Placeholder Image

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

字幕與單字

單字即點即查 點擊單字可以查詢單字解釋

B2 中高級

球形片 (Sphereflakes)

  • 0 0
    林宜悉 發佈於 2021 年 01 月 14 日
影片單字