Too Many Triangles - Numberphile

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...
Henry's hinged doilies were joint work with Geoffrey Irving (naml.us)
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[@ryPish]

This is the best and most intuitive way to teach people about hyperbolic surfaces, yay 3D printing!

[@TunnelDragon44]

I've heard that this one person actually crocheted a hyperbolic surface.

[@brenorocha6687]

If you search "hyperbolic crochet" on youtube you can see some people doing it and even tutorials if you want to make your own.

[@Triantalex]

false.

[@GelidGanef]

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.

[@user-pr6ed3ri2k]

wowok

[@stormsurge1]

That looks like the cloth my grandma has on her TV

[@tomu890]

Yes it does xD

[@sadhlife]

crochet? XD

[@marclarell]

that's what grandmas do, filling empty areas with wrinkly cloth so it doesn't look empty anymore xD.

[@legendoftheaflurflur]

kkarahodzic l

[@Apathetish]

kkarahodzic it;s a doily

[@Snaake42]

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?

[@Kram1032]

4:30 sneaky self promotion!

[@dvoraj20]

Too bad subliminal messages don't work, and certainly not for QR codes :-D

[@Nilguiri]

Audrey did it!

[@unvergebeneid]

If that ugly dog counts as "promotion"...

[@totally_not_a_bot]

You can use the period and comma on your keyboard to frame advance. So if you spot something, you can always find it.

[@Kram1032]

Timothy Warner that's how I did it

[@Hecatonicosachoron]

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.

[@dvoraj20]

Negative curvature everywhere in Euclidean 3-space using intrinsic metric? I would be surprised.

[@Hecatonicosachoron]

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.

[@magnusdagbro8226]

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.

[@ganaraminukshuk0]

And here I am thinking about Vihart just saying "triangles" constantly and "hyperbolic doily" takes the cake.

[@whatthefunction9140]

*"hyperbolic doily"* is my band's name.

[@Snootypriss]

I like everyone, but Segerman's my favorite numberphile guest. I like how he explains stuff and I like the 3D printing models.

[@antoineroquentin2297]

"Triangles are happier in groups. They're like sheep. They get sad and lonely by themselves" --ViHart

[@proloycodes]

i saw that too!

[@danielstephenson7558]

Henry has created what I can only describe as the 'forbidden doily'

[@stuartrockin]

Needs more triangles

[@imveryangryitsnotbutter]

NO YOU FOOL.

[@pedroanitelli]

Nice avatar

[@titaniumO2]

Yes! It does need more triangles. You can never have enough of them.

[@tristenarctician6910]

@@imveryangryitsnotbutter IT IS TO LATE, YOUR WORLD MUST END

[@NA-mg2eb]

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

[@joelshewmaker3567]

I must say, I didn't expect him to name drop the Triforce.

[@LinkAranGalacticHero]

me neither! O.O

[@secularmonk5176]

I just came from another math video that name dropped the Triforce when discussing Sierpinski's triangle: title "Binary, Hanoi, and Sierpinski, part 2"

[@LinkAranGalacticHero]

+Len Arends Wow! I'll watch it later, anyway if you're interested in these topics, drop by my channel ^.^

[@Triantalex]

??.

[@tristanbatchler]

Who else saw Brady's cheeky snapchat handle half way through?

[@qwertyman1511]

i just did, i wonder why.

[@FenrizNNN]

Hmmmmm

[@alonsomartins712]

Me

[@MajikkanBeingsUnite]

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!!!

[@liambohl]

On KZbin on desktop, you can use . or , to move forward or back one frame

[@MultiSteveB]

3-5 = "spheres" 6 = "plane" 7-8 = "quantum foam model"?

[@explosu]

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.

[@f5673-t1h]

7+ is pringles

[@samuelthecamel]

12 = "oh no"

[@tibimose823]

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

[@IJustLoveStories]

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

[@Spekter2500]

someone do that

[@antoniolewis1016]

Do you want them to be sued for malpractice!!

[@Fasteroid]

you are an evil person

[@_____alyptic]

this is awesome!

[@EchoHeo]

*** a mathematician patient steals it from the desk ***

[@ffggddss]

How deep in the cheek was the tongue of whoever wrote this part? : "Henry's *hinged* doilies were *joint* work ..."

