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If you have a paper and pencil nearby go ahead and put four dots anywhere on the paper and connect them to make a quadrilateral
It doesn't matter how weird the shape is and I'll just put a few examples here to illustrate that point
Now put a dot at the midpoint of all of those lime segments you just drew and connect those to make another quadrilateral
Turns out no matter what connecting those midpoints will always yield a parallelogram or shape or opposite sides are parallel to each other
The reasoning behind this is a theorem that says if you connect the midpoints of two sides of a triangle
That line segment will be parallel to the third side
I could then just form another triangle which shares that third side and then connect those midpoints you see here
So again as a result that theorem I mentioned earlier these two line segments. You see here are parallel to the third side
Which means they're also parallel to each other
then if I connect the other two corners
I can make the same argument which gives us two new parallel line segments leaving us with a parallelogram
What we just solved was math in two dimensions something that could be done on a flat piece of paper
We'll soon get to three dimensions though, which we can of course grasp since the space we live in is three-dimensional
But once we move to higher dimensions, you'll see the intuition is mostly gone leaving us to rely on mathematical analysis
Which can still be done without that intuition?
But to set up a lot of what's coming
I'm first gonna ask how many different types of three-dimensional dice are possible to make?
Now I'm using the word dice pretty loosely here because what I really mean is three dimensional polyhedra that meets certain criteria
The first one is that every face must be a regular polygon and the same regular polygon
So for example this cube or a die we're familiar with is an example of this as every face is a square with the same
sides and same angles the second criteria is that add the same number of faces must meet at each vertex for each corner for a
Cube, that would be three and then the third is that shape must be convex. So it can't look like this or anything
So including the cube how many 3-dimensional objects fulfill all these requirements?
Well, I've got a few here that do
Like a tetrahedron which has four faces that are all equilateral triangles. We of course have the cube. Here's the octahedron
That's also made of equal our triangle faces X four phases in me at a corner here not three
There's a dodecahedron that has twelve phases which are all regular Pentagon's then we've also got the icosahedron which has 20 phases
But how many more are possible I?
Mean in two dimensions you can have infinitely many regular polygons or shapes with equal side lengths and angles
But when it comes to putting those on the faces of a 3-dimensional polyhedron what is possible?
Well, this is it
These are the five platonic solids the only shapes that meet the criteria we discussed earlier
That's definitely not obvious. For example, I found it weird that we can't have a die with hexagons as faces
Okay what we can like this one here, but as you can see there Pentagon's as well
So it doesn't meet that requirement having the same regular polygons on each face now
This is the point where I was gonna do a proof as to why there are only five platonic solids
But I decided to a dedicated video just for that. So if you want to see you can click here or down in the description
However, we will do a little more analysis with these shapes
Take a look at the vertices edges and faces of each solid
There's something these all have in common and it's not obvious at first
But if we subtract the edges from the vertices and then add the number of faces in each case the value comes out to two
This relationship of vertices minus edges plus faces is known as Euler's characteristic
Yes, another one of his and it turns out for any convex polyhedron Euler's characteristic is two in
fact even for a sphere with regards to the surface Oilers characteristic is still -
That's because the sphere can be morphed into any convex polyhedron
Aka the shapes are homeomorphic and Euler's characteristic is preserved under homeomorphisms
But things get even weirder when you realize for example that Oilers characteristic for a four dimensional
Klein bottle is 0 which we're going to get to soon
Now in a previous video I showed you what a mobius strip was and this is a very famous shape in the field of topology
Where you take a flat length of paper and put a half twist in it, then connect the ends
Then I also showed how when you cut it right down the middle and go all the way around
You're still left with one piece
But that one piece is not a mobius strip as it has more than a half twist in it
then we also saw how if we start our cut about 1/3 of the way in rather than
And go around twice we end up with two pieces that are linked together
But there are plenty more interesting results that we can get
Like what if now I decided to put one?
