Too Many Triangles - Numberphile
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- čas přidán 27. 11. 2016
- How many triangles are too many? Featuring Henry Segerman from Oklahoma State University.
More links & stuff in full description below ↓↓↓
Check Henry's book about 3D printing math: amzn.to/2cWhY3R
More Henry videos: bit.ly/Segerman_Videos
Henry's hinged doilies were joint work with Geoffrey Irving (naml.us)
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That looks like the cloth my grandma has on her TV
Yes it does xD
crochet? XD
that's what grandmas do, filling empty areas with wrinkly cloth so it doesn't look empty anymore xD.
kkarahodzic l
kkarahodzic it;s a doily
This is the best and most intuitive way to teach people about hyperbolic surfaces, yay 3D printing!
I've heard that this one person actually crocheted a hyperbolic surface.
If you search "hyperbolic crochet" on youtube you can see some people doing it and even tutorials if you want to make your own.
false.
Oh man, I used to draw those little 7-triangle things on my school notebooks!
Just go out farther and farther from the center, making smaller and smaller triangles just as equilateral as you possibly can. Until suddenly you hit a hard limit and you just cant fit anymore in, or you can't see them anymore because they get too small.
It makes a cool design. I'd love to have a 3d printed version to play with now.
wowok
*"hyperbolic doily"* is my band's name.
Henry has created what I can only describe as the 'forbidden doily'
"Triangles are happier in groups. They're like sheep. They get sad and lonely by themselves"
--ViHart
i saw that too!
And here I am thinking about Vihart just saying "triangles" constantly and "hyperbolic doily" takes the cake.
A geodesic dome like that would be a great tool to teach school kids about map projections, and how you can't trust a world map.
Print a world map on one and place it on a matching sphere so it looks like a globe, then let the kids play with the "carpet" of triangles and see how you can never make it flat without distorting something.
4:30 sneaky self promotion!
Too bad subliminal messages don't work, and certainly not for QR codes :-D
Audrey did it!
If that ugly dog counts as "promotion"...
You can use the period and comma on your keyboard to frame advance. So if you spot something, you can always find it.
Timothy Warner that's how I did it
I must say, I didn't expect him to name drop the Triforce.
me neither! O.O
I just came from another math video that name dropped the Triforce when discussing Sierpinski's triangle: title "Binary, Hanoi, and Sierpinski, part 2"
+Len Arends
Wow! I'll watch it later, anyway if you're interested in these topics, drop by my channel ^.^
??.
The fractal-ish nature of the 7- and 8-triangle surfaces and especially the "geodesic dome" version of the 7-triangle surface reminded me of the way the surface of kale, some other cabbages and lettuce are wrinkled (I think he actually mentioned lettuce earlier on in the video). Another natural approximation of a mathematical concept, much like Romanesco broccoli?
I'd like to put one of those on a little table in a psychologist's waiting room and watch all the OCD patients go mad
someone do that
Do you want them to be sued for malpractice!!
you are an evil person
this is awesome!
*** a mathematician patient steals it from the desk ***
I like everyone, but Segerman's my favorite numberphile guest. I like how he explains stuff and I like the 3D printing models.
Needs more triangles
NO YOU FOOL.
Nice avatar
Yes! It does need more triangles. You can never have enough of them.
@@imveryangryitsnotbutter IT IS TO LATE, YOUR WORLD MUST END
You can get closed hyperbolic surfaces, analogous to a sphere. They just have at least two holes in them - but no boundaries.
You can even tile them with regular polygons, if you feel so inclined. It's a great puzzle to think about! It also comes with profound group theoretical consequences.
Negative curvature everywhere in Euclidean 3-space using intrinsic metric? I would be surprised.
Jan Dvořák
I never said in Euclidean 3-space. In Euclidean 4-space or higher.
It can have negative curvature *almost* everywhere in 3-space as well though.
The idea that keeps going through my head with this is that if you could find some way to keep small miniatures attached (velcro? magnets?) even when the area they're in is crinkled up, then these would make excellent battle mats for Call of Cthulhu
Who else saw Brady's cheeky snapchat handle half way through?
i just did, i wonder why.
Hmmmmm
Me
Is that what that less-than-half-a-second subliminal thing with the dots and ghost was‽ At 4:29? Even at 0.25× speed it goes by too fast to pause at it!!!
On CZcams on desktop, you can use . or , to move forward or back one frame
7 triangles = 420 degrees = TOO HIGH
Truls Henriksson and 6 is 360
Oh wait 360 meme is because it's 360, it's full turn so naturally many math here will be MLG
:D :D :D
360 noscope + Dorito = high
MLG math
nice
false.
