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# The cube shadow theorem (pt.1): Prince Rupert's paradox

Welcome to a different kind of Mathologer video. All my students at uni
know that I have a bit of a cube fetish-- mathematically cubes, Rubik's cubes,
anything goes. What I want to do today is introduce you to some very recently
discovered mind-boggling facts about cubes that hardly anybody knows about,
not even mathematicians. Alright, so I've got two warm-up exercises for you. The
first one concerns this. What is this? Definitely looks like a cube, right?
Well, if you're cuber you'll say, nah, nah not a cube, it's a twisty puzzle. It is a Skewb.
So it's a cube-shaped twisty puzzle. And you're right, it's a twisty puzzle, but
it's not cube-shaped. Let me just peel it off the table here and show it
to you. There you see, it's not a cube. The shape here is a
rhombic dodecahedron, so it's twelve rhombuses, very symmetric, really nice
thing, we'll need it for later, so keep that one in mind. I've got a second
warm-up exercise and the second warm-up exercise is, well what is that? Well it
definitely looks like a circle, right, but it actually is not a circle. No it's a
2d shadow of a 666-dimensional cube. Those warm-up exercises were just to whet
your appetite now let's get into it. Alright so the plan is to get this video
done in two parts and the first part I'll introduce you to the basic facts.
Pretty much anybody can appreciate these I think and enjoy them and in the second
part I drill much deeper mathematically, as usual. Alright so our story begins
in 1985. In 1985 mathematician Peter McMullen discovered a fact about the
cube that nobody had seen before, as far as I know.
Imagine a cube like this hovering in midair with the Sun right above casting a
shadow on the ground. okay well I've got a bit of an animation
here for you. What we're going to do is we're going to measure
the area of the shadow. And the other thing we're going to do is we're
going to measure the difference in height between the topmost point of the
cube and the bottom most point of the cube. As I rotate the cube, the shape of
the shadow will change and so will these two numbers. Now for the
really really amazing bit, let's adjust the side length of our cube to be
exactly 1. then the number measuring the area of the shadow and the number
measuring the height difference will become exactly equal. And this is not a
coincidence: as long as we are dealing with a UNIT cube this will always be the
case, no matter how the cube is oriented in space and what shape the shadow pans out
to be. Now people often get confused by this unit cube business. So let me say a
little bit more about this. So usually we measure say in meters
distance and then well if it's meters for distance we usually use meters squared
for areas. So let's just do that. Now if you've got a cube that has side length
one meter, then all this will work out. The shadow theorem will guarantee that
the two numbers we are measuring are going to be exactly the same. Now if you're
measuring in meters but you use a cube that has side length two meters this
will no longer work. And it's also pretty obvious why. If it works for one meter,
because distances and area scale in different ways, it will definitely never
work for anything else. Very cute property but is it useful?
Well yes. For example, if we want to figure out what the smallest possible
shadow is or what the largest possible shadow is, because those two numbers are
the same, the only thing we have to figure out is what's the smallest
possible height difference and what's the largest possible height difference.
Well the smallest possible height difference we definitely get when we orient the
cube like this, just parallel to the ground, right? Then the height difference
is just that, you can't get any smaller than this obviously. What's the shadow?
The shadow is just a copy of this square face, alright. What about
the largest possible shadow? For that you have to arrange the cube in space so that
we get maximum distance and obviously that's the case when we've got one
corner right above the other one like this. And when you do the calculations
or do the experiment we actually find that the shadow is
a perfect regular hexagon which is also pretty neat. So what I've drawn here is
the smallest possible shadow and the largest possible shadow and you can
actually see that the smallest shadow fits inside the largest one and that
suggests something really cute. What you do is you take a cube like this and
drill a hole through it with this square as cross-section. Okay I've actually not
done this but I've done a 3d printed version of this. So here are two cubes of
exactly the same size and one features one of those square holes. So I'll just
take it out. And now what I can do is something
amazing. I can actually pass the second cube through the hole like so. Very neat! So I can actually pass a cube through itself and you can actually do
even better. You can see I can make this square hole even even bigger and then I
can pass an even larger cube through this hole. Now the first to notice that
something like this is possible is was Prince Rupert who lived in the 17th
century. He was a general, so basically in the business of killing
people but he was also quite interested in science and if you are quite
active on YouTube I'm pretty sure you've come across some videos about the Prince Rupert drop
not the Prince Rupert cube but the Prince Rupert's drop and that's,
well you just melt a bit of glass and then you pour into a bucket of cold
water and then you get these strange glass tadpoles which have this amazing
property that if you hit the front part with a hammer
they won't shatter like anybody would expect. But if you just snip the tail off
a little bit the whole thing will explore and basically
to turn into dust, really amazing, especially if it's captured on slo-mo
video. And a couple of people have done this. For example, here's a screenshot
from a video by Smarter Every Day. So you can really see this thing exploding, it's
amazing, check out these videos. And this is the end of part 1. In
part 2 we'll go a lot deeper mathswise. If you are up for that check it out.