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• 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.

• 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.

So say it's the holiday season and you're

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Snowflakes, Starflakes, and Swirlflakes

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林宜悉 posted on 2020/03/30
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