3-5 = "spheres"
6 = "plane"
7-8 = "quantum foam model"?
Hey, that's what I started to think about :D you poke it in one spot, it crinkles up in another. Sounds an awful lot like a complimentary variable in physics.
7+ is pringles
12 = "oh no"
I finally found out why in my grandma's time, there was a hype with "mileuri" (it's a romanian word for something that looks like the 6 triangle flat one, that you put on furniture for decoration). The fascination with maths was real
How deep in the cheek was the tongue of whoever wrote this part? :
"Henry's *hinged* doilies were *joint* work ..."
😂 the description section contains some interesting nuggets
when i was 7 years old i stubled upon this problem while playing with geomag xD
Me too, with those buckyball magnets (but I was like 27)
Milehupen me too!!!
Welp, time to go digging through my closet for my geomags.
Do you still use them? If not why!? Those things are fun!
??.
I had just come back to watching numberphile after a 6 month hiatus. I enjoy this Henry Segerman.
one of my favorite videos from Numberphile
I don't know why but I love when numberphile uploads videos about geometry
3:07 sums up my friends at school and my life.. 😓
"Sub-divide it into 4 like a TRIFORCE."Epic Yes.
This video in particular, going back and watching it again, something is clicking. I understand a bit more about hyperbolic geometry from this video alone than I have fleetingly glimpsed before.
There's a snapchat code at 4:30, I added it. Do I win?
Yes
You can do this quite nicely in software called "magic tiles", it's a software that does all sorts of Rubik's cube equivalents in all kinds of spaces, even hyperbolic, really nice stuff!
This guy is a 3D printing wizard. Seriously, what a skill and knowledge!
"What is this!?" feckin hilarious!
3:13 Is my favourite moment in the video as it gets me laughing everytime.
As Vi Hart could tell you, there's no such thing as "too many triangles".
The answer is -1/12
Best platonic explanation that I have saw!
Where can I buy these doilys? You can never have enough triangles on your table
Hey Brady, can you make a video on how to go about solving a mathematical problem and how to go about proving theorems?
Real pleasure to meet the man at my university after his presentation!
The Zelda in me almost jumped out of my seat when you said the word "Triforce". Awesome.
You can never have enough triangles, Vihart is the proof.
This effect is yielded very easily when crocheting in the round. Just keep adding increases at a given point in the round and you end up with this "hyperbolic plane" styled piece of material. There's a TED talk about crocheting hyperbolic planes :)
This is awesome.
This made me understand the problem of curvature of space.
2:56 I'm pretty sure a hydraulic press would do the job...
Unfortunately it wouldn't work past a certain point. Something would either break or distort, depending on the flexibility/toughness of the material
I appreciated that Zelda reference.
I'm in love with the hyperbolic doily.
4:36 you can't wrap the world with that. That's what he just explained
I love topology. Helps me understand GR.
There's a game that you play on a hyperbolic plane! It's called HyperRogue, it's super fun, and it's available on all platforms. Best part? You can get it without the music so it takes up a single megabyte of space, three on mobile, as compared to around fifty with the music. And it's huge! Makes my brain hurt a little, though.
numberphile can still blow my mind. at least a little bit :)
With the >6 triangles on a vertex. Would the edges fold into an iterative function system such as the Koch snowflake, Hilbert or Dragons curve?
I like the usage of a triforce in the explanation.
These are great ! :'D
3:34 Yes, lettuce - first thing that came to my mind.
5:51 does anybody know where I could read more about this open problem, like the name of the problem or the current research on it?
So, if you add more layers of the seven one, using ideal one-dimensional sides, the outer edge becomes a 3D space-filling curve, (Hilbert curve?) which might or might not run into itself at some limiting number of layers?
Drinking game: take a shot every time he says "Triangle"
Have you done Fermat's Last Theorem? I'd love to have the answer explained.
"What is this trying to be?"
Too relatable.
Simplest solution to Pythagoras: 3 4 5 and those are the possible shapes too
I think Hyperbolic Doily's second album was their magnum opus ... after that they got too big and started getting in their own way.
What is the canonical folding of a surface made with 7 triangles? The floppy doily shape isn't right. The saddle shape is clearly better: A half-circle up, and a cross-wise half circle down. But the stuff at 45 degrees needs work. Do you need to make cuts to make the material end up in the right place? Do you need infinitely many cuts? What does the resulting shape look like?
There's a game called hyperrogue which is a sort of puzzle game on hyperbolic space. very cool.
They should make a vid about arcimeadian solids, goldberg polyhedra, as well as cantalating and stelating said solids.
wow this is a brilliant way to understand the curvature of the Universe.
