Of course most of you are here to find out what a Klein bottle Rubik's cube
looks like. So here straightaway are three different views of the same Klein
bottle Rubik's cube in the last stages of being solved. Sounds mysterious but I'll
explain. Now this Klein bottle Rubik's cube is just one of hundreds of
different Rubik's Cube like puzzles in MagicTile, an amazing piece of software
by Roice Nelson, and it's completely free. So get your copy, link is in the description.
MagicTile contains some of the most difficult twisty puzzles you'll ever see
but also a lot of really easy ones that you can solve completely
intuitively, so without looking up any sort of algorithm. Today's video is an
introduction to MagicTile, to this universe of puzzles and some of the nice
mathematics at the core of it. So, what do these two patches have to do with
Klein bottles.
Well it's really quite easy, you just take a square and bring the two red
sides together like so that gives you a cylinder. Now the open ends we can bring
together like that and that will give us a doughnut or, in math lingo, a torus. That's the
the standard way of bringing the two ends together. There's also a non-standard way and
that goes like this and you've got your Klein bottle. So, really, when you look at
the square you're dealing with a flat Klein bottle, so as long as we specify
how the two opposite sides are glued together.
Well with the red ones we already know they're glued together straight across.
What about the yellow ones? Well, there, if you have a close look, it turns out they are
identified with a twist,
so just like a Möbius strip. So this thing actually does contain lots of Möbius
strips. In other words, what we're dealing with here is a pacman space. So there's
little Pacman. As he hits one of the red sides he re-enters right opposite. As he
hits one of the yellow sides he re-enters diagonally opposite, but also
mirror reversed which is very very important.
Remember that one when you are solving one of those Klein bottle Rubik's cubes and
some pieces just don't seem to fit anywhere. This flat Klein bottle is
actually the nicest Klein bottle there is, much nicer than this one as far as
mathematicians are concerned. Why? Because it's cut from a super nice surface a flat plane.
So what that means is that for a little Pacman that kind of floats around here
this guy here looks the same wherever he is, so points are not distinguishable for a
little Pacman, unlike here. So, when Packman floats around inside this surface
he can actually do little experiments and distinguish between points.
Sne little experiment like
this is draw a little triangle. Do it here and you get not what you would
expect 180 degrees as the angle sum of this triangle but you
actually get something less than180 degrees. On the other hand, if
you draw a little triangle up there, you get more than 180 degrees
and that number will change as you move Packman around inside the surface.
Unlike here, you can draw little triangles whereever you want, the angle sum will
always be 180 degrees, just like in the plane. And, of course, now you know
what it means for this to be a Klein bottle Rubik's Cube. We can actually also
bend this square together into a real Klein bottle as I showed before and then
you get this guy here which is actually also part of MagicTile so you can
actually solve this puzzle using that view, too. It's a lot more complicated and
most people would use that flat Klein bottle view. If you're a four-dimensional
alien or you have access to the fourth dimension you can actually glue the
sides of your square together so that the squares all stay square. So it actually
stays super nice and you can play with this thing without open edges. So here's
a view of our Klein bottle Rubik's cube. We've got nine square faces and we're
turning around those faces.
What about here, how do we turn there. Well there's the turning circle. We know that
things get glued together here, so the turning circle cuts into this part and
really that part attaches here and here we go diagonally across and so this gray
bit attaches here and there's a little corner missing there, that comes from up
there. Now let's turn. There you go. Now this extending business that we just produced
just around here, we can actually extend all the way around and that's something
very very very useful, so
let's just do it and actually extend all the way. What you are seeing here is the square tiling
of the plane colored in with nine different colors and whenever you see a
red tile you are actually looking at the same face. If you're focusing in on any
three-by-three block you get a different view of the Klein bottle. It's basically
moving or turning the Klein bottle Rubik's cube around and
viewing it from different sides when you moving this three-times-three block around.
What about individual pieces?
Well we've got center pieces here and there. The center pieces
don't move with respect to each other both here and there as we manipulatate
the cube. We've got corner pieces, three stickers here four stickers. We've got edge pieces,
two stickers here two stickers there. Now with this guy here we actually get one
additional kind of piece, the Star Trek logo like piece that also gets shuffled
around. This is actually just one possible twisty puzzle that you can put
on a Klein bottle. There are many many more.
So how do you get the other ones. Well you can, for example, shrink that cutting
circle here and when you do this you actually only get two different kinds of
pieces you get center pieces and these lens like edge pieces. Or you can make
the circles bigger. What that does is it destroys the centers here and again the
turn behaves differently but you can also move the circles so that they're
centered at corners. You may be familiar with that principle, if you've got a
Rubik's Cube collection you probably have something like, this is a Rex
cube that also turns around corners and if you've gotta Rex cube chances are you also
have a helicopter cube and with helicopter cubes you turn around edges and of
course you can do the same sort of thing with your Klein bottle Rubik's cubes.
Other variations, you can actually change the number of faces that are playing with. So
this is a six-faced Klein bottle Rubik's cube or here we've got four faces. In
fact you can do any number of faces. That looks just like what we looked at before
but it's actually different so this turning is different from the one we saw before.
