The Fractal Menger Sponge and Pi

[@gery49]

It's a very useful shape when you want to 3D print something but you don't have any filament :P

[@Nusma]

+Gergely H It depends. If you want to 3D print a Menger Sponge in reality then you have to fill the "empty" spaces with something since a total vacuum is rarely practical. So lets say you use plastic A for the sponge and plastic B for the space in between. You would end up with a solid block of plastic B wich, with some abstract thinking, could be rightfully called a Menger Sponge. (Or an inverted one if you are picky)

[@gery49]

Nusm4 :'D

[@ciarfah]

+Nusm4 you, I like you.

[@JNCressey]

+Nusm4, Nah, in reality, you'd just have to accept that you're forced to have a limited resolution and have to print one of finite level.

[@iqbaltrojan]

+Gergely H 181 likes on your comment no reply's intill this one XD

[@Box-of-hats]

To this day, I'm still finding myself laughing at all of the little jokes you throw into your videos. Thank you for bringing so much humor to the world of mathematics!

[@standupmaths]

My pleasure. Thank you for watching the videos.

[@jonahvanke5002]

+standupmaths You probably won't see or answer this, but is there another fractal to get Tau?

[@secularmonk5176]

+Jonah Vanke Not sure if this will be satisfying for you, but ... Draw a diagonal from opposite corners of the current square. Use this length as the sides of a new, larger square, and perform the Wallis sequence on that new square. All areas will be twice as large, compared with the original square.

[@peter7718]

+standupmaths You need to make some videos about τ. =P

[@trobin]

+Laurelindo he supports pi not tau

[@54m0h7]

The second form square having the area of a circle, and 3D a sphere, actually makes perfect sense if you visualise it right. Picture a square. Now cut a square out of each corner. Now cut another square out of the new corners. More and more cuts will define a more and more accurate circle. When you get to infinity you get a proper circle. The same squares you just cut and the same squares you cut out in the other example, just a different location.

[@TheNdoki]

I feel cheated. I really wanted to see a level 4 actually assembled.

[@YouTubist666]

Ditto. I was hoping Matt would say something like "and we asked everyone to ship their assemblies to us, and we put it together and here it is ..."

[@azuritet3]

if they keep up the tradition for 20 years in a row then we could have a level 5

[@maxnullifidian]

I'm trying to visualize in my head what it would look like...

[@skydivingisfun]

Lol let's see a level grahams number

[@brcoutme]

Walt F. it shouldn't be too hard to visualize it would look exactly the same but bigger, well very slightly different but by a level 4 the difference from a level 3 would be minor enough to not be very visually noticeable. From a level 4 to 5 it would more so just look bigger. Still I would love to see one level 4 even never mind level 5.

[@imacds]

Time to build a level 5! Let's get other planets into this project...

[@lavaande]

and after a while it would be great to announce that all the multiverses are now united :D

[@firefish111]

@Whited Out Black holes

[@vasudevraghav2109]

@tshrpl level 12: different timelines Level 13: different timelines clusters with different physics. Oh wait isn't q3 dimensions related to string theory....

[@ChrisLuigiTails]

Level 5! = Level 120

[@R1ckr011]

@Whited Out they just make them out of unit square crystal materials and you can probably get do level 14 or something

[@esyrim]

The Menger Sponge is WITHOUT A DOUBT the scariest fractal in 3D and the friendliest fractal in 2D.

[@TheEvolNemesis]

Pi squeezed out of a Menger sponge? Doesn't sound very appetizing...

[@BlobVanDam]

7:00 Make up your mind on the pronunciation, Matt! :P

[@standupmaths]

+BlobVanDam I try to please everyone! Or annoy everyone. Depends how you look at it.

[@BlobVanDam]

+standupmaths It's always the former for this viewer!

[@foobar7471]

+BlobVanDam It's definitely the former

[@WyattJKR]

+standupmaths you should of showed what a menger sponge looks like when you cut it in half P.s. It looks awesome.

[@Smartlylinked]

+standupmaths For completeness (second comment), here's a Level 4 Menger Sponge and a Level 5 Sierpinski Tetrahedron in Sketchup: drive.google.com/file/d/0B6HS5LXNwmKmZXc5b1ZPLWJNU2s/view?usp=sharing

[@BlightVonDrake]

Someone had to say this. "Dojyaaa~~n!"

