An Odd Property of the Sierpiński Triangle - Numberphile

Featuring Ayliean MacDonald. See brilliant.org/... for Brilliant and 20% off their premium service & 30-day trial (episode sponsor)... More links & stuff in full description below ↓↓↓
Discussing the Sierpiński Triangle, Fermat Primes, and Constructible Polygons.
Ayliean's KZbin channel: / ayliean
More from Ayliean: linktr.ee/Ayliean
More Numberphile videos with Ayliean: • Ayliean MacDonald on N...
Ben Sparks on Chaos and the Sierpiński Triangle: • Chaos Game - Numberphile
Ayliean on the Tower of Hanoi and the Sierpiński Triangle: • Key to the Tower of Ha...
Constructible Polygons...
With David Eisenbud and the 17-gon: • The Amazing Heptadecag...
With Zsuzsanna Dancso: • Euclid's Big Problem -...
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[@numberphile]

See brilliant.org/numberphile for Brilliant and 20% off their premium service & 30-day trial (episode sponsor)

[@MonsieurBiga]

Ayliean : you can timelapse this Brady : don't tell me what to do

[@soilnrock1979]

At what point do they have this conversation??

[@liambohl]

0:53

[@soilnrock1979]

​@@liambohlOh OK, the second part is in filming not in saying :-)

[@jayluck8047]

I love the way she can draw triangles with so much equilaterality.

[@johnjeffreys6440]

very isosolece

[@jayluck8047]

@@johnjeffreys6440 - did you mean, isolesence? But I got you.

[@johnjeffreys6440]

@@jayluck8047 yust a yoke 🤤

[@Jivvi]

​@@jayluck8047 *isoscelescence

[@betoneiracromadarebaixada8187]

the Sierpinski triangle really just randomly jumpscares people when it feels like it

[@jimi02468]

I thought it was pi that did that

[@McLir]

Take Pascal's Triangle and dot out all the odd numbers - that also gives a Sierpinski triangle. Seashells can also produce Sierpinski-like patterns.

[@rogerkearns8094]

Amazing. The moment you said, convert to binary, I saw it - but the effect not continuing forever, I didn't see.

[@leif1075]

But for gods sake theres no resson to think of binary..its contrived and out of nowhere rught?? No obewould rver think of that no.matter how smartbyou are

[@andrasszabo1570]

@@leif1075 But you just saw that someone has thought of that and did convert it to binary. It's not about being smart. It's about having the affinity and the time to play around with numbers.

[@Yulenka-]

​@@leif1075When you're dealing with Fermat numbers, there's all the reason in the world to use binary haha 😅 You can easily derive binary representation of their products since each number only has bits in two positions. I'm sure the pattern will reveal itself very quickly if you continue down this path

[@JuusoAlasuutari]

​@@leif1075Binary is the smallest integral base. Binary is arguably the fundamental number system and everything else is arbitrary.

[@rogerkearns8094]

@@leif1075 Thanks for the comment, mate, but I don't think I'm that smart. Hope your hangover's not too bad. Cheers :)

[@element54_]

My heart broke at 9:48 "until row 33".

[@RealCadde]

For a moment, my brain heard "until rule 33" and I was like "don't you mean rule 34?"

[@Testgeraeusch]

@@RealCadde "If we mathematicians can write it down, it must exist."

[@Sharxee]

And I was thinking Parker for some reason.

[@sk8pkl]

33 degrees in free masonry. Not a coincidence.

[@ArawnOfAnnwn]

@@Testgeraeusch Calm down there, Max Tegmark.

[@hepiik.8822]

It might be weird, but, as a Pole, seeing a properly written polish name made me smile

[@ChknKng]

What does the accent over the n do to the pronunciation?

[@MichalGlowacz86]

@@ChknKng It turns n into a nasal consonant. Polish ń sounds similar to Spanish ñ. Edit: it was pointed out to me, that n is already a nasal consonant. If I got the terminology right, then n will be voiced denti-aleovar nasal consonant while ń will be voiced palatal nasal. But feel free to correct me again! Soundwise though my analogy to Spanish ñ holds, with Polish ń being maybe a bit shorter.

