I'm assuming you've all just watched part 1 of this video and that you know all about
the shadow theorem in three dimensions. It turns out that our shadow theorem
has counterparts for hypercubes living in abstract four- and higher-dimensional
spaces. The maximal 2d and 3d shadows of these abstract hypercubes turn out to be
fantastic bundles of real-world symmetries. In fact, a lot of what's nice
and symmetric about the 3d world we live in is actually captured by these
hypercube shadows. Ok, so here's a bit of a summary of these higher-dimensional
extensions of our shadow theorem. It's also supposed to be a bit of a lead-in
to the next video which will be about how we construct and use higher-
dimensional space in mathematics and apply these higher dimensions in the
real world. It's going to be really hands-on, you know, how do you deal with
higher dimensions. Should be good. Anyway so what we have to do is we have to have
another look at this very basic set up here. There's the cube hovering
above this plane and mathematically the plane is just xy-plane and this rod
sticking out this is the z-axis and we're casting a shadow onto the xy-plane.
Now let us take an arbitrary point in space that point has coordinates (x,y,z).
So what's the coordinate in the xy-plane of the shadow of this point?
Well, obviously, xy, so we are basically forgetting about z.
So, in a sense, to get the shadow of our cube here we just take all its points
and just throw away all their z-coordinates. That's like (x,y,z), throw away z,
what remains is just (x,y). You may also have wondered what this black region
on the z-axis stands for. Well that's the shadow of the cube onto the z-axis.
Just like we get the 2d shadow down there by throwing away the z-coordinates of
the points of the cube, we get the 1d shadow by throwing away the x and y
coordinates. x,y,z, throw away xy you just get all of the z-coordinates. So that means
that the height difference that we talked about before is really just the
length of this second shadow. So we're measuring the length of a shadow and
an area of shadow. So every point in 3d is
captured by (x,y,z) coordinates and so in a way 3d space is really just a collection
of all number triples (x,y,z) and things like distances, areas, angles,
spheres, cubes, etc. are captured by various mathematical formulas and
equations involving coordinates. That's what we do in maths. Then we define
abstract four-dimensional space just to be the set of or quadruples (x,y,z,w)
and we extend the definitions of distances, areas, cubes etc. by just
generalizing what these things pan out to be in two and three dimensions.
So, what we're doing is we're really creating a fourth dimension. Four
dimensional space, five dimensional space, etc. these things have no physical
reality attached to them and so we are really creating them. It's very important
to stress this. I'll show you exactly how all this is done in the next
video but to understand what the shadow theorem pans out to be for for 4D unit
cubes for example you don't need to know any more details.
In three-dimensional space there are three coordinates and there's essentially only
one way to split these into two parts, that's exactly the way we've done it,
right? So the 2d part, say x,y, and and the 1d parts say z.
Then we simultaneously cast shadows into the 2d part and the 1d part
and we find that in the case of unit cubes the numbers we get when we measure
these shadows are always equal. In 40 we can split up the 4d coordinates in two
ways: a 3d and a 1d part x,y,z and w or two 2d parts x,y and then z,w.
I've prepared a couple of animations again. The first one is an animation that
shows the shadow play of a rotating 4d unit hypercube casting one 3d shadow
into x,y,z space, that's our universe, and one 1d shadow on the W axis.
So what it really is is, on my computer I'm rotating a 4D hypercube, that's in
abstract space and I'm just kind of forgetting about
the last coordinate and that gives me a shape in three dimensions, in the world
that we live in. On the other hand I just cancel out the first three
coordinates and that gives me a shadow on this last coordinate axis the w-axis.
And then I compare and I find when I measure the volume of the
shape and when I measure the length of this other shadow they're going to be
exactly the same no matter what happens. And then again, as in the case of a just
a regular cube we just have to figure out what's the maximum distance of
two points in a hypercube. Once we've got that we also know what the maximal 3d
shadow of a hypercube is. And that's what it is and you should recognize it. It's
just from the first warm-up exercise this rhombic dodecahedron. It's actually
a really fantastic shape and I could say like an hour's worth of stuff about it but
there's even more fantastic stuff coming up so let's just move on. So I said
there is a second way of splitting up 4D space, into x,y and then z,w
On my computer I'm rotating my virtual hypercube and I'm forgetting about the
second two coordinates and what that gives is something flat so something in
the xy-plane. That is this shape on the left. And then I forget about the first
two coordinates and that gives the shape on the right. You see they're totally
different shapes but again if you're rotating a unit hypercube the areas of
these two shapes are always going to be exactly the same. And then we can ask
again, well what are the maximum shadows here and these are also super nice.
The are just coming up in the second here. They are regular octagons. So with this
one here as a maximal shadow we had a regular hexagon, with the
four-dimensional hypercube we have a regular octagon and actually that
continues throughout all dimensions. So we always have a (maximal) 2-dimensional shadow
for example of a 666-dimensional cube. What do we expect here? Well the rule is
we're in n dimensions we have to double up, that's what we get, a 2n-gon, a
regular 2n gon, so for a 666-dimensional hypercube we get a regular (666 times 2 =) 1332-gon
and that was the second warm-up exercise. So something like this is
indistinguishable from a circle, so I fooled you there, I hope. Alright so now
we can do the same sort of thing for 3-dimensional shadows, maximal ones.
