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.