two
Three half twists in this strip and then cut it down the middle
Well, let's see what happens
And
Here we have
Holed up I'm gonna need a flat surface for this
The bends in the paper are making it not behave well, so I just kind of press everything down
Now what we have here if you can tell is a knot you
can see if we start somewhere and follow the paper all the way around we get to the original spot and the loop just keeps
Going so it is one connected piece knot rings like before
This is actually a specific type of knot known as the trefoil knot
Here, let me try rearranging it. So it looks more like the graphic
Now this isn't a knot like in your shoelaces where they have loose ends because in this case
There's nothing I can do to unknot the paper without tearing it
But if I could tear it or move the knot through itself in just one location
Like here, it would become totally unknotted. Otherwise known as the UH knot to demonstrate moving it through itself
I will just cut it in one location and retake the pieces on the other side of that crossing and
Now if you just rearrange the paper a little you'll see we no longer have a knot
Since I was able to do that in one cut one is the unknotting number of our trifle knot
That's not too difficult to see but it's not as obvious that for example this
Stevedore knot with six crossings rather than three also has a nun knotting number of one
Well, the cinquefoil knot with five crossings has a nun knotting number of two
We can see several other unknotting numbers here as well
But it turns out that determining those numbers is actually not so easy to do mathematically
We know it for several of these simpler knots before example. We haven't determined it for this knot here
When you move into higher dimensions, you'll see there's a lot of unsolved problems there actually
But let's go back to that mobius strip and answer another question
What would I have to cut in half in order to get to moe strips?
Or what shape would I get if I could sew these together?
Well, the answer is that Klein bottle I mentioned earlier which is a shape that only exists in four dimensions
That means if you really did connect to mobius strips
Like numberphile has shown before you only get a 3d representation of the Klein bottle
Which is the best we can do?
to make a real Klein bottle you need access to a fourth spatial dimension if you're wondering why it's because right here it appears like
the client bottles intersecting itself and in three dimensions it is
But a true Klein bottle does not self intersect and that's only possible if this is embedded in a higher dimension
However, the Klein bottle is still a two dimensional manifold
Meaning close up or locally. It looks like a two dimensional plane since this really is just a surface
See, this is just like the surface of the earth
Of course, we know the global structure of Earth but to something small like us standing on that surface. It does look flat and two-dimensional
Because of this the surface of the earth is also a two dimensional manifold as is the Klein bottle a torus and plenty more
Locally they appear to be two-dimensional but they all curve or are embedded in higher dimensions
Then when you get to relativity space-time and the curvature of the universe, you move into those three and four dimensional manifolds
But that's beyond this video
Let's now get into how to think about a four spatial dimension though and to do that
We should first imagine a two-dimensional object living in flatland that can only see along this completely flat surface
if a three-dimensional sphere entered that world all the flat creature would see is the
Two-dimensional cross-section or a circle?
If the sphere said hey, I'm a three dimensional object and the flatlander said prove it move in this third dimension
You speak of all they'd see is that cross-section gets smaller and the same goes for us. I
Mean if there were some two-dimensional world living on this piece of paper
I could see everything that's going on and there's nothing they could do to hide for me
Then if I wanted to at any point, I could enter their world
But all they would see is the two-dimensional cross-section of my fingers in this case
And this is exactly how it be for a four-dimensional object visiting us
They could see our three-dimensional world and there's nothing we could do to hide from this
They can then enter at any point and we would see a three-dimensional
Cross-section of that object. So if it was a four dimensional sphere for example
We would see a 3d sphere just appear out of nowhere. It would get bigger and bigger and then smaller and smaller and eventually disappear
See our brains aren't really made to understand how that object is moving through that higher dimension
But as you can see we can still draw comparisons and do the mathematical analysis
Even if we don't have the geometric intuition to back it up, for example
We can answer the question. How many?
Four-dimensional dice are possible. And again what I mean by that is how many four-dimensional platonic solids are possible that meet those same criteria
We discussed earlier we saw before how only a few regular polygons could be put on the faces of those platonic solids
But can we connect these shapes to each other in the fourth dimension to create 4d dice
The answer is yes
in fact
every platonic solid in three dimensions has a four-dimensional counterpart like if you join
Tetrahedrons together at three to an edge you get the hyper tetrahedron
There's also the hypercube hyper octahedron hyper dodecahedron and the hyper icosahedron as well
But there is a six platonic solid that emerges in the fourth dimension and it's the octa cube or hyper diamond
The analog of this constructed in three dimensions is the rhombic
Dodecahedron something which is not a platonic solid do the irregular faces
But in four dimensions amazingly the necessary criteria are all met and you'd think it gets even crazier as we move up
But guess what happens in five dimensions?
