Tier lists have been a staple of YouTube for a few years now, at this point, you can find
a ranking of just about anything.
TierZoo was the first channel I saw to combine the format with science and learning.
But it's already spread into physics and maths as well.
For example, Andrew Dotson recently did a ranking of his favourite physics equations.
Today, I thought it would be time to create an ultimate mathematical shapes tier list.
I've got a non rigorous point system to give each shape a fair chance at that coveted S-tier
position.
So each object will be judged on symmetry, singularities, the significance in other disciplines
outside of mathematics, how difficult it is to define or create and the aesthetics.
For aesthetics, we're really asking, does this shape deserve its own mood board on Pinterest.
Now, that's of course going to be subjective, but it's worth including aesthetics as a criteria,
because it's kind of important to math.
The beauty of the shape may indicate that we've got some kind of elegant solution, or
at least to help inspire interest in the topic.
So let's start with a nice simple shape, the common sphere, I'll be using the software
SURFER to model some of these shapes.
This algebraic surface is as old as it gets, it is perfectly symmetric, which makes it
awesome to work with.
But at the same time, it might be a bit boring and uninspiring because it looks the same
in all directions.
However, this hasn't stopped the sphere from becoming a major player in other disciplines.
You see it in art and sports and engineering all the time.
And even in astrophysics for things like planets, and even black holes.
Spheres are very effective for modelling all sorts of things.
Probably because the sphere is very easy to understand.
Aesthetics wise, the sphere might not be the most interesting shape.
But because of its heroic performance in all sorts of disciplines, and how easy it is to
construct, I think that B-tier is more than deserved.
Next up is another classic, the Mobius strip.
It's a two dimensional object with only one side and one edge that can be embedded in
a three dimensional space.
There are no notable things to say about it symmetry or singularities.
But just like with the sphere, where a mobius strip shines is in its applications to other
disciplines.
Surprisingly, there are actually some applications of this strange little shape.
One of them is that in the days of industrialization, a lot of factories had one central engine
to provide rotational energy to operate all the machines, a common thing was to use flat
leather belts to transmit the central power.
Some were instead given a half twist before joining the ends and this would allow for
the wear to be evenly distributed over both sides of the belt.
You can make a mobius strip pretty easily yourself, just take a long piece of paper,
give it a twist and glue the ends together.
If you start at any point and draw along the strip, you end up we started.
In mathematics, we call the shape with a property like this non orientable.
Since you can't really define an up or down on a surface like this.
I think the simplicity of being able to create such an interesting shape makes it very intriguing
and worthy of an A-tier spot.
Now this next crazy critter is the Barth Sextic, which I recently made an entire video about.
It's going to belong in S-tier because it's got very interesting symmetry, record breaking
singularities and an intriguing history.
It is one of the world record surfaces since it holds the record for having the largest
number of singularities for a polynomial of degree six, that record that it holds is 65
singularities.
And it was proven that it can never be beaten, that 66 singularities isn't possible.
Now there are some other members of the world record squad and they ought to join it in
here as well.
So we have the Togliatti Quintic, which holds the record for a polynomial of degree five,
and that record is 31 singularities.
I think they belong one tier below the Barth Sextic since the record is for a lower degree.
Then Likewise, we have the Kummer Quartic, which holds the record for a polynomial of
degree four.
And that record it holds is 16 singularities.
I'll put that one in B-tier, and then we'll have the Kayley Cubic, another world record
surface, this time of degree three, coming in with four singularities, and getting placed
in C-tier.
The next one is also a record holder, it's the double cone coming in with one singularity,
for degree two.
I talked about this one a bit as well in my video about the Barth Sextic, you can really
see that prominent singularity in the middle of the shape.
This shape is also quite interesting in special relativity in physics, as you might have seen
it in something called a light cone.
It shows the causal future and the past.
You can think of the singularity in the middle as an event in space time, and all other events
inside the double cone can have a causal relationship with that first event, everything outside
the cone would be too far away to have any impact, or to be impacted by the original
event in space time.
And this angle of the cone is governed by the speed of light.
That's a very interesting application but because the record this one holds is not as
impressive as some of the other records we've seen, it's going to sit in C-tier.