So we take something higher- dimensional, we can't really see it but
can manipulate it on our computers, we can throw away everything apart from x,y,z
and that gives us a solid shape and
whenever we're looking for maximum ones these solid shapes are usually going to
be something really really, really nice. We already had one example. For the 4D hypercube the
maximal three-dimensional shadow was this rhombic dodecahedron that's
spinning there. Very impressive but gets even more impressive when you up the
dimensions. Now I've actually had a look at the literature and it seems like we don't
really know much in terms of these maximal shadows beyond dimension 6 so we
know the maximal shadow for four dimensions, for five dimensions and for
six dimensions and let's just go for six dimensions okay. That maximal shadow, that
maximal 3d shadow of a 6D unit hypercube looks like this and it's got a
name. It's a very famous shape in geometry, it's called a rhombic triacontahedron.
Looks impressive and let me just tell you a couple of the amazing
features of this thing. Alright here we go:
This is one of the views of this rhombic triacontahedron. Now let's split it up
into three views, three particular symmetric ones. On the left you see a
five-fold symmetry, there you see a three-fold symmetry and just in the
middle there is a mirror symmetry. Now , every one of the 30 faces is a
golden rhombus. How's it golden. Well if you take the long
diagonal and divide it by the short diagonal you actually find its always
golden ratio. That makes it a golden rhombus.If we just look at a wireframe
of this triacontahedron you get this and inside you find all the regular
solids of 3D. For example, here we find the cube, there's actually quite a few
cubes in there. There's a dodecahedron, there's a tetrahedron, again lots of
those in here. There's an octahedron, again lots of those in here. There's an
icosahedron and well much much more stuff to be found in here. I'm sure
everybody here has seen pictures or animations of higher-dimensional cubes.
These are all shadows but these are usually not shadows of solid higher-
dimensional cubes, they're shadows of wireframes, so like this. What's spinning
up there is a shadow of a spinning four- dimensional cube. We can
play the same sort of game for our six- dimensional cube and see what happens.
You get a lot more structure inside, it's just not just a blob with an outline
there's going to be lots of structure just like in this guy here. I just fill in
those extra bits that you get here. So here is what you get, I'm actually getting
ready for 3D printing these things, that's going to be quite exciting. So all
these struts are of the same length. Inside there you find a lot more of
these golden rhombuses and they form other incredible shapes. I
just want to highlight one of them, one that you've probably seen before. See
whether you can guess it, I'll just zoom in on it. You recognize this one? Well it's
actually the logo for Wolfram Alpha. it's called the hexecontahedron.
What else is in there? Well I'll just show you one more although there's
really zillions more. It's to do with this guy here. This is something pretty
amazing, right? It's somewhere in there, so maybe you have another close look at
this rhombic triacontahedron and see whether you can find this shape in
it. Now just recently Matt Parker
actually did a whole video, really fun video about the rhombic triacontahedron.
Also check that one out, really really a lot of fun. Usually I try to
give you a bit of a insight into how these amazing things that I'm talking
about are actually proved. Now it turns out that it's quite hard to do this for
the shadow theorem, but it's quite doable for the 2D part and the 3D part. So let me
just show you the 2d part. So what a 2d cube? A 2d
cube is just a square. And how can you split up 2D? Well really only in one
way, into two one-dimensional bits, basically the x-axis and the y-axis. So
here we've got the square and what are the shadows on the x-axis and the y-axis?
There they are. As I rotate this thing around it's pretty
obvious that the shadows are always going to be the same length. Why? Well because
the square has 4 fold rotational symmetry. Pretty easy but it
suggests a nice question: Are there other shapes that have this strange shadow
property. In 2D not a big deal. (For example) anything with the four-fold symmetry
should work. For example, a regular 8-gon like this guy, you know I can rotate it
around, pretty obvious this is going to have the shadow property. Great, what
about 3D? It gets a lot more tricky there.
Apart from the cube are there other shapes that have the same shadow
property? Answer is yes. One of them is really easy to find. Beyond that first
easy one pretty hard. See whether you can find any of them. Pretty
happy with the first one. If anybody finds any of the other ones I'll be
really impressed. So leave your findings in the comments. Now just in case you are
familiar with the cross product you can actually try to solve one of my first year
assignments that I often give to my students, so this guy here. I'm also going
to put it in the description so you can try and solve this. A lot of my 1st year
students can do it so see where they can do it too and if you can, again, leave
your solution in in the comments. What else?
Well what about Prince Rupert, Prince Rupert in higher dimensions? For example,
can a hypercube pass through itself? Answer is yes but really this video is
already getting reading way way too long as usual, so I better stop here and I
promise I'll talk about it in another video. So I think this is it for today
and well I hope you enjoyed it and if you did say so in the comments. And I'll
see you next time.