"huh, the way the triangles have to get smaller and smaller to fit the further out it goes kinda reminds me of a hyperbolic plane"
30 seconds later
0:39 *feels inspired to make a doily*
Cool, I used to play with shapes like these when I was a kid, I was making them with triangles from a building kit.
This helped me understand negative curvature.
hay Do you remember The 3x+1 problem? well I was messing around on my calculator and I think I found a similar problem. It has 3 rules. If even dived by 2, If divisible by 3 dived by 3, IF the number isn't divisible by 2 or 3 the multiply by 5 and add 1. do this and It always seems to get stuck at the loop 6, 3, 1, 6, 3, 1. Or depending on If you divided by 3 or 2 first 6 ,2, 1, 6, 2, 1. I've tried tones of numbers and I can't find anyone that brakes this rule. I've tried huge numbers too like 54673.
Could you maybe do a video on 'Knight's Tours'?
Do a video on the lagrangian equation and what it could help solve, how it connects with the particle accelerator and so on please!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
I would've thought that you'd Want the hyperbolic surface to go into itself and form a closed shape. Because then, even if it curves the "wrong way", wouldn't it still count as a platonic solid? All vertices have the same amount of triangles on them?
that 6 sided mesh has got to be on the top 5 most frustrating toys ever list
What would happen if you subdivided each of the triforce triangles into four again and again, for an infinite number of times? I know that with each iteration, the triangles would become increasingly less equilateral, but would the sheet tend towards a structure with zero curvature overall? How many times could you subdivide before the constituent triforce shapes became unworkably distorted?
could you make some sort of toroid or loop with the negative curve?
If it saddles, the top parts would extend along a curve eventually meeting up. Then if you extend the other sides outward along this curve they would also meet up making a torus shaped object. In theory...
neat how the hyperbolic stuff _can_ be organized into a saddle
"This Traingle is too much to handle" RIP Zyzz always be mirin'
All these squares make a circle
Can you do a spot on the moving sofa problem?
I saw Oklahoma State University in the description...
And then I cried a little inside
And was confused a little too...
What happens if you do 5 squares around each vertex? Or 4 pentagons around each vertex?
Interesting that it forms a saddle. Can you extend it to become a torus??
That's sooo cool. I want crinkle triangles
So if 17 would that then be again positive curvature meaning it would form a closed surface like two spheres but perpendicular to each other?
Part of me wonders if as you continued the saddle if it would loop back into itself and create a torus kind of shape
They look like something you grandmother would knit.
HarbourOfMarbles there are crocheting patterns available for hyperbolic surfaces.
is there a way the rhino/stl files to 3d can be shared so we can do these experiments at home?
I may be wrong about this, but the exponential f(x) = e^x actually seems to beat the cubic g(x) = x^3. It takes a while, but I'm fairly certain it means that you could theoretically expand this far enough and it might eventually exceed that mark, and suddenly, you could extend it outward forever.
3D printing is quite amazing
It would be really pleasing to see Henry create the surface which it could laid "flat" upon. i.e. Every triangle being tangent to the surface.
I tried pretty hard to make something like this - there are some serious problems. First, it's hard to make a smooth surface with constant negative curvature that has much area - the hinged surfaces have a lot of area in comparison to things like the pseudosphere. Second, the triangles of a geodesic dome are all inside of the circumscribing sphere, while the triangles of these hinged surfaces want to be intersecting a smooth constant negative curvature surface. Which makes it hard to put them together in real life.
The open question seems to be about the planc length basically isn't it? Like using 2d triangles, how many could you fit, youd have to know the thickness , so whats the minimum thickness and that's youre limit to whatever size you can grow that ball too without it crashing. This problem wouldn't be hard to simulate on a graphics engine using 2d polygons - triangles no?
Bradyharen is the snapchat username that flashes on the screen for one second at 4:30. Just to save other people time trying to get it.
actually, it's bradyharan not bradyharen
Keep trying to catch glimpses of that awesome shirt
6:35 Triforce
Omar Velázquez That was awesome!
How do you simulate hyperbolic space in the source engine
In guessing you use world portals but that's only in portal 2
is it possible to get or purchase 3d models or physical representations of the various hyperbolic doilies?
It's not quite as high-tech as 3d printing, but you can pretty easily crochet hyperbolic doilies by crocheting in the round and increasing (putting multiple stitches into a single stitch) fairly often. Margaret Wertheim has been doing this for years.
That fast SnapChat of bradyharan with a little Chihuahua (Chee-hoo-uh-hoo-uh) dog at 4:30 .