Welcome to another Mathologer video. Have a look at this -- four equally spaced

circles in a box. Let's fill up the box with circles by stacking them row by row

like this: stack, stack, stack. Okay let's

start again and have a closer look at what's happening here. Because the first

row of circles is perfectly level and the circles are equally spaced the same

is true for the second row and for the third and for all subsequent rows. Right?

Let's now do the stacking again but before we begin we'll slide the two

inner circles a little along the base like that. With the circles in the first

row no longer equally spaced, the second row ends up all crooked. And so does the

third, the fourth, the fifth, the sixth and ... well, not the seventh. The seventh row

ends up being perfectly level just like the first row. Surprised? I sure hope so.

And this is not a fluke. Let's move the middle two circles of the first row

around a little more and see how the stack changes.

Pretty mesmerizing isn't it? So why is row seven always perfectly level? What's the

secret? In fact, this row seven business is just

the tip of an iceberg of amazing circle stacking phenomena.

So let's first go for a bit of a tour of these phenomena and then I'll give you a

super nice animated explanation of what is going on. Alright

here we go. This time let's begin with six circles instead of four. When we move

the inner circles at the bottom you can see that the rows are again all over the

place, ... until we get to row 11 which remains level no matter how we begin.

In fact, no matter the beginning number of circles on the bottom row there will

always be a corresponding magic number M with the Ms row guaranteed to be level.

First challenge for you: if we start with three circles in the bottom row, what's

the corresponding magic number? Too easy? Well then, what's the general rule? So if

we start with N circles at the bottom what is the corresponding magic number?

Maybe also experiment with some coins in a box or even more fun some bottles in a

wine rack. Just make sure it's the rows that are out of order and not just your

wobbly wine sight. Anyway it's really extra magical when you see this kind of

order arise out of chaos with real world objects. Okay, here's the next stunning

highlight of our tour. If you don't build the whole stack but instead just this

pyramid here, then the center of the tip of the pyramid will always be exactly

the same distance from the two walls. The red circle can move up and down a bit

but its center will always be on the pink line in the middle. That gives a

pretty bizarre way to find the middle between two walls: just build any weird

circle pyramid like this and the tip will end up right in the middle. Crazy, hmm?

Next highlight. If you do build the whole stack, then the red circle at the pyramid

top is not only exactly halfways between the walls

it's right smack at the center of the whole stack. Even better, it turns out the

stack always has a half turn symmetry with the red circle as its center.

So when you rotate the stack 180 degrees around the red circle it comes into

coincidence with itself like this. There's plenty more amazing stuff here, some of

which I'll mention further down the line but I assume I've already shown you

enough to have you hooked and craving for an explanation for these circle

stacking phenomena. Right? Okay, so let's begin with a visual proof from the BOOK

as to why the tip of the pyramid is always smack in the middle of the two

walls. As usual I'll focus on a special case that illustrates how things work in

general. Let's make a copy of the pyramid and flip it so that its tip is

pointing down. So flip. Now connect the centers of all touching circles like

that and let's have a really close look at the resulting grid. The first thing we

notice is that all the segments of the grid have the same lengths. This length

is twice the radius of the circles. Cool. Second, because all the segments are

the same lengths, all the 4-sided cells of the grid are rhombuses. And what do we

remember about rhombuses from school? Well quite possibly nothing but once upon

a time many of us knew that opposite sides of a rhombus are always parallel.

And so these two segments here are parallel and so are these two. But then

opposite sides of this adjacent rhombus are also parallel and so all three red

segments that you see here are parallel. But why stop? Of course by now it should

be clear that all four red segments that you see here are parallel. All clear? Great :)

But then these blue segments are also parallel. Right? And these

green segments are parallel as well. Piecing it all together we see that the

two jagged multicolored curves are parallel. Third, we're getting there, the

line through the center of the tip of the pyramid also passes through the tip

of the pyramid that's pointing down. Pretty obvious since we flipped the

pyramid vertically like this. Now can you already see just by looking at this

diagram why the pink line is right in the middle of the two walls? Well because

of all that parallel business this orange distance here can also be

found down there and therefore also here. And now it's completely obvious that the

pink line is the same distance from the two walls, the distance that's here is

the same as that one there. Perfect :) Do you like this proof? Now I want to go one

step further and show you why row 7 is always level and why our stacks have a

half-turn symmetry. For this the first crucial observation is that the original

base circles are level and so these points here are horizontally aligned.

That means that the other pairs of corners in these rhombus cells are

vertically aligned. So they are vertically aligned, vertically aligned,

vertically aligned. Okay, make a copy of the grid, explode the copy like this and

notice that we can reassemble the pieces in reverse order. In this new grid

it is now these corners here that are vertically aligned. Right? Now combine the

two grids and make a second copy of this super grid. There we go.