[@Decimator69420]

D4C’s favourite puzzle cube

[@Periwinkleaccount]

What’s this a reference to?

[@GeekyAsItGets]

I've build a level 4 Menger sponge by hand several years ago in Minecraft. A few years ago, I used the ComputerCraft mod to program a robot to build all possible levels that can fit in Minecraft, level 0 to level 5.

[@jsmunroe]

So does that mean we have successfully squared the circle? And well cubed the sphere?

[@needarandomname4330]

+Jordan Munroe lmao!

[@PhilBagels]

+Jordan Munroe Yes! In an infinite number of steps.

[@DustinRodriguez1_0]

+Jordan Munroe Heh, that was my first immediate reaction as well, though I believe part of the rules of that kind of construction is that you have to be able to do it with only finitely many steps. I could be wrong about that, though.

[@Shadow81989]

+Jordan Munroe I think we didnt square the circle, we rather circled the square :D

[@jasonbagley3712]

+Jordan Munroe I think the fact that we've done an infinite number of steps means that we haven't truly squared the circle. As with anything involving the finding of Pi, it cannot be done in a finite number of steps when involving rationals. This is why Pi is called Transcendental.

[@denascite2029]

the fact that I like most about the menger sponge: if you have one right in front of you(so that the line of your sight makes a 90º angle with the surface (I hope you can understand what I mean)), you can look through it (you don't see it at all) but if you want to go through it you will hit a hard surface. and if you turn it a little bit so that you don't look directly at it you can't see through it anymore.

[@Crlarl]

9:53 Shouldn't it be "80/81" not "63/64"?

[@Henrix1998]

Probably

[@DaveScottAggie]

+dimmddr1 Yes I searched on using the Wallis products, and all of the denominators are squares of odd numbers: 3², 5², 7², 9², 11², 13², ... Here is the site I found, which I believe is the same one that Matt scrolls through in the video community.wolfram.com/groups/-/m/t/822984

[@juggernaut93]

+dimmddr1 Yeah, I thought the same thing and checking on Wikipedia it seems so.

[@jesusthroughmary]

+dimmddr1 Yes, of course. Besides the fact that you can look up the Wallis products to confirm, it's a matter of simple logic - a square divided into an even number of squares doesn't have a center square.

[@Paul_Kielty]

+jesusthroughmary that was kinda condescending, they noticed the mistake, and pointed it out politely in case it was intended. For all you know they used that logic when the decided to make the comment.

[@XiphosCreates]

10:10 If you look at the square it makes sense it has the same surface area as a circle. If you moved all the holes to the corners from big to small, and do that infinitly many times, you'd end up with a circle.

[@Michaelonyoutub]

+Lucas Mulder exactly what i was thinking

[@xnick_uy]

+Lucas Mulder Does not look so clear to me: you could attempt the same with a part of the first "carpet" and you won't get a circle.

[@alejandronq645]

I was just thinking the same! I got to try it on my computer

[@alejandronq645]

+x nick since the area of the first "carpet" is 0 you'll end up with nothing

[@XiphosCreates]

x nick Matt said the area of the first carpet is 0, it's a different square.

[@ValanceJ]

Level 3 Menger Sponge... Am I the only one imagining a cube going through an RPG dungeon, beating up enemies, and progressively getting more complicated while it levels up?

[@anthonycourte1384]

A gelatinous cube is a fictional monster from the Dungeons & Dragons fantasy role-playing game. It is described as a ten-foot cube of transparent gelatinous ooze, which is able to absorb and digest organic matter. Thanks wikipedia. They're really nasty things btw.

[@rewrose2838]

No, eventually the cube will get an option to change its class and become a Sphere Knight... where it's volume would remain unchanged but it unlocks a new skill tree involving spherical geometry! (for the sake of completeness, it'll also unlock unique skills- Squaring The Circle & Squeezing Pie)

[@grassy556]

Good idea for a video game

[@MasterC2012]

This is why I love fractals so much. They always have a weird quirk hidden somewhere that blows your mind away. My favourite fractal is the dragon curve. It's really fun to draw and see how the shape of the curve evolves each step.

[@KryssAA]

"Send money for researches, please, we need more, for project GigaMenger !"

[@SaHaRaSquad]

+Christophe Abi Akle But why stop there? TeraMenger is just a bit larger!