[@rewrose2838]

😂 the description section contains some interesting nuggets

[@AaronRClark]

I had just come back to watching numberphile after a 6 month hiatus. I enjoy this Henry Segerman.

[@Kebabrulle4869]

7 triangles = 420 degrees = TOO HIGH

[@Xeverous]

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

[@pvanukoff]

:D :D :D

[@StarTheTripleDevil]

360 noscope + Dorito = high MLG math

[@sawyer8297]

nice

[@Triantalex]

false.

[@zlac]

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!

[@WondrousHello]

There's a snapchat code at 4:30, I added it. Do I win?

[@samuelthecamel]

Yes

[@null1449]

I don't know why but I love when numberphile uploads videos about geometry

[@edibletwix]

3:07 sums up my friends at school and my life.. 😓

[@Milehupen]

when i was 7 years old i stubled upon this problem while playing with geomag xD

[@Platanov]

Me too, with those buckyball magnets (but I was like 27)

[@phibsie6494]

Milehupen me too!!!

[@failatlife1]

Welp, time to go digging through my closet for my geomags.

[@AidenOcelot]

Do you still use them? If not why!? Those things are fun!

[@Triantalex]

??.

[@AlanKey86]

"Sub-divide it into 4 like a TRIFORCE."Epic Yes.

[@SpyridonJohn1633]

"What is this!?" feckin hilarious!

[@nerdbot4446]

Where can I buy these doilys? You can never have enough triangles on your table

[@911gpd]

The answer is -1/12

[@adizmal]

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.

[@Rurexxx]

This guy is a 3D printing wizard. Seriously, what a skill and knowledge!

[@HontubeYT]

3:13 Is my favourite moment in the video as it gets me laughing everytime.

[@ricardoabh3242]

Best platonic explanation that I have saw!

[@robertbauer499]

one of my favorite videos from Numberphile

[@davidq.1321]

Hey Brady, can you make a video on how to go about solving a mathematical problem and how to go about proving theorems?

[@DKQuagmire]

The Zelda in me almost jumped out of my seat when you said the word "Triforce". Awesome.

[@qwertyTRiG]

As Vi Hart could tell you, there's no such thing as "too many triangles".

[@neoqueto]

This made me understand the problem of curvature of space.

[@totally_not_a_bot]

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.

[@nahailyenvanakkor]

2:56 I'm pretty sure a hydraulic press would do the job...

[@Kaiveran]

Unfortunately it wouldn't work past a certain point. Something would either break or distort, depending on the flexibility/toughness of the material

[@scowrules]

I'm in love with the hyperbolic doily.

[@meesvandenberg9468]

4:36 you can't wrap the world with that. That's what he just explained

[@grahams5871]

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?

[@secularmonk5176]

I think Hyperbolic Doily's second album was their magnum opus ... after that they got too big and started getting in their own way.

[@orbik_fin]

3:34 Yes, lettuce - first thing that came to my mind.

[@feynstein1004]

I love topology. Helps me understand GR.

[@ConorFenlon]

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 :)

[@Nixitur]

I appreciated that Zelda reference.

[@kenjinks5465]

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?

[@ughsomenonsense]

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?

[@anthonycannet1305]

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...

[@Henrix1998]

Simplest solution to Pythagoras: 3 4 5 and those are the possible shapes too

[@GarryDumblowski]

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.

[@LupeFenrir]

This helped me understand negative curvature.

[@zzasdfwas]

There's a game called hyperrogue which is a sort of puzzle game on hyperbolic space. very cool.

[@AliHSyed]

wow this is a brilliant way to understand the curvature of the Universe.

[@dogwithsocks]

Real pleasure to meet the man at my university after his presentation!

[@HontubeYT]

Try 12 around a vertex. It's beautiful

[@ianflanagan8744]

They should make a vid about arcimeadian solids, goldberg polyhedra, as well as cantalating and stelating said solids.

[@invictus127]

I like the usage of a triforce in the explanation.

[@ratoim]

0:39 *feels inspired to make a doily*

[@E1craZ4life]

What happens if you do 5 squares around each vertex? Or 4 pentagons around each vertex?

[@WiseSquash]

6:35 Triforce

[@gold4963]

Omar Velázquez That was awesome!

[@flamencoprof]

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?