[@SylveonSimp]

Grzegorz Brzeczyszczykiewicz

[@drenz1523]

​@@MichalGlowacz86 But n is already a nasal consonant...

[@MichalGlowacz86]

@@drenz1523 Damn, you're right. It seems n and ń are different sub-types of nasal though. Ń will be voiced palatal nasal I think, while n is denti-aleovar.

[@NatiNugasu]

Ayliean: timelapse this Brady Brady: 👍 *awkward silence*

[@thatonedynamitecuber]

That is the straightest triangle i have ever seen. To clarify I mean by hand not by any other means

[@Ayliean]

Pretty much the straightest thing I've ever done.

[@bagelnine9]

💀 💀💀 💀 💀 💀💀💀💀 💀 💀 💀💀 💀💀 💀 💀 💀 💀 💀💀💀💀💀💀💀💀

[@thatonedynamitecuber]

@@bagelnine9 nice italic sierpinski you got there. SKHULLLLEMOJIIIII

[@catmacopter8545]

​@@bagelnine9isnt this Wolfram automaton rule 90

[@cosmiccake791]

​@@thatonedynamitecuberno. No matt rose here...

[@JohnRunyon]

I love the shell tattoo while talking about pretty math drawings 😂

[@kirillsukhomlin3036]

And if you just take Pascal triangle mod 2, there would be proper infinitely growing Sierpinski triangle.

Yes. I think it's utterly fascinating that people can avoid seeing that. On the other hand it's something you might have to anticipate to look for.

[@JeroenBou]

Was looking for this comment. That's my favorite way to generate it.

[@oliverfalco7060]

4:25 Looking at someone making a pentagon with compass and ruler is always so exciting :3

[@TheArizus]

Fun side note, one of the problems on the 2023 British Algorithmic Olympiad was related to finding rows of the Sierpinski triangle when written in binary (similar to this)

[@morganconnelly5734]

Ah Ayliean coming back again with the amazing content! I love seeing her come back to the channel with her incredible mathematical story telling

[@TonboIV]

The thumbnail immediately grabbed my attention because there was something weirdly familiar about 65537. And of course the reason is because it is one more than 65536, which 2^16, which is the number of possible values in two bytes, and _that_ number can't help coming up in computer science quite often. So, came for the eerie number, learned something interesting-and vaguely annoying-about constructable polygons. Thanks I guess?

[@cybore213]

That's what drew me in as well.

[@jaymanier7286]

"Timelapse this." "...No." 😄

[@josephpk4878]

Neat to see this geometry again. I just designed a 3d model based on Sierpiński's Triangle, which is a 3D rendered pyramid of the 2D fractal, but I took it a step further and actually modelled the negative space, then printed out these interesting cubes composed of negative and positive 3-sided pyramids - beautiful things, especially when printed with clear materials.

[@jellorelic]

Gonna tease us like that and not offer photos? Maaaaaan...

[@genghiskhan6688]

yeah I wanna see that too!

[@HoSza1]

Ok, let's just name it The Parker Triangle.😂

[@user-hr8fj5ve3s]

There’s always something so captivating about watching a person explain something that they’re truly interested in and excited about

[@happyvirus6590]

5:08 and the length from that point to the edge of the circle is the *golden ratio*

[@stickfiftyfive]

and the length from that point to the edge of the circle *is the side length times the Golden ratio*. It's only the Golden ratio itself if the sidelength is 1. Worth clarifying.

[@waltercisneros9535]

Good to see a old style video, without the animations instead the very draws of our favorites mathematicians

[@kappasphere]

I think an interesting way to generate an image of a sierpinski triangle is to take every pixel coordinate (x, y), and color the pixel if x & y == 0, where "&" is the bitwise and operator.

