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.