[@krischan67]

5:30 - The number of triangles is NOT going up exponentially with each way around the center. When assuming that a turn means finishing those corners which have been started during the previous turn (and turn 1 being the start), doing it with M triangles means adding (1+(M-3)*(N-1))*M=(MN-3N-M+4)*M of them on the N-th turn. Summing up the added amounts leads to a quadratic increase. With M>6, the total area in the plane is still more and more falling behind the area taken by the triangles and the difference is going towards infinity, however.

[@kurtilein3]

exponential does not mean that its simple, as in, with a straightforward whole-number exponent.

[@krischan67]

I'm not sure what you mean with "simple". All I can do in this context is describing it in a mathematical way. Exponential growth means that the derivative can be described as f0*e^(k*x) with f0 and k being non-zero constants. It can't, so it's not exponential growth.

[@garrett3883]

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.

[@kashgarinn]

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

[@davidwuhrer6704]

I once read that the brain resembles the surface of a seven-dimensional hypersphere.

[@aaronvoelker311]

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.

[@CertifiedDoc]

The brain also resembles a bowl of Ramen noodles.

[@JustinVZyl]

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?

[@TheLolle97]

numberphile can still blow my mind. at least a little bit :)

[@johnbatsch7938]

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.

[@syaba2327]

actually, it's bradyharan not bradyharen

[@ody10able]

You can never have enough triangles, Vihart is the proof.

[@rhubarbcheese]

Do a video on the lagrangian equation and what it could help solve, how it connects with the particle accelerator and so on please!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@chasemarangu]

neat how the hyperbolic stuff _can_ be organized into a saddle

[@Radditz770]

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?

[@mikejones-vd3fg]

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?

[@hiimapop7755]

This is awesome.

[@aurachanneler8396]

7:19 420 NOSCOPE!!!!

[@yaslerfuj]

?

[@Djorgal]

4:43 No it wouldn't go around the world because you can tile a plane with it, not a sphere.

[@CptGallant]

But you could make a giant cylinder that DOES wrap around the world like a belt and it would still have zero Gaussian curvature.

[@Djorgal]

Katrina S It doesn't have to be noticeable, it'll add up if you try to cover it entirely. I agree that it could be be shaped like a cylinder, but not a sphere, no matter the size.

[@agr.9410]

The earth is flat tho

[@Stickycomix]

"What is this trying to be?" Too relatable.

[@Miitchyy]

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.

[@henryseg]

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.

[@Fasteroid]

Part of me wonders if as you continued the saddle if it would loop back into itself and create a torus kind of shape

[@LordOfFlies]

All these squares make a circle

[@midvvolf]

could you make some sort of toroid or loop with the negative curve?

[@audreyrasmussen540]

I saw Oklahoma State University in the description... And then I cried a little inside

[@audreyrasmussen540]

And was confused a little too...

[@jpphoton]

doesn't it breakdown to whether or not the infinitesimal slice has width? Or is it a question of 2d planar collisions? Or not.

[@goodchris13]

So what would happen if you kept dividing the triangles infinitely, would you get a flat surface?

[@timh.6872]

Binary Bits: The prospect of placing 7 Sieperneski triangles around a common vertex is interesting, but as usual, I suspect throwing infinity at the problem isn't going to work without some creativity.

[@Brandlin]

Can you show the seven and eight models with regularly spaced rigid joints between the triangles rather than the current flexible ones. The flexibility of the structures you have makes them difficult to compare to the rigid joints of the regular solids.

[@tiagotiagot]

I'm not sure it would make any difference...

[@ArtanisOwns]

the transitions and stuff are so jarring and hurting my eyes stop that

[@t0m_mcc]

Keep trying to catch glimpses of that awesome shirt

[@xXSkyXDragonXx]

Damn, Inb4 NumberPhile becomes part of Illuminati

[@Wargon2013]

3D printing is quite amazing

[@tristenarctician6910]

How do you simulate hyperbolic space in the source engine In guessing you use world portals but that's only in portal 2

[@goodstormsgames9744]

That's sooo cool. I want crinkle triangles

[@WintermuteVR]

Can you do a spot on the moving sofa problem?

[@philipsalter934]

Interesting that it forms a saddle. Can you extend it to become a torus??

[@_brutalistsbible_5049]

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?

[@sullygrowel1574]

Could you maybe do a video on 'Knight's Tours'?

[@Czeckie]

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.

[@henryseg]

I use Shapeways' SLS machines. I think these would be very difficult to make on an FDM machine.