[@lawrencecalablaster568]

+SaHaRaSquad I prefer the great new dawn of human-fractal understanding, YoctoMenger.

[@DomenBremecXCVI]

+Christophe Abi Akle Lets just make a Menger Sponge from a cube big enough to fit the wholr Earth in, it shouldn't be that hard.

[@irvanluhung3326]

when Matt posts a Video, I press like even before I watch it

[@standupmaths]

I hope you adjust that like if conflicting data later comes to light.

[@Punk4kids]

Since you didn't actually put toghether the actual lvl 4 Menger Sponge, can we say that you guys parker squared it ?

[@HylianGirl56]

They Parker *Cubed* it.

[@ethangates5860]

+T Alex LOL

[@Fox_RZK]

It was a bit of a Parker sponge

[@nathanliu2031]

T Alex If Matt sees this, he will probably go insane

[@Kebabrulle4869]

PARKER SPONGE CONFIRMED

[@tggt00]

Every week I discover another project you've been working on, a year ago I thought you were just a standup math lover who was featured on numberphile, now I can't even describe you besides being a math madman.

[@asp-uwu]

var val=4;for(var i = 3;i

[@standupmaths]

That's my kinda π calculation!

[@lawrencecalablaster568]

+standupmaths :D Matt, you've coupled together two of my favourite mathematical phenomena! I thank you profusely.

[@Guil118]

+standupmaths I'm trying here to find a way to calculate the result of the product of an infinite series ( just like the sum ), but I can't seem to find any info on how to do this kind of calculations. Of course if you multiply it by yourself one by one, you will realise it goes to pi, but I need to test with a valid mathematical method.

[@asp-uwu]

+ILYES brb *steps away to write a program to draw a graphic sponge*

[@ELYESSS]

+Eric Pratt I'll be waiting

[@BloodyHand29]

I had goosebumps when you said 4/3 Pi.

[@ViktorLox]

I want them to actually assemble a level 4, then burn it!

[@spicytaco2400]

I like how you unintentionally poke fun at the location in more than one way.

[@TPRJones]

"Distributed" seems like cheating. I mean we could observe that almost 30 million business cards are printed every day, which could be considered roughly 2.5 "fully distributed" level 6 menger sponges. That's more than 2 "fully distributed" level 8 menger sponges every year.

[@Paul_Kielty]

Yeah but only the 4th iteration is distributed, which to me is a whole lot better than the 1st.

[@TPRJones]

+Paul Kielty That's fair.

[@tacobor]

It was this video that inspired me to build a magnitude 5 Menger sponge in Minecraft. It took the better part of 4 months. I calculated a level 6 would take just over 2 years.

[@0xBADFECE5]

Squeezing Pi out of the square also makes geometric sense because if each of the holes you punch gets moved to the closest corner, the shape actually approaches a circle quite fast. Same happens with the cube.

[@kenhaley4]

Well, not exactly. You will still end up with holes inside after level 2. It's not clear how to methodically shift the remaining bits to fill those gaps. But if you could...

[@0xBADFECE5]

you might be right; it was 1am when i wrote that lol

[@danjones4492]

This channel is even better than Numberphile (which is also really good). Best channel on KZbin! Great stuff Matt!

[@z-beeblebrox]

Someone needs to create a MegaMenger manga

[@ourgloriousgodoursaviourbe2757]

Yeah, about that...

[@ourgloriousgodoursaviourbe2757]

Also hello person from 5 yrs ago

[@lawrencecalablaster568]

:D Matt, this is amazing! You've married together my favourite fractal & one of my favourite numbers in an awesome way. I salute you.

[@GuiltyGearRockYou]

LET's GO FOR A LEVEL 4 MENGER! =)))

[@branthebrave]

I like Mandelbrot, Julia set, and the dragon curve.

[@johnny141093]

We did this at my university (Leeds)! We have desks made out of them all over the maths department now!

[@standupmaths]

+John Pearmain Yes, I have put my coffee on one of the Leeds fractal coffee tables.

[@luukderuijter1332]

Ladies and gentlemen, we have squared the circle

[@DiCasaFilm]

Is there any feasible way that this level 4 can actually made?? We have to unite the clans!

[@vlogdemon]

Cargo ships?

[@YouTubist666]

I think the resulting structure would just collapse under its own weight.

