
very high squishification.
In the thumbtack definition of an ellipse based on a constant sum of the distances from
each point to two foci, this eccentricity is determined by how far apart focus points
are.
Specifically, it’s the distance between the foci divided by the length of the longest
axis of the ellipse.
For slicing a cone, the eccentricity is determined by the slope of the plane.
And you might justifiably ask: Why on earth should these three definitions have anything
to do with each other?
I mean, sure, it kind of makes sense that each should produce some vaguely oval-looking
stretched out loop, but why should the family of curves produced by these three totally
different methods be precisely the same shapes?
In particular, when younger, I remember feeling surprised that slicing a cone produces such
a symmetric shape.
You might think the part of the intersection further down would sort of bulge out more
to produce a lopsided egg-shape.
But nope!
This intersection curve is an ellipse, the same evidently symmetric curve you’d get
by stretching a circle or tracing around the two thumbtacks.
But of course, math is all about proofs, so how do you give an airtight demonstration
that these three families of curves are all the same?
For example, let’s focus our attention on just one of these equivalences, that slicing
a cone will gives a curve which could also be drawn using the thumbtack construction.
What you need to show is that there exist two thumbtack points somewhere in the slicing
plane such that the sum of the distances from any point on the intersection curve to the
two points remains constant, no matter where you are on the ellipse.
I first saw the trick to showing why this is true in Paul Lockhart’s magnificent book
“Measurement”, which I’d highly recommend to anyone young or old who needs a reminder
of the fact that math is a form of art.
The stroke of genius comes in the first step, which is to introduce two spheres to this
picture, one above the plane and one below it, each one of them sized just right so as
to be tangent to the cone along a circle of points, and tangent to the plane at just a
single point.
Why you would think to do this of all things is tricky question to answer, and one we’ll
turn back to.
For now, let’s just say you have a particularly playful mind that loves engaging with how
different geometric objects fit together.
But once these spheres are sitting here, I actually bet you could prove our target result
yourself.
Here, I’ll help step you through it, but at any point you feel inspired please do pause
and try to carry on without me.
First off, these spheres have introduced two special points inside our curve, the point
where they’re tangent to the plane, so a reasonable guess might be that these two tangency
points are the focus points.
That means you will want to draw lines from these foci to some point along the ellipse,
and ultimately you want to understand the sum of the distances of these two lines.
Or at the very least, to understand why that sum doesn’t depend on where you are along
the ellipse.
What makes these lines special is that each one does not simply touch one of spheres,
it’s tangent to that sphere at the point where it touches.
In general for a math problem, you want to use the defining features of all the objects
involved.
Another example here is what defines these spheres.
It’s not just the fact they are tangent to the plane, but that they lie tangent to
the cone, each one at some circle of points.
So you’re going to need to use those two circles of tangency points, but how exactly?
Well, one thing you might do is draw a straight line from the top circle to the bottom one
along the cone.
There’s something about doing this that feels vaguely reminiscent of the constant-sum
thumbtack property, and hence promising.
It passes through the ellipse, and so it can be broken down as the sum of two line segments,
each hitting the same point on the ellipse.
And you can do this through various different points of the ellipse, always getting two
line segments with a constant sum; namely, whatever the straight-line distance from the
top circle to the bottom is.
So you see what I mean about it being vaguely analogous to the thumbtack property; every
point of the ellipse gives us two distances whose sum is a constant.
Granted, these lengths are not to the focal points, they’re to the big and little circle,
but maybe that leads you to making the following conjecture:
The distance from a given point on the ellipse straight down to the big circle is, you conjecture,
equal to its distance to the point where the big sphere is tangent to the plane, our first
proposed focus point.
Likewise, perhaps the distance from that point on the ellipse to the small circle is equal
to distance from that point to the second proposed focus point, where the small sphere
touches the plane.
Well, yes.
Here, let’s give a name to that point we have on the ellipse here, Q.
The key is that line from Q to the first proposed focus point is tangent to the big sphere,
and the line from Q straight down along the cone is also tangent to the big sphere.
Here, let’s take a look at another picture for some clarity here.
If you have multiple lines draw from a common point to a sphere, all of which are tangent
to that sphere, you can probably see just from the symmetry of the setup that all these
lines will have the same length.
I encourage you to try proving this yourself or to otherwise pause and ponder on the proof
on screen.
So back to our cone slicing setup, your conjecture would be correct; the two lines extending
from the point Q on the ellipse tangent to the big sphere have the same length.
Similarly, the line from Q to the second proposed focus point is tangent to the little sphere,
as is the lin from Q straight up along the cone, so those two have the same length.
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 to the big circle passing through Q, which
clearly doesn’t depend on which point of the ellipse you chose for Q!
Bada boom bada bang, slicing the cone is the same as the thumbtack construction, since
the resulting curve has the constant focal sum property!
Dandelin This proof was first found by Germinal Pierre
Dandelin in 1822, so these two spheres are sometimes called “Dandelin spheres”.
You can use the same trick to show why slicing a cylinder at an angle will give an ellipse.
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 that shape, this also shows why
the definition of an ellipse as a stretched circle is the same as the other two.
More homework!
So why do I think this proof is such a good representative of math itself; 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 such a good candidate.
The obvious reason is that it’s substantive and beautiful without requiring too much background.
But more than that, it reflects a common feature of math that sometimes there is no single
“most fundamental” way of defining something; that what matters more is showing equivalences.
And even more than that, the proof itself involves one key moment of creative construction,
adding the two spheres, while most of it leaves room for a nice systematic and principled
approach.
This kind of creative construction is, I think, one of the most thought-provoking aspects
of mathematical discovery, and you might understandably ask where such an idea comes from.
Talking about this particular proof, Paul Lockhart says “How do people come up with
such ingenious arguments?
It’s the same way people come up with Madame Bovary or Mona Lisa.
I have no idea how it happens.
I only know that when it happens to me, I feel very fortunate.”
Where does genius come from?
I agree, but I think we can say something a little more than this.
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 particular spheres.
First, a common tactic in geometry is to relate one length to another.
And in this problem you know from that outset that being able to relate these two lengths
from the foci to some other two lengths, especially ones that line up, would be useful.
Even if it’s not clear how exactly you’d do that, throwing spheres into the picture
isn’t all that crazy.
Again, if you’ve built up a relationship with geometry through practice, you’d be
well acquainted with how relating one length to another happens all the time when circles
and spheres are in the picture, because it cuts straight to their defining feature.
This is obviously a very specific example, but the point is that you can often view glimpses
of ingeniousness, both here and in general, not as inexplicable miracles, but as the residue
of experience.
And when you do, the idea genius goes from being mesmerizing to instead being actively
inspirational.