Having a world record is a pretty great achievement. Having a world record that's
mathematically unbeatable is even better. Today I'm going to show you some world record surfaces,
which are part of the field of algebraic geometry. That topic is really a way to connect equations
with the beautiful shapes that they correspond to. Mathematicians have always been eager to visualise
their work and get an idea of what an equation looks like. Without computers around, they went
to great lengths to calculate lots of values, and used plaster, wire and paper to handcraft those
models. World Record surfaces are particularly baffling shapes that stand out because of their
number of singularities. I'll get to what that means in just a bit. But first, let's take a look
at how an equation forms any shape at all. This is a free programme called SURFER,derived from
the word surface, there's a link to it down in the description. It allows us to visualise the shapes
from equations in real time. More precisely, those equations are polynomials in the three variables
x, y, and z. We don't have to worry about logarithms, roots, or negative exponents here,
algebraic geometry cares about sufficiently nice equations. Let's take a look at the equation x
squared plus y squared plus z squared minus one is equal to zero. Having Pythagoras in mind,
the solution here are just those points that are a distance of one from the origin. And indeed,
we can see a perfect sphere on our screen. If I rotate the sphere around, you can't even notice.
This is because of its perfect symmetry. And we can see that the symmetry is anchored in the
equation already, we do the same thing to x, y and z, which is raising the variable to the power of
two and adding them all up. We can't distinguish between the variables in terms of mathematical
operations, so we wouldn't expect to see anything unsymmetric in this picture. But let's try and
break the symmetry, we can keep x squared plus y squared, and just write minus z squared. So we
still can't differ between x and y. But something should happen in the z direction. And lucky
enough, that's exactly what we see. This double cone is still symmetric in the x and y directions.
But we see a different story when we look up and down. But there is something else going on here.
You can see that very spiky point in the middle. Well, this is what we refer to as a singularity.
If the first thing that pops into your mind, upon hearing, the word Singularity
is a black hole or the movie Interstellar, then you're on a great path to becoming a physicist.
In more abstract mathematics. This term could mean a lot of different things, and usually describes
points where something interesting happens. It is more than good enough for this video to keep
thinking of that double cone intersection when you hear the word singularity. For those of you
eager for a more technical definition, these points have vanishing partial derivatives.
Now, how do these singularities relate to the world records we spoke about before? Well,
it wouldn't really make sense to just try and find the most amount of singularities
because there wouldn't really be a limit and instead what is quite a natural question
is to ask how many singularities can you find for a given degree of a polynomial equation.
The degree just counts how many variables get multiplied together
in each term of the equation. For example, the degree of our sphere, or our double cone
would be two. Since two variables get multiplied in the terms, the degree of x times y times z
equals zero, would therefore be three. If two terms have different amounts of those variables,
the degree would be the bigger number. So x squared times y times z plus z squared
has a degree of four, and so on. As it turns out, the double cone is already a world record surface
itself. As it turns out that you cannot have more than one singularity for degree two. But
what about higher degrees? Mathematicians asked themselves this exact question for centuries.
And yet we still only know the exact number of singularities up to degree six. So there are still
lots of records still to be snatched out there. And it might just be one of you watching now
he gets one. Mathematicians have been able to sort of sandwich the maximum number of singularities
a surface of a given degree could have by examples for the lower bound, and some more
abstract arguments for the upper bound. And it's finally time to get to the star of this video,
a surface of degree six, simply called a sextic that cannot be bettered, the Barth Sextic.
Six is currently the highest degree for which an unbeatable record exists. And the history
of the surface is quite interesting. It's a surface that really shouldn't be possible.
Here are two research papers by well known Italian mathematicians, one from 1946, and one from 1982.
They both went through a thorough review process for the reputable journals they will published.
The older paper claims that the maximum number of singularities for a sextic is 52. While the other
one claims it is 64. Yet they are both wrong. They're wrong because the Barth Sextic published
in 1996, 16 years after the second paper actually has 65 singularities. One is quick to think that
something mathematically proven would outlast any debate. However, since we're human after all,
we consider a mathematical proof correct if we don't find any mistakes in it. In the same year,
as Barth introduced his new shape to the world, two other researchers proved that 66 singularities
wasn't possible, making Barth's record unbeatable, at least if there are no mistakes this time.
So let's first take a look at the equation for this Barth Sextic. It is said to enclose in its
symmetric frame, many 100 years of geometrical insight and skill. We can see that the first
part has three factors that are quadratic in the variables. Hence, we have a term of degree six
there, ignoring phi, which is a constant. The second term looks familiar. We just have the
formula of our sphere squared, and multiplied with some other number. The mysterious number phi in
this equation is actually the famous Golden Ratio. If you are somewhat baffled that this number makes
an appearance here, then you're no different than me. And if you're unaware what the Golden
Ratio is, there's a link with more information in the description below. So at first glance,
the Golden Ratio shouldn't have anything to do with the singularities we're concerned about here.
