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 :)
23 of us think one of the best Mathologer videos ever is worth an up vote as opposed to the math equivalent of a cat picture which gets 2.6k. This is so depressing.
So does this circle stacking idea extend to 3-D? If so, it gives a way to think about the structure of embedded crystals inside another crystal.