There are now only three platonic solids. We move to the sixth dimension
They're still three and as we continue to go up we unfortunately only find three platonic solids now moving on
Here's another example of something. We know very well for up to four dimensions
But afterwards we know very little and this has to do with kissing numbers or the maximum number of units spheres that can be arranged
to barely touch or kiss another common unit sphere
For example in two dimensions the kissing number is six
That's because if you have a single circle at most six other circles can be touching it in this way
This wasn't too hard to find but in three dimensions it already got difficult as it took
Mathematicians a while to figure out that in fact 12 is the kissing number and after much more effort
We have determined that in four dimensions. The kissing number is 24
However, five dimensions, we don't know the kissing number
We just know as of this video that's between forty and forty four and four six dimensions and seven dimensions
We also only have a range of numbers. It could be but funny enough
We have figured this out for eight dimensions as the kissing number is 240 and even stranger
We have determined that in twenty four dimensions. The kissing number is a hundred ninety six thousand five hundred and sixty
It turns out something weird does happen in eight and twenty four dimensions when it comes to spheres, but I'll get to that soon
the last thing I want to mention about four dimensions though is that there are no knots like the ones we're familiar with in the
fourth dimension
Well other than the trivial knot which isn't really knotted
the reason is because a knot can be thought of as a
One-dimensional string embedded in three-dimensional space as it needs that third dimension to exist
You couldn't make a knot on flat paper without self-intersection
But remember how I said I could unknot something by having it essentially passed through itself, which I demonstrated by cutting it
Well in the fourth dimension a knot would be capable of passing through itself through that extra dimension
Meaning it's not really a knot. It'd just be a loop in disguise and in mathematics, this is just the trivial not nothing that's special
However, in four dimensions, it would be possible to have knots made out of two-dimensional sheets, though
No clue what that would look like, but we need that difference of two dimensions in order to create a true knot
So using that same logic, we could technically not a three dimensional object
but we'd have to move it through five dimensions to do so and
With that let's now transition to something else interesting that happens in the fifth dimension
Which goes back to the video I made about adding up all the volumes of even dimensional units fears
Which comes out to e to the PI for anyone who hasn't seen it?
Many of you guys want more of an explanation and the biggest thing to know is that since the series?
Converges that must mean the volume of these spheres gets smaller as the dimension goes up
For a one-dimensional quote sphere or really just a line segment with radius one
the length is of course to a two-dimensional sphere or circle with a radius one has an area of pie or
3.14 a
Three-dimensional sphere with radius one has a volume of 4/3 pi or four point one eight nine
then from the equation for the volume of an n-dimensional sphere we can calculate that a
Four-dimensional sphere of radius one has a volume of about four
Point nine three five and a five dimensional unit sphere has a volume of five point two six four
Notice that the number keeps going up with every dimension
But once we get to six dimensions a sphere with radius one has a volume of five point one six eight
It's finally lower than the previous dimension for seven dimensions. It goes down again and continues to do so
If we look at the plot for the volume of a unit sphere versus the dimension
We're in we see that the fifth dimension is the peak and it eventually does approach zero but geometrically why is that the case?
Well, I'll give the best explanation I could find in
Regards to two dimensions. Let's say we perfectly rest a circle inside a square which has side lengths of 1
Well, it turns out the circle takes up
78.5% of that squares area
When we move to three dimensions and put a sphere inside a cube also a side length of one that sphere takes up only
52.3% of the volume adding the extra dimension decrease the ratio of those volumes
If we go to four dimensions and embed a hyper sphere inside a hypercube, the sphere only takes up 31% of that volume
But notice that the cubes volume is not changing as you move up in dimensions for two dimensions
The area of the square is one squared or one in three dimensions. The volume is one cubed or one and in four dimensions
It'd be one to the fourth or of course one
Yes, we are comparing completely different units
So it's not the best physical comparison
But the sphere is taking up less and less of something that always has a volume of one
Thus the spheres volume will go to zero
Another thing that notice is the diagonals of the cube in two dimensions. The diagonal has a length of root two
Well, the circle has a diameter of one just a little less in
Three dimensions though. The diagonal of the cube goes to root three
well
The spheres diameter again stays at one in
Any dimension n the diagonal of the cube has a length of the square root of n as in those corners get further and further
apart without bound as we move up in dimensions
The sphere on the other hand is maintaining the same diameter and thus taking up less and less of the space as the dimension goes
up
Okay. Now I know before I showed how the fifth dimension strangely has that peak volume
But actually this was nothing special and that's because we are specifically using it's fierce
If we look at spheres with the radius of three the peak would happen at a different dimension
But since spheres with a radius of one are just so common. I decided to highlight it now
Let's look at sphere packing for centuries. Mathematicians have been curious about the optimal way to fill space using identical spheres
for example in two dimensions the question was how can we arrange pennies on a table such that they are as dense as possible and
It turns out the best arrangement takes up about 91% of the space
For spheres it was much harder
But we found the optimal arrangement takes up about 74% of the space now for most dimensions higher than three
We know very little about the best way to arrange spheres
However, as we go from dimensions 3 to 4 to 5 and so on the gaps between the spheres start to increase from the 3d
case
but for some reason in dimension 8
There are new gaps that are just large enough to hold spheres that can be perfectly locked into place resulting in that optimal arrangement
Which takes up about 25% of the 8 dimensional space?
