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Course: Math for fun and glory > Unit 1
Lesson 9: Other cool stuff- What was up with Pythagoras?
- Origami proof of the Pythagorean theorem
- Wau: The most amazing, ancient, and singular number
- Dialogue for 2
- Fractal fractions
- How to snakes
- Re: Visual multiplication and 48/2(9+3)
- The Gauss Christmath Special
- Snowflakes, starflakes, and swirlflakes
- Sphereflakes
- Reel
- How I Feel About Logarithms
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Sphereflakes
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! Created by Vi Hart.
Want to join the conversation?
- did anyone notice that sphere made out of letters? how is that made? anyone know? :P(11 votes)
- it said gardn or something right(1 vote)
- How did she cut the ball and put it inside another one?(8 votes)
- The first one, Vi cut it up to be like a snowflake. Then she opened it, and put the second one inside of the first one, and blew up the the second one that was inside of the first one.(8 votes)
- I wish Vi can slow down and actually take the time to show us how to cut the paper specifically. Anyone know how to cut those?(6 votes)
- just cut out holes and or lines then make sure u dont cut nothing that connects or do what that person said just go on youtube but in stead of looking up vi look up how do you make a snow flake(2 votes)
- How does she make those?(4 votes)
- the ones she made during the video were made by cutting one of the balls up like the snowflakes, then she put another deflated ball inside and inflated it.
Some of the ones in the background were made by George Hart, Vi Hart's Father(6 votes)
- What does Euclidean mean❓(3 votes)
- Euclidean basically means planar. Some postulates in geometry are only true for when they are located on planes. For instance, a triangle when enclosed by a plane has internal angles which add up to 180 degrees. That is a Euclidean triangle.
But a triangle doesn't have to be enclosed by a plane. It could be enclosed on a sphere, in which case the interior angles always add up to something between 180 degrees and 540 degrees. It could also be enclosed on a hyperbolic plane, in which case the interior angles always add up to something between 0 degrees and 180 degrees.(4 votes)
- How do you make a sphereflake? Vi might have gone over this in the video but she talks too fast and I like to read instructions off paper.... it kind of looked like she took a balloon and cut stuff off of it. Thanks in advance.(3 votes)
- You take an inflatable ball, fold it, then cut patterns into it. Then unfold it, put another ball inside the first one, then inflate it.(2 votes)
- Why is every video are in fast motion?(1 vote)
- Was It hard saying all those words fast ?(2 votes)
- Vi actually records herself talking at a normal speed, but she can speed it up using a computer.
--Blue Leaf(1 vote)
- I wish Vi can slow down and actually take the time to show us how to cut the paper specifically. Anyone know how to cut those?(2 votes)
- I don't, but there is an option button underneath the video and you can slow the video down.(1 vote)
- how does she get the cut ballon with air(2 votes)
Video transcript
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.