If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:4:04

So say it's the holiday
season and you're supposed to be all
festive and jolly but you're more of
the grinchy type. And all you want to do is wield
sharp objects against things. So you're doing
your holiday partly by making paper snowflakes. Regardless of your feelings
about the holidays, slicing things into tiny
bits is an art and one that you are taking
very seriously. Like, some people make
so-called paper snowflakes by folding a piece of paper
in half, and then in half again and again and
then cutting it up. But paper snowflake connoisseurs
know that real snowflakes have six-fold symmetry and this
thing has four-fold symmetry. That would never
happen in nature. So you do it the real
way by folding your paper in half twice, and then
folding it in thirds. Or I suppose you could do it
by halves, and then thirds, and then halves again. But you can't do thirds, then
halves, then halves again, because what would
that even mean? But you do notice
your first halves can be at any angle you want. You don't need to line
it up or anything. It's kind of funny because
to get six-fold symmetry, you need to fold it
into 12 sections. And if you fold
into six sections, you only get
three-fold symmetry. Which is actually a way that
snowflakes occasionally form, so those are allowed. And then there are
even sometimes 12 fold symmetric snowflakes
in nature, which means you can fold
again to make that. But never four-fold. The problem with
folding paper is that the thickness
starts to get in the way. This makes points uneven, which
might actually be more natural. Most real snowflakes
are actually pretty lumpy and
flawed, just those aren't the ones people
take and share pictures of. And that's not the
kind of snowflake you want to make either. I mean, when you fold
this angle into thirds, this flap is under this one. So it has to be
a little shorter, at least if this
edge lines up here. But maybe if you folded
one in front and one in back accordion style,
then all the sections could be the same. In fact, then instead of
folding it in half like this you could do each
section back and forth. And that's much better. Or maybe you get bored
of six-fold symmetry and decide to make
a five-fold one. Well, if we need five lines of
symmetry, that's 10 sections. So first you fold
it in half and then you need to fold
this into fifths. You can use a protractor or just
kind of eyeball it and adjust. There, five-fold snowflake. In fact, if you
get good at folding this initial
five-fold wedge, you can do a single straight cut on
it to get a star super quickly. Or you can slice it
and get lots of stars, or cut fancy stuff in
there for fancy snowflakes. Stars count as
holiday spirit, right? And you can do seven-fold
symmetry in a similar way but you're probably going to
need your emergency protractor. But you could do nine-fold
without a protractor because you can do thirds,
and then thirds again. And if you can do fifths
without a protractor, you can do tenths too, because
it's a fifth times a half. Look, I said happy holidays
but I never said which one. Valentine's Day is
totally a winter holiday. 11 is prime, though, so time
for the protractor again. Look, prime factorization. OK. So now theoretically you can get
all sorts of end-fold symmetry, but what about
rotational symmetry? There's no mirror lines,
which means no folding, so does it even make
sense as a question? Cutting a snowflake
design efficiently is all about putting the same
cut lines on top of each other so you only have
to cut them once. So how do you take a
rotationally symmetric design like this and put all the
layers on top of each other without overlapping
anything else? Maybe it's not surprising
to see that to get stuff with rotational symmetry
to line up, you rotate it. If you make a cut to the center
then you can rotate all the way and roll the symmetry up
into one unique thing. It's hard to draw accurate
rotational symmetry by hand. But now I can symmetrize this
badly-drawn swirl design. So to cut out a paper swirl
flake, start with a cut, then curl your
paper into a cone. You can swirl around
once or twice or more. But the important
thing is to make sure the cut lines up with itself,
because as far as symmetry is concerned, that
cut doesn't exist. I like to tape it in place so it
doesn't unroll, then cut stuff out. I find that spiraly
things work well. Folding the paper is a
good way to start a cut, but remember that
folding creates symmetry. So I like to use it just to
get the scissors in there and then do
something asymmetric. Voila, snowflake. For a starflake
swirlflake you'll have to curl your paper around
five times, or four times. It's funny because I think
of this as going around once but really it's
going around twice, and a flat sheet of
paper goes around once. Anyway, yeah, do that. And then give it a nice
spiraly arm or two. You can make a nice fancy
starflake swirlflake snowflake, awesome flake. Of course, from snowflakes it's
only one small step to folding and cutting freeze patterns,
and then wallpaper patterns and, hey, what kind of
patterns do you get if you start by folding
stuff into a [INAUDIBLE] strip? And then maybe you'll want
to start folding and cutting spheres and everything
will be a mess, so you'd better just stop now.