So we do know the math in eight dimensions and guess what the math of optimal sphere packing was determined in 24 dimensions as well
Just like the kissing number problem. By the way for anyone saying what's the point of this?
it turns out sphere packing and the hypercube graph have
Applications and information and coding theory when it comes to correcting errors and data that show up in noisy communication channels
for example
Let's say if I want to transmit the word yes to someone I'll send them a one and if I want to transmit an O
I'll send a zero now. This is simple enough but errors could come up when it comes to long distance communication
So instead I'll send three 1s for a yes and three zeros for a no
That way veneer does occur like in a message that says 1 0 1
You can still say that's more likely to be a yes since two of the bits are one
Well, we can think of each possible message as a point in 3d space located on the corners of a cube
We'll put a sphere centered at the one-one-one message and one at the 0 0 0 message if the radius of those spheres
Matches the length of the cube then whatever messages they cover will be interpreted as the message at the sphere center
Like here the 1 0 1 message would be interpreted as the 1 1 1 or yes, since that sphere includes both of those corners
But if I want to include more words than just yes or no, I need to pack more spheres into the space
Yes, this example is pretty simplistic
But openly it shows how packing steers tightly into space does have direct applications
And as we add more bits, we do the analysis on higher dimensional hypercube graphs. Meaning we're dealing with higher dimensional sphere packing
Now, let's see something weird in 9 & 10 dimensions
Imagine the square that is cut into four smaller squares and then draw a unit circle in each one of those as shown
Then draw another circle in the center that is tangent to the four others
You'll notice that the center circle is pretty small relative to the entire square and it's far from the edges
We'll do the same thing in three dimensions
Which would involve a cube being cut into eight pieces and those each being filled with a unit sphere?
Then we'd pack a sphere in the very middle such that it barely touches the other eight
Here you also notice that the sphere is much smaller than the cube
But as the dimensions go up that central sphere gets larger and larger and in nine dimensions that central sphere actually
touches the edge of the nine dimensional cube then in ten dimensions
It breaks through the sides of the cube
Even though the cube well looks like it should be the boundary of the inner sphere holding it inside
However, when the sphere breaks through the cube, it still has a smaller volume than the cube that is until dimension
262 after 262 dimensions the hyper sphere becomes larger than the hypercube volume-wise
Okay, I don't officially quote me on that as I found it in various forum posts rather than an official source
But once I came across that information, I really couldn't exclude it
Now let's look at an example. That's kind of similar to the sphere packing we saw earlier
Let's say we have a bunch of oranges or spheres that we need to wrap using saran wrap or whatever
The question is what's the best way to arrange these such that you use the least amount of that saran wrap?
Now I only have three oranges here so you don't have many options
But in general the question is should you align the oranges in a line or should you clump them?
Just like we saw the sphere packing
Well at the moment
Both of those answers are right because it does depend on how many oranges we have
When you only have a couple like four or five it is best to just line them up
however
For some reason when you get to 50 seven oranges
It is now better to clump them and the same goes for any number of spheres greater than 57
However, what's the best way to arrange them in four dimensions?
Well, it turns out again you should arrange them such as their centers are on the same line
Until you get to somewhere between 50 thousand and a hundred thousand four-dimensional oranges
because we have determined that somewhere in that range is when you should finally start to clump them in order to minimize the amount of
Wrapping paper used and after that we have no idea what the optimal packing method is in terms of higher dimensions
Until we get to dimension 42 in
42 dimensions and everything after
we have determined that it is best to again align the spheres such that all their centers are on the same line and
there is no breaking point for this so that it's always the best case no matter how many
Very high dimensional oranges that you have. So just in case you guys need to wrap some forty two dimensional fruit
You now know the best way to minimize the cost of that gift wrap
Now if this kind of math interests you there are quite a few areas that you can look into further things such as hyperbolic geometry
differential geometry or abstract algebra
I'll deal with complex surfaces or higher dimensional symmetries and so on but the main branch you want to look into is probably topology
This is a branch of math dealing with properties of space that are preserved under continuous deformations
So like homeomorphisms, which we saw earlier are an important topic in this field
Now this is a course that most people outside math and physics majors will never see
but it is usually offered an undergrad along with some of those other courses I mentioned so if you're not in those majors you could
Always try to tailor your electives to these higher-level math courses or you get a math minor
Even if these don't directly apply your field of study
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