[@rossthebesiegebuilder3563]

+YouTubist666 Only one way to find out...

[@chrismofer]

no, they're quite strong with all those stacked edgewise cards in box form not to mention adhesive. If they were only assembled up to level 3 then they could be shipped in less volume to the combination site, where they would be assembled into level 4 form. If a smaller standard for level 0 cubes was determined (like half business cards) then universities could build small level 4s, making a merged level 5 even more impressive.

[@nikkirennardo5100]

I’d suggest private flights to make sure it doesn’t get crushed in shipping

[@watcherfox9698]

One of the things I love about maths is how things are connected in such surprising ways.

[@doggonemess1]

When I drop my pi, I always use a Menger Sponge to clean it up.

[@jeffirwin7862]

You just created a set that is uncountably infinite yet dense nowhere, you damn magician.

[@nikosaarinen3258]

0:15 You could call that a Parker square

[@glmathgrant]

A while back, inspired by your love of the Menger sponge, I actually assembled my own level 2 Menger sponge out of playing card-sized cards rather than business cards (including many actual playing cards, some Magic: the Gathering cards, and some Pokémon cards). Because of all that hard work I did, I can't help but also love the Menger sponge. :)

[@UCreations]

Correction on the correction: "8/9 × 24/25 × 48/49 × 80/81 × 120/132" should have 120/121 as 5th fraction, if I'm correct ;)

[@code_explorations]

At 11:38 you mention the equality of area with a unit circle. I think there could be a nice animation there. The biggest black squares could move to the corners. Then the next- biggest, then the next-biggest, etc. It should end up looking like a square with enough bits removed from the corners to make it look like a circle. This could be an actual geometric demonstration of Wallis's product.

[@elzearcontelly2651]

"for completeness" talking about 3D fractals...

[@chraffx8217]

I liked the way you presented the sierpinski carpets. Nice video!

[@standupmaths]

Thanks! I was very pleased with that.

[@trudyneo]

it would make more sense if you took a third of the cube out from the center rather than from each side. Although you wouldn't be able to see it, it would still be the correct higher dimension up. if you were in the forth dimension, you would be able to see it.

[@cananamanda]

Great Flying Pi in the Sky, I'm in one of those photos (5:51)! I was a student from one of the high schools that the Perimeter Institute in Kitchener/Waterloo, Canada, outsourced to do the tedious segment construction. My partner and I were two of the hand-full of students representing our school at the final assembly, and we made one of the level 2 cubes. How have I not found this video sooner!? By the way, I still have a single cube made from the spare cards! I doubt Matt will see this almost two years later, but I had a blast! Thanks Matt & friends!

[@huash9179]

5:51 : Matt, youre insulting canada :'(

[@GroovingPict]

Doesnt this really highlight an inherent problem with the way we work with limits and infinity? It very obviously has a surface area, otherwise it would all be just one big black box: we can still see some white. And yet the maths tells us it has no surface area. So which is more likely to be broken and in need of fixing here: reality, or the maths used to describe reality?

[@ebrahimalfardan8823]

That's amazing. You know Matt what would be more amazing that this, that is to repeatedly take the corner cube of the fractal and fill the holes in the middle and transform the square/cube into a circle/sphere. The animation would be neat.

[@Ollervo100]

Classic pi.

[@frankharr9466]

Happy Birthday fractals! Hey, he pronounced Oregon properly!

[@David-uc4hc]

What's with pi anyway? Always showing up uninvited and acts like it's the life of the party. It's just rude.

[@SmolSnake]

Thank you for pronouncing "Oregon" correctly! :D

63/64? Or 80/81? How can you remove the center square of an 8x8 square?

[@jesusthroughmary]

+Víktor Bautista i Roca Beat me to it.

[@andrewolesen8773]

+Víktor Bautista i Roca That threw me off as well, the denominator of each fraction should be the square of an odd number.

[@PJoriginal]

was thinking the same

[@carsonwood1513]

I think he screwed up on it.

[@nicholaswallingford3613]

+Víktor Bautista i Roca you are correct www.wolframalpha.com/input/?i=4*product(((2k%2B1)%5E2-1)%2F(2k%2B1)%5E2),1..inf)

[@TheMikkelOLaursen]

Great video as always, love your content. Awesome graphics, the video doesn't feel rushed to me!

