Snowflakes, starflakes, and swirlflakes

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.