And now I've got two more shapes to quickly mention here, which also currently hold world
records but perhaps not for long.
These are the Labs-Septic, and the Endrass-Octic.
These currently hold records for seven and eighth degree polynomials.
Although these records haven't been proven to be unbeatable in the way that our other
record holders were.
For the Barth Sextic, we've proven that you can never have more than 65 singularities.
But here for the septic, and this optic, it's very feasible that someone might be able to
find a shape which will beat it.
For that reason, they're both going to be banished to D-tier.
Let's see what we can do next, we have Gauss Bell surface.
The surface might look unspectacular at first, since there are no interesting points that
really stand out, no singularities.
But don't underestimate this gem of mathematics.
It is used all the time in statistics and is the foundation of the normal distribution.
Furthermore, what's very interesting is that the formula defining the surface shows a really
nice connection between the fundamental constants pi and e.
This one is difficult to rank because the aesthetics are lacking a bit and the competition
is harsh, but we'll place them here in C-tier.
Next up, we can do the Torus, or you might like to call it a doughnut.
It is a rotational symmetric shape of genus one, since it has one hole, and this shape
is often brought up when talking about the topic of topology.
Everything with one hole is said to be topologically equivalent to a Torus, and the classic example
is a coffee mug with a handle or drinking straw.
Of course, its applications are very wide from hair ties, to rings, to edible donuts,
or even pool toys.
It has vast applications and because of that, we're going to put it in B-tier.
Next we have a shape which is a close relative of one that we've already seen.
This is the Klein bottle, and it's very similar to the Mobius strip since both surfaces are
non orientable.
The Klein bottle also has only one surface but unlike the strip, the two dimensional
bottle can't be embedded in three dimensions, it always looks like it intersects itself,
when in reality, it doesn't.
You might have noticed this shape in my profile picture here on YouTube.
The beanie that I wear is actually a Klein bottle hat that I made by knitting it.
Although of course, it's not an accurate Klein bottle since that can't exist in 3D.
If you see the way it intersects, that wouldn't happen if we had the luxury of a fourth spatial
dimension.
Unfortunately, making nice hats and glass bottles might be the limit of applications
that I know about the Klein bottle to the world outside mathematics.
But the Klein bottle definitely deserves to be ranked alongside the Mobius strip up here
and A-tier.
Especially because if you actually were to split a Klein bottle in half, what you would
have would be two Mobius strips.
This little twisty boy is called the Trefoil knot qnd it does seem to be the poster child
for knot theory.
It is named after the three leaf clover, the Trefoil plant and it has some nice triangular
sort of symmetry to it.
The knot is one continuous loop and easy enough to make yourself using a string.
I also happened to have this sculpture lying around, which is another example of this kind
of knot.
So clearly, it's gonna rank pretty high on aesthetics, if they're literally selling it
as a decoration.
I don't have a great deal of knowledge about knot theory but apparently this is the first
non trivial knot.
And one of the reasons why knot theory is so interesting is that it seems to have many
applications in biology and chemistry with things like proteins and DNA that are knotted
in some sense.
I'm going to rank it in B-tier for now, it's got some points for aesthetics, some points
for a possible usefulness even though I might not totally understand how this note is useful,
but it is still a fairly basic shape.
We're back to a heavy hitter with the next one, Flamm's paraboloid.
This one's going to get ranked right up at S-tier.
It doesn't perhaps look that spectacular, but it is pretty cool.
This shape mimics the outside of a black hole in the space time called the Schwarzschild
solution.
In 1915 when Einstein published his famous field equations of general relativity, he
didn't actually solve them himself.
Instead, a German scientist named Karl Schwarzschild found the first interesting solution in relativity,
a spacetime containing a black hole.
He calculated this while he was stationed in Russia during the First World War.
And sadly, he died not long after his groundbreaking discovery.
He calculated that spacetime stops behaving normal at some radius, this is the inner circle
in the middle, now, we call this the event horizon.
The surface shows how everything gets attracted to that circle, and what happens beyond this
point is anyone's guess at the moment.