[@maxbuskirk5302]

Hello everyone

[@smokeweed9901]

hi

[@Bella_Stend]

+Max Buskirk What's up?

[@GeneralKronosRocks]

+michael ruoff he has a fractal as his profile picture, i believe thats why he commented

[@Bella_Stend]

+Amir Allidina I know. Looks like the mandelbrot set. I was just greeting him back

[@Ottmar555]

+Max Buskirk hOI

[@Scubadooper]

The solution to squaring the circle

[@philosofickle]

Try to bring all these lvl 3 sponges together and build that level four member sponge. Please try

[@MrGallagher]

Fascinating! And also that seems to be about as close to naturally squaring a circle as one can get!

[@gojoubabee]

Hey Matt, today is 4/8/16 (written the American way) aka 2^2/2^3/2^4. happy powers of 2 day!

[@CASTCorp]

Cool! It's my birthday tomorrow

[@gojoubabee]

+CAST Corp Happy birthday!

[@gojoubabee]

+CAST Corp Wow, your birthday will be on 4/9/16 which is 2^2/3^2/4^2

[@branthebrave]

4/4/16 was square day 4^2=16 4*4=16

[@VittorioMass]

Today it's 2day

[@Victor-sw4ne]

the cantor's conjunt in 2D, and an actual proof that it has zero meazure... loved it!

[@prawtism]

#lifegoals: Make a Menger sponge with your friends :>

[@timetodowhatever]

i am blown away. in highschool my friend had shown my how to make an aragami puzzle piece witch is by far on of my favorite things to make. it started of with using 6 of them to make a simple box. after making a few of them i realized that i could make a 6 sided trangle and a bunch of other complex shapes, for a wile i was making a 24 sided cube, then i got board. i make over 1000 of the pieces and made this manger spounge without even knowing this is so cool

[@SebastianLopez-nh1rr]

Doesn't this counts as squaring the circle?

[@joshhyyym]

No. Squaring the circles is a Greek construction, ie it must be completed with a pair of loose compasses and a straight edge, in a finite number of steps. This takes an infinite number of steps.

[@roberteospeedwagon3708]

+Joshua Mcateer Yeah, Numberphile did a video on it, they did say getting a square to be the area of a circle is possible, but the nature of pi means the squares length will be off because we can't know 100% of pi, and they said doing it the old fashion way like the Greeks with rulers and compasses only, would as this guy says, take an infinite number of steps.

[@DanDart]

Then this must be circling the square!! XD

[@engineer_cat]

+Sebastián López squaring the circle would be constructing a square with the same area. The Wallis sieve isn't a square, although each step is (I think) constructible.

[@MisterHunterWolf]

Cubing the sphere

[@you_just]

You can squeeze pie out of an infinite sponge Finally, math I can get behind

[@kordellcurl7559]

What about a hyper cube factual of the same geometric series what is the "volume" of it. I have in quotations because I don't know what a hyper cube would be volume in the 4 dimension.

[@mercatorpsi]

The Menger Sponge is my fave fractal too, and I'm very impressed with all the people who helped out on the project! See (I tell my cynical self), humans CAN do cool things! And that teasing pi out of the modified carpet was the icing on the cake. Super groovy!

[@unvergebeneid]

2:20 More like a 1.8928D shape, amirite?

[@standupmaths]

I think Hausdorff dimensions deserve their own video!

[@chrisdrew1768]

but how do fractals exist if that dont have whole dimesions arrrrgggghhhhh

[@unvergebeneid]

Chris Drew Well, sure it's mind-bending but on the other hand, it does make some kind of intuitive sense. How could it be a 2D object if it has no area? 1D objects have no area. Same for the 3D sponge without a volume. Sounds more like a 2D shape then, doesn't it? So those fractals really are somewhere between two dimensions.

[@unvergebeneid]

standupmaths I'm looking forward to it!

[@chrisdrew1768]

Penny Lane this is why I do physics, things are simpler here.

[@n.l.4025]

Saying 4 Serpinski Carpets and 8 Serpinski Cubes have an area of Pi and a volume of Pi*4/3, respectively, definitely makes this a authentic Parker Video! There was just a bit of some fallistic (Fallistic is not a word, I know and don’t care. New words are created almost daily.) thinking and calculation. Read one of my other comments and you find out why if you haven’t already realized it yourself by some means.

