Season 7 Episode 5

Do two lines always meet at a point? What would a really big parabola look like? In this episode, Miles shows us what projective geometry is all about.

Further Reading

Conic Sections

Consider the cone $x^2+y^2=z^2$. That describes a surface for which, at different values of $z$, you have circles of different radii. A little bit of Pythagoras shows that the radius is $z$ if $z>0$, or it's $-z$ if $z<0$, so we have a pair of cones. The vertex of each cone is at the origin; one is pointing up the $z$-axis, and the other is pointing down.

Those circles I mentioned just now are the intersection of the surface with a particular plane $z=c$ for some constant $c$. What happens if we take a different plane?

What's the intersection of that surface with a plane like $x=1$? If we substitute $x=1$ into the equation for the surface, we get $1+y^2=z^2$ which you might recognise as the equation of a hyperbola in the $y$ and $z$ coordinates. (Technically, the information $x=1$ is also part of the specification of where the hyperbola is in 3D space).

If we intersect the surface with a plane like $x=2z+1$ then we get $3z^2+4z+1+y^2=0$. It's a bit harder to see what's going on here, but if we complete the square for the $z$ terms, we get $$3\left(z+\frac{2}{3}\right)^2+y^2=\frac{1}{3}$$ which is the equation of an ellipse. (Again, technically the information $x=2z+1$ is also part of the specification of where the ellipse is in 3D space. We're looking at the shadow of the intersection on the $(y,z)$ axes, whereas the actual intersection is on a particular inclined plane... but it's still an ellipse).

What about the intersection with the plane $z=x+1$? We get $x^2+y^2=x^2+2x+1$ which simplifies down to $y^2=2x+1$. That's a parabola!

These shapes (hyperbola, parabola, ellipse, circle) are known as conic sections.

They turn up in planetary motion, which is quite a surprise (where's the cone?!)

This gives you a nice way to construct an ellipse at home; take a glass of water, tilt it a little, and the surface is an ellipse! This works for both cylindrical and conical glasses (ones that are a bit wider at the top than the bottom), but not if your glass is square or something.

If you tilt the glass far enough then you would get a hyperbola, but you'll spill water everywhere before that happens (hyperbolas are unbounded!).

Projective Geometry

The name for the space we've been exploring on the livestream is the real projective plane.

We talked a lot about adding in points "at infinity", but there are other ways to construct it (or at least, something topologically equivalent).  

One way is to consider points on the surface of a sphere, with the condition that antipodal points (points exactly opposite each other on the sphere) are the same. I quite like this way to construct it, because nothing is infinitely far away.

The Wikipedia link above has some diagrams about how to glue together the edges of a square in order to make a projective plane. Warning; we're using a topologist's square here, the sort that we imagine to be made of very stretchy rubber, but even so it's impossible to make this in our 3D universe without getting the square to intersect itself.

To make a projective plane, you glue the left side to the right side with a half-twist (like you would to make a Möbius strip). Now glue the top side to the bottom side with a half-twist (like you would to make a Möbius strip).

Equivalently, you could take a Möbius strip, and a circular patch of the same material and glue the edge of the circle to the (singular!) edge of the Möbius strip.

There are some attempts to draw what this surface might be like in this StackExchange thread.

Fano plane

The thing with seven points at the end of the stream is the Fano plane. That's the smallest finite projective plane (a projective plane with finitely many points!).

If you've seen the card game Dobble, you've interacted with a finite projective plane before. If you interpret the points as symbols and the lines as cards, then the property "each pair of lines intersect in exactly one point" becomes "each pair of cards has exactly one symbol in common". The actual game of Dobble uses a larger finite projective plane than the Fano plane though. See this Wikipedia page for a diagram.

Matt Parker has a video on this; How does Dobble (Spot It) work?.