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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 :)