[@themeeman]

MUSIC. SOUNDCLOUD. PLEASE. I LOVE YOU MATT.

[@quaglemy]

+Clingfilm Productions Seconded, we beg you Matt the music is amazing!!!

[@Scy]

That is so wicked sick. I think they just levelled up fractals.

[@frikkthoen]

Drop some LSD, and you'll see plenty of fractals, they're beautiful.

[@hanniffydinn6019]

Unfortunately mathematicians into fractals are too fucking dumb to do so. I mentioned this on a fractal forum and got banned. If you want proof reality and God is just an infinite fractal, take psychedelics. Totally fucking amazing! I knew God and reality were a fractal, lsd will prove it to you!

[@Ganpan14O]

Neat thing: Minecraft actually added a menger sponge in the April fool's update.

[@hweigel528]

Shouldn't it be 80/81 at 9:53? Looks like you're supposed to take the product of (n^2 - 1)/n^2 for all ODD n.

[@standupmaths]

Absolutely correct! I've added you to the corrections.

[@Coldo3895]

+standupmaths Oh I was so happy to have found a mistake !! But I am not the first.... ;)

[@teaser6089]

I feel that Matt tries to escape the Pi, but the Pi always catches up with Matt.

[@colinjava8447]

Very interesting, but he can't count, it should be: 4 * 8/9 * 24/25 * 48/49 * 80/81 * ... * ((2k+1)^2 - 1)/(2k+1)^2 * ... And also, its no longer a fractal since as your removing a smaller and smaller fraction each step, the shape is not self-similar at all scales. You would techinically be able to determine how far zoomed in to the shape you were by just looking at the shape in front of you, which is impossible to do in the menger sponge or carpet.

[@YounesLayachi]

thank you

[@justinlasker6269]

+Colin Java I guess we could call it a Parker Fractal

[@ferdiaobrien1500]

Fractal =/= self-similar. All it means is fractional dimension. Some fractals are self-similar, some aren't.

[@colinjava8447]

Yes, you're right, I'm used to infinite sums, so just put a + out of habit. Thanks, I have edited it now.

[@colinjava8447]

Well there's not really a strict definition, some say a true fractal must be self-similar, which makes the Mandelbrot set not a true fractal, and also the fractal in the video. If one wants to call it a fractal, I guess that's legal.

[@ifyoubelieveanythingmatter8924]

A friend directed me here ... beautiful ... this helps explain so much about the MacDonald Codex ... a work in progress for all who like an adventure.

[@superliro100]

10:42 why isnt it 80/81? Shouldn we remove the square of odd numbers?

[@tomprogramming]

+superliro100 I was wondering that myself. The graphic a) doesn't show the center third (ninth?) being taken out. Then the outer squares have their third taken out, then a fifth, then a seventh, then an eighth? Such weirdness.

[@nBasedAce]

Great video Dr. Mengerla! That's a sponge Elaine Bennis would think worthy of using!

[@jonnivuorinen8374]

Suomi mainittu

[@mikkoholopainen5112]

Torilla tavataan!

[@tube71000]

Mikko Holopainen Tortillat avataan!

[@JoelGrayson]

0:15 Parker Square

[@SergeofBIBEK]

Awww, you were here in Atlanta and I didn't know. :(

[@standupmaths]

+SergeofBIBEK Sorry, it was a flying visit! I barely left the conference hotel.

[@SergeofBIBEK]

standupmaths That's too bad. Oh well, I'm sure I'll get over it eventually. ;)

[@jensdevries6532]

Incredible... Currently studying infinite sums, but this infinite product just blows my mind! I really want to take a look at the proof of this convergence to pi and 4/3pi.

[@PlasmaHH]

I wonder what would happen if you get your hands on a high speed high resolution 3d printer...

[@cosmicjenny4508]

+Dennis Lubert L E V E L F I V E M E N G E R S P O N G E S

[@IndigoGollum]

Wouldn't even need to be high speed.

[@gabor6259]

The beauty of maths never ceases to impress me.

[@donfolstar]

No pic of a level 2 menger sponge? Stop slackin'.

