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.
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?
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.
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.
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.
I just came from another math video that name dropped the Triforce when discussing Sierpinski's triangle: title "Binary, Hanoi, and Sierpinski, part 2"
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
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.
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!!!
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
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.
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!
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.
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 :)
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?
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.
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...
are these things printed using homestation 3D printer, or are they actually done professionally? The details looks very intricate, not sure if Prusa i3 reprap would be able to do this.
The brain has similar structure to pack in as many cell as possible, but in a specific interconnected way. Can you find a similar pattern that emulates the brain I wonder
Also for the 6-triangle case, you get the same hexagonal lattice that the brain uses to represent spatial location via "grid cells". This discovery recently won the Nobel Prize.
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?
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.
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?
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
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
I like everyone, but Segerman's my favorite numberphile guest. I like how he explains stuff and I like the 3D printing models.
And here I am thinking about Vihart just saying "triangles" constantly and "hyperbolic doily" takes the cake.
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?
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.
*"hyperbolic doily"* is my band's name.
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.
"Triangles are happier in groups. They're like sheep. They get sad and lonely by themselves"
--ViHart
i saw that too!
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
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 ^.^
??.
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.
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
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
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 had just come back to watching numberphile after a 6 month hiatus. I enjoy this Henry Segerman.
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 UA-cam on desktop, you can use . or , to move forward or back one frame
I don't know why but I love when numberphile uploads videos about geometry
Henry has created what I can only describe as the 'forbidden doily'
This guy is a 3D printing wizard. Seriously, what a skill and knowledge!
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
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 ***
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!
??.
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
one of my favorite videos from Numberphile
Best platonic explanation that I have saw!
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.
"Sub-divide it into 4 like a TRIFORCE."Epic Yes.
The answer is -1/12
3:07 sums up my friends at school and my life.. 😓
This made me understand the problem of curvature of space.
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!
As Vi Hart could tell you, there's no such thing as "too many triangles".
There's a snapchat code at 4:30, I added it. Do I win?
Yes
Where can I buy these doilys? You can never have enough triangles on your table
You can never have enough triangles, Vihart is the proof.
Real pleasure to meet the man at my university after his presentation!
I appreciated that Zelda reference.
3:13 Is my favourite moment in the video as it gets me laughing everytime.
Try 12 around a vertex. It's beautiful
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.
All these squares make a circle
The Zelda in me almost jumped out of my seat when you said the word "Triforce". Awesome.
I'm in love with the hyperbolic doily.
"What is this!?" feckin hilarious!
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 :)
I love topology. Helps me understand GR.
I like the usage of a triforce in the explanation.
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?
Damn, Inb4 NumberPhile becomes part of Illuminati
This is awesome.
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
This helped me understand negative curvature.
Simplest solution to Pythagoras: 3 4 5 and those are the possible shapes too
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?
3D printing is quite amazing
Hey Brady, can you make a video on how to go about solving a mathematical problem and how to go about proving theorems?
4:36 you can't wrap the world with that. That's what he just explained
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.
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?
numberphile can still blow my mind. at least a little bit :)
"This Traingle is too much to handle" RIP Zyzz always be mirin'
wow this is a brilliant way to understand the curvature of the Universe.
"What is this trying to be?"
Too relatable.
6:35 Triforce
Omar Velázquez That was awesome!
neat how the hyperbolic stuff _can_ be organized into a saddle
I hated those frilly 7 triangle structures I just wanted to tear them apart
So what you're saying.... Is that 420 is too high?
that 6 sided mesh has got to be on the top 5 most frustrating toys ever list
I saw Oklahoma State University in the description...
And then I cried a little inside
And was confused a little 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.
"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
What happens if you do 5 squares around each vertex? Or 4 pentagons around each vertex?
3:34 Yes, lettuce - first thing that came to my mind.
Keep trying to catch glimpses of that awesome shirt
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?
If I could subscribe twice I would.
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...
They look like something you grandmother would knit.
HarbourOfMarbles there are crocheting patterns available for hyperbolic surfaces.
are these things printed using homestation 3D printer, or are they actually done professionally? The details looks very intricate, not sure if Prusa i3 reprap would be able to do this.
I use Shapeways' SLS machines. I think these would be very difficult to make on an FDM machine.
The brain has similar structure to pack in as many cell as possible, but in a specific interconnected way. Can you find a similar pattern that emulates the brain I wonder
I once read that the brain resembles the surface of a seven-dimensional hypersphere.
Also for the 6-triangle case, you get the same hexagonal lattice that the brain uses to represent spatial location via "grid cells". This discovery recently won the Nobel Prize.
The brain also resembles a bowl of Ramen noodles.
I like how he just casually uses the Triforce as an analogue, because it is implied that anyone who watches numberphile knows the Legend of Zelda :D
That's sooo cool. I want crinkle triangles
Cool, I used to play with shapes like these when I was a kid, I was making them with triangles from a building kit.
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?
the transitions and stuff are so jarring and hurting my eyes stop that
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.
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?
I suddenly want to follow numberphile on Snapchat...
I was wondering if anyone else had commented on that :P
The whole sequence is only 4 frames long and snapchat pic is a single frame D:
There is no such thing as too many triangles.
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
Congrats to the one that stopped the video at 4:30 and saw the snapchat account with a dog profile.
, and . for those who can't stop that fast!
Thanks! Downloaded it and used VLC but this is much better :)
Numberphile needs to collaborate with the hydraulic press channel. They'll stop the models from bulging up. Problem solved! You're welcome.
Does anybody else get goosebumps looking at the 7 and 8 constructs?
He has discovered the mathematics of the 16th century ruff.
There's a game called hyperrogue which is a sort of puzzle game on hyperbolic space. very cool.
Drinking game: take a shot every time he says "Triangle"
These are great ! :'D
I like being so early that no one has made a constructive comment yet
accepted*
excepted*