[@bogdan_ostaficiuc]

xd ur imagining it

[@ulob]

​@@bogdan_ostaficiuche's not

[@bogdan_ostaficiuc]

@@ulob how? can you please explain? i'm dumbfounded

[@maksymisaiev1828]

@@bogdan_ostaficiuc it won't build exact sierpinski triangle but more like something area of sierpinski triangle. Here is easy python code to check: for i in range (0,40): for j in range(0,40): if i&j == 0: print(0, end="") else: print("_",end="") print("") You can play in numbers and still see that it is building triangles if you play with range numbers.

[@maksymisaiev1828]

​@@bogdan_ostaficiuc somehow youtube removed my comment. But idea is that bitwise operator gives 1 only in case when x and y share the same binary 1 at that position (in other words, it is binary multiplication). If we look at rows only, first row will be filled with 0, second row will have flappening 0 and 1, well because we compar numbers X1 and X0 and only X1 will return non zero. The third row is also similar. We compare 10 (binary 2) with numbers like X00, X01, X10, X11 and only last 2 numbers will return non zero bitwise response. Same for further rows. But the same picture is for columns, because we just flip x and y coordinates.

[@ggb3147]

I really appreciate keeping an acute over the letter N. Greetings from Poland ;)

[@xinpingdonohoe3978]

I'm not even offered it. Just ñ.

[@Bronzescorpion]

The 15 in binary mistake was somewhat funny considering Ayliean pointed out how close it was to 16. Even without giving it much thought, one could easily conclude that it must then be a row of ones, as all the numbers that are 2^n-1 must follow this pattern, before the next number ie. the number that is a power of two rolls over and becomes a number with a 1 followed by a string of zeroes (equal to n).

[@topherthe11th23]

I guess that if making an x-gon (where "x" is some positive integer) is impossible, then making a yx-gon is impossible where "y" is also some positive integer. IF the 28-gon were possible, then by connecting every fourth vertex going around it (skipping three vertices between any two connected vertices), you could make a 7-gon. By the law of contrapositives ("if A is possible then B is possible" means "if B is impossible then A is impossible") the impossibility of making a 7-gon proves the impossibility of making a 28-gon. Is that correct?

[@Nawakooo0]

It's always a delight to see Ayliean on Numberphile 💜

[@jaspermcjasper3672]

3:00 - I was afraid of this. Another Roof does the same. Euclid does NOT allow you to set a compass to a length for the purpose of transporting that length by lifting both legs and moving to another place in the diagram. A compass in strict Euclideanism loses its length-setting if you stop pressing it into the drawing-surface. Allowing a compass to transport a length is the same as allowing the marking of the edge of the straight-edge. Now, it can be proved that if a person can't do something using a Euclidean compass, then they also can't do it with a length-transporting compass either. Under the Law of Contrapositives, then, if you CAN do something with a length-transporting compass, then, you can ALSO do it with a stricter Euclidean compass. So, demonstrating a proof with a length-transporting compass proves that you could ALSO prove it with a stricter Euclidean compass. The universe of theorems that can be proved by Euclid isn't EXPANDED by adding length-transporting compasses, so it's not "cheating" in THAT sense. You can't "sneak in" any invalid theorems using length-transporting compasses. HOWEVER, and this is the key point that everyone misses, the diagram you construct of the proof, using a length-transporting compass, IS NOT THE SAME DIAGRAM as you'd construct of the proof of the same theorem using a stricter Euclidean compass, even though the existence of the former PROVES the existence of the latter. People are just missing the fact that if you stand on dry land and prove that there IS a method for doing something underwater, you haven't shown anyone how to do that thing underwater. You've merely shown that somewhere somehow there is some way to do it. That's NOT THE SAME!

[@rosiefay7283]

You are correct to say that marking the straight-edge enables constructions that are not possible using classical tools. "Allowing a compass to transport a length is the same as allowing the marking of the edge of the straight-edge." This isn't true. In Book 1 Prop. 2, Euclid shows how to use a classical compass to transport a length. This proves that anything constructible using straight edge and length-transporting compass is also constructible using straight edge and classical compass. So allowing a length-transporting compass is *not* the same as marking the straight-edge.

