## 字幕列表 影片播放

• So in my last video I joked about folding and cutting

• spheres instead of paper.

• But then I thought, why not?

• I mean, finite symmetry groups on the Euclidean plane

• are fun and all, but there's really only two types.

• Some amount of mirror lines around a point,

• and some amount of rotations around a point.

• Spherical patterns are much more fun.

• And I happen to be a huge fan of some of these symmetry

• groups, maybe just a little bit.

• Although snowflakes are actually three dimensional,

• this snowflake doesn't just have lines of mirror symmetry,

• but planes of mirror symmetry.

• And there's one more mirror plane.

• The one going flat through the snowflake,

• because one side of the paper mirrors the other.

• And you can imagine that snowflake suspended

• in a sphere, so that we can draw the mirror lines more easily.

• Now this sphere has the same symmetry

• as this 3D paper snowflake.

• If you're studying group theory, you

• could label this with group theory stuff, but whatever.

• I'm going to fold this sphere on these lines, and then cut it,

• and it will give me something with the same symmetry

• as a paper snowflake.

• Except on a sphere, and it's a mess, so let's

• glue it to another sphere.

• And now it's perfect and beautiful in every way.

• But the point is it's equivalent to the snowflake

• as far as symmetry is concerned.

• OK, so that's a regular, old 6-fold snowflake,

• but I've seen pictures of 12-fold snowflakes.

• How do they work?

• Sometimes stuff goes a little oddly

• at the very beginning of snowflake

• formation and two snowflakes sprout.

• Basically on top of each other, but turned 30 degrees.

• If you think of them as one flat thing, it has 12-fold symmetry,

• but in 3D it's not really true.

• The layers make it so there's not a plane of symmetry here.

• See the branch on the left is on top, while in the mirror image,

• the branch on the right is on top.

• So is it just the same symmetry as a normal 6-fold snowflake?

• What about that seventh plane of symmetry?

• But no, through this plane one side doesn't mirror the other.

• There's no extra plane of symmetry.

• But there's something cooler.

• Rotational symmetry.

• If you rotate this around this line, you get the same thing.

• The branch on the left is still on top.

• If you imagine it floating in a sphere

• you can draw the mirror lines, and then

• 12 points of rotational symmetry.

• So I can fold, then slit it so it

• can swirl around the rotation point.

• And cut out a sphereflake with the same symmetry as this.

• Perfect.

• And you can fold spheres other ways to get other patterns.

• OK what about fancier stuff like this?

• Well, all I need to do is figure out the symmetry to fold it.

• So, say we have a cube.

• What are the planes of symmetry?

• It's symmetric around this way, and this way, and this way.

• Anything else?

• How about diagonally across this way?

• But in the end, we have all the fold lines.

• And now we just need to fold a sphere along those lines

• to get just one little triangle thing.

• And once we do, we can unfold it to get something

• with the same symmetry as a cube.

• And of course, you have to do something

• with tetrahedral symmetry as long as you're there.

• And of course, you really want to do icosahedral,

• but the plastic is thick and imperfect,

• and a complete mess, so who knows what's going on.

• But at least you could try some other ones

• with rotational symmetry.

• And other stuff and make a mess.

• And soon you're going to want to fold and cut

• the very fabric of space itself to get awesome,

• infinite 3D symmetry groups, such as the one water

• molecules follow when they pack in together

• into solid ice crystals.

• And before you know it, you'll be

• playing with multidimensional, quasi crystallography,

• early algebra's, or something.

• So you should probably just stop now.

So in my last video I joked about folding and cutting

# 球形片 (Sphereflakes)

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林宜悉 發佈於 2021 年 01 月 14 日