Sphereflakes
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0:00 - 0:01So in my last video,
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0:01 - 0:03I joked about folding and cutting
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0:03 - 0:04spheres instead of paper.
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0:04 - 0:05But then I thought, why not?
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0:05 - 0:07I mean, finite symmetry groups
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0:07 - 0:09on the Euclidean plane are fun and all.
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0:09 - 0:11But there are really only two types:
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0:11 - 0:12some amount of mirror lines around a point,
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0:12 - 0:15and some amount of rotations around a point.
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0:15 - 0:17Spherical patterns are much more fun.
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0:17 - 0:18And I happen to be a huge fan of some of
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0:18 - 0:22these symmetry groups – maybe just a little bit.
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0:22 - 0:24Although snowflakes are actually
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0:24 - 0:25three dimensional. (3-D.)
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0:25 - 0:26This snowflake doesn't just have
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0:26 - 0:27lines of mirror symmetry –
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0:27 - 0:29but planes of mirror symmetry.
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0:29 - 0:32And there's one more mirror plane –
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0:32 - 0:34the one going flat through the snowflake –
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0:34 - 0:36because one side of the paper mirrors the other.
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0:36 - 0:38And you can imagine that snowflake
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0:38 - 0:39suspended in this sphere –
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0:39 - 0:41so that we can draw the mirror lines more easily.
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0:41 - 0:43Now this sphere has the same symmetry
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0:43 - 0:45as this 3-D paper snowflake.
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0:45 - 0:46If you're studying group theory,
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0:46 - 0:48you could label this with group theory stuff.
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0:48 - 0:49But whatever.
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0:49 - 0:51I'm going to fold this sphere on these lines,
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0:51 - 0:52and then cut it – and that will give me
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0:52 - 0:53something with the same symmetry
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0:53 - 0:55as a paper snowflake – except on this sphere.
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0:55 - 0:56And it's a mess.
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0:56 - 0:57So let's glue it to another sphere.
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0:57 - 1:00And now it's perfect and beautiful in every way.
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1:00 - 1:03But the point is it's equivalent to this snowflake –
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1:03 - 1:04as far as symmetry is concerned.
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1:04 - 1:07Okay, so that's your regular old six-fold snowflake.
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1:07 - 1:09But I've seen pictures of twelve-fold snowflakes.
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1:09 - 1:10How do they work?
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1:10 - 1:12Sometimes stuff goes a little oddly
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1:12 - 1:14at the very beginning of snowflake formation,
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1:14 - 1:16and two snowflakes sprout, basically,
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1:16 - 1:18on top of each other – but turned 30 degrees.
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1:18 - 1:20If you think of them as one flat thing,
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1:20 - 1:21it has twelve-fold symmetry.
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1:21 - 1:23But in 3-D, it's not really true.
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1:23 - 1:25The layers make it so there's
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1:25 - 1:26not a plane of symmetry here.
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1:26 - 1:28See. The branch on the left is on top.
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1:28 - 1:29While in the mirror image,
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1:29 - 1:31the branch on the right is on top.
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1:31 - 1:32So, is it just the same symmetry
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1:32 - 1:34as a normal six-fold snowflake?
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1:34 - 1:36What about that seventh plane of symmetry?
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1:36 - 1:37But no, through this plane,
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1:37 - 1:39one side doesn't mirror the other.
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1:39 - 1:41There's no extra plane of symmetry.
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1:41 - 1:43But there's something cooler –
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1:43 - 1:44rotational symmetry.
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1:44 - 1:46If you rotate this around this line,
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1:46 - 1:47you get the same thing –
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1:47 - 1:49the branch on the left is still on top.
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1:49 - 1:51If you imagine it floating in a sphere,
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1:51 - 1:52you can draw the mirror lines,
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1:52 - 1:54and then you have twelve points
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1:54 - 1:55of rotational symmetry.
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1:55 - 1:58So I can fold, then slit it, so I can [indistinct]
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1:58 - 1:59around the rotation point.
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1:59 - 2:01And cut out each 'sphereflake'
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2:01 - 2:04with the same symmetry as this. Perfect.
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2:04 - 2:05And you can fold spheres
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2:05 - 2:07other ways to get others patterns.
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2:07 - 2:09Okay. What about fancier stuff like this?
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2:09 - 2:10Well, all I need to do is figure out
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2:10 - 2:12the symmetry to fold it.
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2:12 - 2:13So, say we have a cube.
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2:13 - 2:15What are the planes of symmetry?
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2:15 - 2:16It's symmetric around this way,
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2:16 - 2:18and this way, and this way.
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2:18 - 2:19Anything else?
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2:19 - 2:22How about diagonally across this way?
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2:22 - 2:24But in the end, we have all the fold lines.
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2:24 - 2:25And now, we just need to fold
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2:25 - 2:26the sphere along those lines
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2:26 - 2:29to get just one little triangle thing.
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2:29 - 2:30And once we do,
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2:30 - 2:31we can unfold it to get something
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2:31 - 2:33with the same symmetry as a cube.
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2:33 - 2:34And, of course, you have to do something
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2:34 - 2:35with tetrahedral symmetry,
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2:35 - 2:37as long as you're there.
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2:37 - 2:38And, of course, you really
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2:38 - 2:39want to do icosahedral.
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2:39 - 2:40But the plastic is thick and imperfect,
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2:40 - 2:41and a complete mess.
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2:41 - 2:42So who knows what's going on.
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2:42 - 2:43But at least you could try some other ones
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2:43 - 2:46with rotational symmetry and other stuff –
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2:46 - 2:47and make a mess.
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2:47 - 2:48And soon, you're going to want to fold and cut
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2:48 - 2:50the very fabric of space itself to get awesome
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2:50 - 2:51infinite 3-D symmetry groups,
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2:51 - 2:53such as the one water molecules follow
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2:53 - 2:56when they pack together into solid ice crystals.
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2:56 - 2:57And before you know it,
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2:57 - 2:58you'll be playing with multidimensional
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2:58 - 2:59quasi-crystallography,
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2:59 - 3:00or lie algebras or something.
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3:00 - 3:03So you should probably just stop now.
- Title:
- Sphereflakes
- Description:
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Folding and cutting spheres.
Awesome symmetry balls include card constructions by George Hart, which you can learn about here (and others are on georgehart.com): http://youtu.be/YBUEYrzMijA
and the Gardner Ball by Oskar van Deventer with design by Scott Kim made in honor of Martin Gardner, which you can learn about here: http://youtu.be/rVHXlltXIlI
and the Wolfram rhombic hexacontahedron thingy, and others! - Video Language:
- English
- Duration:
- 03:05
Mike Ridgway edited English subtitles for Sphereflakes | ||
Mayank Singh edited English subtitles for Sphereflakes | ||
Mayank Singh edited English subtitles for Sphereflakes | ||
Mike Ridgway edited English subtitles for Sphereflakes |