But to understand why it is here, we can dive into the construction of Barth's Sextic. What was he
thinking about whilst constructing this now very famous surface? Well, probably about symmetry,
and some ancient platonic solids. These regular polygons have been known since antiquity,
and are classified as platonic solids. They are highly symmetric. So take a look at the
dodecahedron for example, if we were to rotate it a little bit a certain way, it looks exactly like
it did before. Another reason why we might find these solids so aesthetically pleasing, might
be their deep relation to the Golden Ratio. In this colourful picture, you can see this relation
displayed nicely. The Golden Ratio pops up almost every time when a lot of regularity is involved,
as in the regular pentagons on screen right now. Since regular polygons are what make
up the faces of platonic solids, imagining a connection between the famous ratio and the
solids doesn't seem so far fetched anymore. This is isocahedron has six symmetry planes.
If we multiply all of those planes together, we get the starting point of the Barth Sextic.
We can see this shape in SURFER as well. These are the six symmetry planes that you saw before,
they all look like discs because SURFER clips big infinite surfaces
by a sphere in order to display them. This is not yet what Barth's Sectic looks like though.
What he did was to subtract a sphere from these planes to construct the singularities. And you
can see a little slider on the side. The reason this is here is because Barth hasn't actually just
created 1 sextic, but instead a whole family of them, depending on this little parameter
a. And when playing around with it, we can see the shape change in real time. If it is just right,
some new singularities appear. And there it is, the world record surface. As it turns out,
this parameter a has to be deeply connected to the Golden Ratio and only then will you get the full
65 singularities. The setup we chose here just shows a from zero to one, we can zoom in and out,
change colours, and play around with the parameter a. Now if you were able to count all of these
double points here, you might notice that there are only 50 singularities. And you might wonder
what's going on since there's supposed to be 65. Well, we're only able to see 50 of them,
because the other 15 singularities are hiding at infinity, but maybe what that means can be a topic
for another time. A cute thing to realise is that the shape of the dodecahedron is in this sextic,
we can see that each of the singularities actually fall on a vertex of that platonic shape. But
perhaps that is actually less of a surprise now that we know how Barth's surface was constructed.
The shape also has radial symmetry, in some sense, due to the subtraction of the sphere. I have here
a 3D printed Barth Sextic, it's not the version with all 65 singularities as that would be pretty
unstable, and this was hard enough to print as it was. But if you do have access to a 3D printer,
there's also a link to this template down in the description. And this leaves us with the future.
The best we can do for a polynomial of degree seven
is 99 singularities. But there might be a surface out there of degree seven with 100 singularities,
all it takes as a curious mind to find it. And that search may start with a lesson on Brilliant.
Brilliant have kindly sponsored this video, so let's take a look at their quiz on platonic solids
part of their course on 3D geometry. They start by asking how many faces could possibly meet at a
single vertex of a three dimensional shape if all of those faces are squares? Well, a cube is such
a shape. And if you look at the corner, you'll see that three faces are able to meet at that vertex.
So we can plug in our answer as three. Two squares would only fold to make a flat shape. And the four
squares cannot meet at a single vertex, because there's no space to be able to fold them up.
Now there are lots of shapes where every vertex is identical, such as the ones in the pictures
here. But we want to consider these shapes with identical vertices that are also made out of only
a single kind of regular polygon. So their faces are either all squares or all triangles are all
some regular shapes. And the shapes with these properties are called platonic solids. What is
the largest number of sides a regular polygon can have, if three of them meet at a single vertex?
This one's a little tricky, but the correct answer is five. A five sided shape, the pentagon,
is the most that can fit on a single vertex. If you look at the picture that leaves a little bit
of room to be able to fold it up. But with the hexagons, they fit together perfectly,
leaving no room to be able to fold them. And with the heptagons, which is seven sides,
or anything with more sides than that, can't share a vertex, because their angles will add to
more than 360 degrees. And you can see you've got that little gap left and then it can't be filled.
The next question is which of the following nets of equalateral triangles can fold up successfully
to make a vertex of a three dimensional figure? A won't work because it will just fold over on
itself. And E doesn't really work either unless you want your shape to have a concave part.
However, all the others can fold to make a vertex.
And so putting it all together, we've found that there are three ways to make
a three dimensional vertex using the equalateral triangles. Those are the ways we just saw. And
there's only one way to make a vertex using the squares. That was that three sided intersection.
And we know that a five sided shape is the maximum that you'd be able to use to make a vertex at all.
So there are exactly five ways to arrange regular polygons around a single vertex to form a net that
will fold into a three dimensional shape. And for all five of these cases for the vertices,
they can be extended to make the full shape where every vertex is identical. That gives us the five
platonic solids, the tetrahedron, the cube, the octahedron, the dodecahedron and the isocahedron.
And this course goes on to explore how to think about paths that travel through or on the faces
of these polyhedra. If you would like to try out Brilliant for yourself and explore the rest of
thecourse, you can head to Brilliant.org/Tibees and that link will be on screen and down in the
description. Thank you to Brilliant. And also thank you to my Patreon supporters
who've made this video possible. A special shout out to today's Patreon cat of the day Phoenix.