It is amazing to see how clearly general relativity predicted the existence of black holes like
this, when no one could imagine this before.
This is a cornerstone of science wrapped up in one picture and that's why it deserves
to be in S-tier.
Second to last, here we have the dodecahedron.
Now, this is supposed to represent all five of the platonic solids.
We've all seen them before, and they're so iconic that they definitely deserve to be
ranked here.
All of these solids have some very special symmetry to them.
And in fact, these solids define what symmetries other shapes can have.
For example, back in my Barth Sextic video, it was the symmetry of the icosahedron that
allowed us to get that shape.
What makes the platonic solids so special is the uniqueness they have.
These shapes are the only options if you want a polyhedron with identical faces and corners.
And the fact that there are only five seems quite surprising.
You see these shapes, of course, a lot in architecture, or anywhere really throughout
history.
And so considering this A-tier is a fitting spot.
And this list would of course, not be complete without at least one fractal.
I've chosen to include the Mandelbrot set.
This relatively new shape finds its home in complex analysis, and is known for its crazy
boundary.
It is one of those fractals which are inherently self similar.
Meaning that you can zoom in on the border as much as you want and would always end up
with the same thing on screen.
There isn't too much to talk about in terms of symmetry, and singularities with this one.
And I'm not sure about any usefulness in other scientific fields, but it is used a lot in
art.
And as I'm sure you can imagine, or might have seen, there are plenty of nice, psychedelic
like zooming videos using this shape, plenty of which you can watch here on YouTube.
The construction of this shape is surprisingly simple.
And I'm going to explore it using Brilliant who are also a sponsor of this video and have
been a huge supporter of my channel.
I do really like their explanation and a lot of their visuals for the Mandelbrot set.
So let's have a look at how they say we can define it, what you do is you start with a
complex number, call it z, then if you square that and add it to itself, look at what you
end up with, then square that and add z to that and keep going like that forever.
You'll notice that one of two things would happen; either the magnitude explodes up towards
infinity, and in that case, the complex number z is not in the Mandelbrot set.
If the magnitude is bounded or tends towards zero, then the numbers z is in the Mandelbrot
set.
We can ask ourselves is the complex number i in the set?
Well, it is because after the first few iterations, we would have i, then minus one plus i, then
minus i, then back to minus one plus i.
And this set of answers will repeat, we're in a bit of an alternating cycle.
And since we're not going to end up going out to infinity, the value of i is in the
Mandelbrot set, so it will be plotted somewhere on that interesting picture.
Another example is the real number one.
With that we would have one, then two, then five, then 26.
And because each time we're squaring that answer and adding one to it, we're just going
to keep getting larger and larger until we get to infinity.
So one is not in the Mandelbrot set.
Another couple of examples here we have that minus two is in the Mandelbrot set, whilst
minus two i is not.
This here is a full image of the Mandelbrot set and you can zoom in on parts of the boundary
to see its chaotic nature.
The set has a main cardioid, which is this heart shaped section here, as well as several
attached bulbs.
And the position of a number within either the cardioid, or these numbered bulbs here
tells us something about recursion, or I think the fate of the number, whether it reaches
an equilibrium or a fixed point.
I love looking at this fractal because every piece of it is loaded with mathematical meaning.
And one of the reasons that it might be so appealing is because you get such a complicated
surface with a relatively easy mathematical construction.
To this date, there are still plenty of open questions surrounding the set and it has led
to the discovery of many equally astounding shapes, such as the Mandelbulb, or the Buddhabrot.
It is pretty legendary so I'm going to put it up in S-tier.
Let me know what you think of my ranking.
I'm sure there will be some that you disagree with but it's just for fun.
And I'm sure there's also some interesting shapes that I've left out, because how could
I include all of them in one video, but tell me some of those main ones you think should
be included next time.
Thanks to Brilliant for sponsoring this video and if you're interested to check out their
content a bit more, they have courses on mathematics, science, including physics, astronomy, computer
science, biology, a lot of interesting stuff, you can head to brilliant.org/tibees and that
link will be down in the description.
Thanks to my Patreon supporters for making these videos possible.
And a special shout out to today's Patron Cat of the Day, Maureen.