Why? Because here we glue together across both in terms of the red sides
and in terms of the yellow sides. So that's actually a flat torus Rubik's
cube and if you actually do this physically you get this view here
which is also part of MagicTile and what you can also play around with. What we've
looked at so far only account for 28 of the 811 puzzles in MagicTile at the
moment. So what else is there?
Well lots more and it's all based on tilings. So our Klein bottle Rubik's cube here that's
based on a square tiling of the plane that's colored in a special way. Now
there's other tilings of the plane, other nice regular tilings or the plane actually
there is two more.
There's one with regular hexagons, for example, so this one here and there's a
special kind of coloring of this tiling and we've got the Rubic's cube cuts on
top and so when you have a close look at this
you'll find there's nine colors here and we get this patch and when you have a close
look you see that the sides of the patch are glued together straight across this
way and with a twist that way. So this gives us another kind of Klein bottle
Rubik's Cube and actually for me this is really THE Klein bottle Rubik's Cube. Lots of
different variations but which one is IT? This is IT. Everything you see here has a
counterpart in the real Rubik's cube. There's the center pieces, one sticker
edge pieces, two stickers, corner pieces three stickers. So if you don't want to
solve anything else and you are in for a challenge you accept the Mathologer Klein bottle
challenge. So Roice and I agreed that there would be a limited-edition Hall of
Fame just for this video here. So what you can do is you can download the
program, choose this puzzle, do a full scramble, solve it,
save your log file, send it to Roice and then he puts your name into the limited
Hall of Fame. It's got space for 100 people. It's a fairly hard puzzle, especially since
there's no ready-made solutions available for it, you have to make them
up yourself. (If you don't know how to do this check out that video there.) But even for
people who are kind of experienced in terms of making up algorithms you'll
find lots of interesting stuff here that you won't find anywhere else.
Okay so check it out. I am interested in seeing how you go with this.
So we've had regular tilings with hexagons, regular tilings with squares
now we've also got these guys here, regular tilings with equilateral triangles and all
of these you can use to make up the Klein bottle and torus Rubik's cubes.
What I've talked about so far accounts for one-third of MagicTile. MagicTile
also has the regular tilings of the sphere and there's five of them. They
correspond to the five Platonic solids. These tilings here very different from the
ones that we just looked at importantly they only have finitely many times. This
only has four tiles, this only has eight tiles and of course with the Cube we've only got as many as there are faces:
six tiles. So what we don't have to worry about to start with is coloring
multiple faces here with the same color, we could just have one tile play the
part of one face and when you do this you get all these sorts of puzzles. So
that's a standard Rubik's Cube that's a Gigaminx,
that's a Jing's Pyraminx and those ones you can probably not find in shops and
there are hundreds of them in MagicTile. We've covered two-thirds of MagicTile
this way. Apart from the sphere and the plane there's one more supergeometry
which is the hyperbolic plane which you can use for making up these puzzles.
There are some incarnations of it here. For example, this coral that's part,
closely part of a hyperbolic plane that actually goes on forever, so it's kind of
really hard to see what's going on,
it gets more and more crinkly on the outside it's not as obvious that this sort of
thing should have the same property as the sphere, with the little Pacman kind of
being able to slide around in here and not being able to distinguish were he is but
they have this property and also in terms of tilings the hyperbolic plane
actually admits infinitely many different tilings. In fact, even better you
can just take any regular n-gon like an equilateral triangle and can make up
infinitely many regular tilings with this triangle or infinitely many with
a square or infinitely many with a regular pentagon and so on. These are bit of a
mess to look at so how do we actually represent them in MagicTile.
Well there's nice pictures
that are actually very symmetrical and that, you know, represent a lot more the symmetry
and the niceness of this plane than the object itself and so just like this is a
picture, a nice picture of this tiling here which is completely flat that is a flat
picture of a regular tiling with regular 7-gons. So, of course, all
these things look very different
so this looks like a regular 7-gon but this one doesn't. Why?
Because we've got a distorted view here we've got a picture of the real
thing but they're all regular 7-gons and I can actually turn this thing around
so let's just turn it around. So if I rotate one of those other 7-gons into the
middle it actually will appear as a regular 7-gon, so all of these things
are regular 7-gons, there is infinitely many of them and I said there is infinitely many
tilings of this thing. Now this particular one is of course one of the special
colored ones. So now we can turn this into a Rubik's cube by putting in the cutting circles.
We're using 24 colors here. Let's just isolate a patch. When we glue this patch
together as the colors prescribe we actually get this 3-torus which with
the special kind of tiling in the background is called a Klein quadric. So
this is a Klein quadric Rubik's cube. So also get that one and play with. It's
actually easier to solve then the Klein bottle Rubik's cube, mainly because of the
absence of the twists. There's a lot more things I could talk about here in terms
of the math. There is orbifolds and there is projective planes and all kinds of other
things but I let you find this out for yourself. Now there will also be another
video on Mathologer 2 in which I talk about MagicTile itself, the interface,
how your program algorithms, how you find algorithms and what I'll focus in
particular are the easiest types of MagicTile puzzles and these are the
Harlequin puzzles.
These are edge turning puzzles. So very very nice puzzles, easy to solve
check that out too if you are really keen. And that's it for today :)