[@mrphlip]

Hey, an excuse to talk about one of my favourite properties of the Sierpinski Carpet... So, just looking at the final carpet, you can see that _most_ of the points in the unit square are missing. Indeed, _almost all_ of the points are missing... by definition, since the area of the final carpet is zero. But of the points that are still there, many are still connected together, and you can find lines of them that are completely within the carpet. For the basic case, you can draw a vertical line at x=1/3 or x=2/3, or horizontal lines at y=1/3 or y=2/3. Or 1/9, 2/9, 7/9 or 8/9 (but not 4/9 or 5/9, as those hit the centre hole). For a more interesting case, vertical and horizontal lines at any point in the Cantor set still work. And this all makes sense, the carpet kinda looks like a complicated grid, it's made up of lots of horizontal and vertical lines. But, and this is the part that blew my mind when I first saw the proof: there are also _diagonal_ lines in the carpet. The line from (0, 1/2) to (1/2, 0), and then line from (0, 1/2) to (1, 0) are both, _in their entirety_, contained within the Sierpinski carpet. Even though the carpet just looks like a fractally-dense gridlike shape, these diagonal lines manage to avoid hitting _any_ of the infinitely many holes that get punched out of the carpet. And then the more fun part: these two lines (and their reflections, rotations, etc) are the _only_ lines that can be drawn. Lines at any other angle will always hit a hole somewhere at some point (indeed, they'll hit infinitely many of them). You can't just draw any old line through and hope it works. Except for these two exceptions

[@jaakkohintsala2597]

5:49 SUOMI MAINITTU TORILLA TAVATAAN!!!

[@ukko1998]

+Jaakko Hintsala Tampere mainittu!

[@fakedeltatime]

If I may ask, why are you people always like this?

[@ukko1998]

Neko Haxor since this is fun :D

[@KingArthurDent]

Suomi Manitu Tortilla Leviathan?

[@wolfbd5950]

12:10 I unashamedly shouted "No way! That's awesome!" Out loud.

[@StudioArrayMusic]

did you say 'recreational math'?!

[@Gold161803]

Uncle Jesse Look up a guy named Martin Gardner.

[@MinusPi-p9c]

Yup. Math is extremely fun if you're doing it for yourself.

[@howardg2010]

He meant 'recreational meth'.

[@IndigoGollum]

I know, it sounds like recreational spreadsheets.

[@Pete-Prolly]

@2:55, it's a Rubik's Void Cube, but by 3:09 it's a puzzle by Oskar Van Deventer but project mega menger takes the cake!!

[@FLOABName]

well, guess what i'm building in minecraft now

[@Ganpan14O]

They added a menger sponge dimension in the April fool's update, so if you give up on making one just boot that up!

[@mestiarcanus]

I actually built a level 2 Menger sponge out of Magic: the Gathering cards years ago (though using Magic cards instead of the Menger pattern for the cladding). It was a nice way to keep my hands busy while I watched KZbin videos. Took up a fair bit of space too. Sadly I ended up recycling it when I moved out of the apartment I was in at the time.

[@infrabread]

What would a 4D Menger Sponge look like?

[@pokestep]

+infrabread You'd be taking out a tesseract out of a tesseract, I assume. It's a really interesting thought though and I'm intrigued now.

[@eekee6034]

Speaking as a lifelong sci-fi nut, "scalesick" is the best new word I've learned this century. :D

[@thytom8534]

Would you call the Pi sponge a method of squaring the circle?

[@treufuss-yt]

+Thyt0m No. For that you have to construct it in a finite number of steps.

[@thytom8534]

Ahh.

[@g.seangourlay2593]

absolutely. except it's not. you can't repeat the process infinitely on paper. but you can get close.

[@Nicegeist]

Well you would be actively squaring the circle... but the circle would never be squared.

[@moraigna66]

+Thyt0m Circling the square?

[@andycohen3365]

This was the most awesome thing I've ever seen

[@jrt99b]

The natural out working of this? A menger sponge big enough to encapsulate the solar system. Whenever some future generation does achieve this I hope they will call it a Menger-Dyson cage.

[@SpaghettiToaster]

It's not a cage, it has no volume

[@dlevi67]

But it encloses one.

[@kristianhaverasmussen8558]

Matt is the kind of person who is haunted by an idea of doing something until he’s done it. And he end up making all sorts of wierd math things

[@thomaskraus7077]

Math, go home, you're drunk.

[@alibengali1150]

Alcohol, go home, you're on math

[@thomaskraus7077]

Math Not even once

[@lucca7716]

A square with area 0 and infinite perimeter. I’m gonna call that a Parker Square.