Give the second copy a 180-degree twist. The two super grids now combine into a

mega grid. Little bit of a challenge: try to justify as concisely as possible why

the two super grids combine seamlessly. let us know in the comments.

Anyway after all his mega grid making we now reinsert circles centered at the

vertices of the mega grid. These are the horizontally aligned circles we started

with. These circles determined a position of everything else here. Right? Now these

points are vertically aligned and the corresponding circles obviously hug the

right wall. Remember to build our mega grid we take

this part here, make a copy and now turn 180 degrees. But of course anything

horizontal stays horizontal under a 180 degree turn and anything vertical stays

vertical. But that means that the top row of circles is horizontal because it

comes from the horizontal row of circles at the bottom. As well, this column of

circles on the left is vertical and hugs the wall because it comes from the

vertical column of circles on the right. Now fill in all other circles in the

middle and so our mega grid corresponds to a stack that fits perfectly within

the walls of our box. But of course since we began with a stack that fits

perfectly within the walls of our box and both stacks are completely

determined by the common four circles at the bottom the two stacks must be

identical. Magic :) We've already seen how the internal structure of the stack

forces the top to be level. Also remember that we combined a super grid with a

half-turned (copy of this) super grid into our mega grid like this. Alright, this means that the

mega grid has a half turn symmetry and so the associated stack must also have a

half turn symmetry. Just gorgeous how everything in this proof fits together

and reveals the topsy-turvy inner structure of our stack, isn't it?

Still with four circles at the bottom why is it row number 7 that is special?

What does 7 have to do is 4? This is actually pretty easy to see given the

symmetric structure of the stack. I'll leave it to you to nut out the

arithmetic in the comments. Wonderful we're done :) But now are you ready for the

foundations of your mathematical universe to be shaken? Well, ready or not

here we go. This is the setup we started with. Let's now make the box a little bit

wider. Moving the inner green circles around the seventh row stays perfectly

level. No surprise there. But now watch this ... Damn, the top is no

longer level !!! But we just proved that the top is always level. So what happened?

We broke maths? Here's the same sort of meltdown with a wider stack. Happy to

stop here? Shall we just declare maths is broken and leave it? Or shall we figure

out what's going on? Alright Watson the game is afoot, let's have another look at

the first meltdown. Definitely the nicely level top is broken but notice that

things didn't completely break. The pyramid inside is still behaving exactly

as predicted. In particular the tip is still smack in the center of the two

walls. Anyway, let's reconstruct our mega grid starting with this pyramid. Okay

there's the mega grid. Put in the rhombuses. Hmm, as you can see things

still line up perfectly at this stage, no meltdown yet. But now have a look at this

region down there and see what happens when we try to complete our construction

of the stack by putting in the circles. As you can see, the circles in this

region overlap. Of course, that does not happen in the real stack. And why do the

circles overlap? Well because the rhombus at the centre of this area is too thin

to keep the two green circles from overlapping. But this rhombus is the same

as that one there. And that rhombus is too thin because we moved these two

green circles at the bottom too far apart which made the orange circle in

between almost fall through to the bottom. But, it turns out that's all that

can go wrong. To be absolutely sure that all

our stacks exhibit all the super nice properties I've been raving about

all we have to do is guarantee that the gap between any two adjacent circles at

the bottom is within this range here. Opening any of the gaps wider means our

proof no longer applies and the resulting stacks go all wacky. Of

course if you open the gap really wide the orange circle will just fall through

to the bottom and with this extra circle at the bottom the stack can return to

behaving nicely, like in this example here. And so what's happening here is

that the ill-formed stack is sort of a phase transition between nicely behaved

stacks with different numbers of circles at the bottom. Beautiful stuff, isn't it?

By the way, our circle stacking theorem was

discovered by Charles Payan in 1989, not that long ago. The proof that

I showed you is by the Mathologer. Are you proud of me? All this is very beautiful but are

there any practical applications? I don't know of any. So have you got any

suggestions? To finish off here a few more closely related to circle stacking

marvels. Turns out that even if the walls of our box are not perfectly

vertical, the circles in the critical top row will still line up, just usually not

horizontally. Things even stay nice if you use different sized circles for

alternating rows which is also really cool. And, finally, something super nice.

I just spent five weeks in Germany. While I was there I showed my dad some of all

these little circle stacking miracles and we ended up building a physical

model of the grid underlying a circle stack from wood and metal. Let me show

you this model in action. As I change the position of the two screws at the

bottom row, you will see how the grid changes dynamically. Also note that the

top row will always be horizontal and that the whole grid always exhibits a

very pretty half turn symmetry. And that will be all for me for today. Enjoy :)