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

by birtanpublished on August 23, 2020

so welcome to a different kind of mythology video now all my students and unit know that I have a bit of a cute fetish so mathematically cubes Rubik's cubes anything goes so what I want to do today

Is introduce you to some fairly recently discovered mind-boggling facts about cubes that hardly anybody knows about not even mathematicians alright so I've got to warm-up exercises for you the first one concerns this what is this

Well it definitely looks like a cube right well if you cuber you say no no it's not a cube it's a twisty puzzle it's a scoop so it's this cube shaped twisty puzzle well and you're right it's a twisty

Puzzle but it's not cube shaped let me just have a peel it off the table here and show it to you and there you see it's not a cube it's actually 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 so I've got a second warm-up exercise and the second warm-up exercise is what is that well it definitely looks like a circle right but it actually it's not a circle

No it's a 2d shadow of a 666 dimensional cube well those exercise were just to whet your appetite now let's get into it all right 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 all right so our story begins in 1985 1985 mathematician Peter McMullen discovered effect about

The cube that nobody had seen before as far as I know all right so imagine a cube like this hovering in midair there's the Sun right above casting a shadow on the ground okay well I've got a bit of an animation here for

You all right so what we're going to do is we're going to measure the area of the show okay and the other thing we're gonna do is we're gonna 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 all right okay now for the really really amazing bit let's adjust the side lengths 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 okay so

Usually we measure say in meters distance and then well if it's meters and distance we usually use meters squared for areas so let's just do that so now if you've got a cube that side lengths 1 meter then all this will work

Out the shadow theorem will guarantee that the two numbers were measuring are going to be exactly the same now if you're measuring in meters but you use a cube that has side lengths to meters this will no longer work and it's also

Pretty obvious if it works for 1 meter because distances and area scale in different ways it will definitely never work for anything else so very cute property but is it useful well yes so 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 of urns well 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 where the shadow is just a copy of this square face all right what about the large possible shadow for that we have to arrange the cuban space so that we get maximal distance and obviously that's

The case when we got one corner right above the other one like this and when you do the calculations or when you do the experiment you actually find that the shadow then 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

Sort of square hole so I'll just take it out yeah and now what I can do is something amazing I can actually pass the second cube through the hole like so very neat John 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 notice that something like this is possible is a Wars Prince Rupert who lived in the

17th century and he was a general so basically in the business of killing people but he was also quite interested in science and if you are like 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 drop and that's well you just melt a bit of glass and then you pour it 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 was expect but if you just snip the tail off a little bit the whole thing will explore and basically to

To dust really amazing especially if it's captured on slow-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 exploring it's amazing so

Check out these videos alright and this is the end of part 1 in part 2 we'll go a lot deeper Matt's wise so you know if you up for it check out part 2 you

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