[@christopherfaytvlarknsassi9013]

@@rosiefay7283 Obviously I will have to look into this more. My belief was that by marking the straight-edge you were being non-Euclidean, but I didn't know that by making marks on the straight-edge you could prove things that you couldn't prove being strictly Euclidean. I had thought that you could only prove Euclidean things but were, as you would be if you used a compass to set, lift, and transport a length, taking non-Euclidean shortcuts.

[@xenontesla122]

Even though the pattern stops, it's satisfying that it at least stops at the base of a full triangle!

[@DeclanMBrennan]

I didn't know this cool connection but another one is coloring the numbers in Pascal's triangle according to whether they are even or odd.

[@WAMTAT]

Heck yeah, more triangles!!!!!

[@johnjeffreys6440]

Isosceles!

[@lornasmith5571]

I love to watch you draw shapes, and explain interesting stuff!

[@itioticginger9520]

I noticed at 6:35 that either side of 2^2^X were consistently constructible, as in either side of 2^2=4 meaning 3 and 5, then 2^4=16, and 15, 17 both worked, then 2^8=256, with 255, 257, then 2^16=65536 with 65535 and 65537 working and the final one shown was 2^32-1 This is too convenient to not be a pattern, and no one has ever been wrong when thinking a pattern holds true after a few iterations Edit: I did not expect to be immediately disproven

[@jamesknapp64]

it has to do with the fact that the product of all up to "nth" Fermat Numbers is 2 less than the next Fermat Number 3 x 5 = 15 = 17 - 2 3 x 5 x 17 = 255 = 257 - 2 3 x 5 x 17 x 257 = 65535 = 65537 - 2 3 x 5 x 17 x 257 x 65537 = 4294967295 = 4274967297 - 2 ; etc Note this another way to show that there are infinately many primes. Since all Fermat Numbers are odd and due to the product relationship above the only common factor could be 2 that means they all have different prime factors. Since we have infinate fermat numbers there are infinately many primes.

[@Buzk_4]

Patterns fool ya

[@Rubrickety]

How they fool ya…

[@David_Last_Name]

"This is my favorite way to draw a serpinski triangle." "Great. I need 34 rows." "No."

[@yiannchrst]

damn! I had accidentally discovered this some day while bored at school! I didn't go far enough to see that the pattern brakes though! Cool to see!!

[@volodyadykun6490]

4:15 I wonder how many people will scream this isn't allowed (in any case, you can find the center with these rules)

[@JohnDoe-ti2np]

Alternatively, start with the center.

[@esajpsasipes2822]

You can first draw the line, choose any point on it, and draw a circle of any lenght with that point as a center. Then you end up in the starting position without "breaking rules".

[@acaryadasa]

I didn't "scream", but yeah I noticed and posted. I suggest drawing the circle, creating a chord, make a perpendicular bisector of the chord to create a diameter, then create a perpendicular bisector of the diameter for the center.

[@rmsgrey]

Technically, you needed to identify the center in order to draw the circle in the first place.

[@jaspermcjasper3672]

Well, you could just start the whole construction by drawing any two random lines that aren't parallel. They must intersect, so when you make the circle, start there. Then you always have the circle's center when you need it. But I agree, you're not allowed to look for the hole that the compass-leg poked through.

[@DustinRodriguez1_0]

The Sierpinski Triangle is pretty wild, and that it shows up in so many weird places.

[@thirstyCactus]

Come for the math, stay for the dazzling hair and makeup! :D

[@panzer1896]

You used to sell the brown papers on eBay…do you still sell the used brown papers? These ones would be pretty cool to get.

[@martingallagher1780]

Ooh! Two helpings of Ayliean in one day. What did we do to deserve this? 🎉

[@ianstopher9111]

It's not the only time we get a finite list of terms. Finite normed division algebras have dimensions 2^n for n=1,2,3,4 and that's it. The general solution in radicals of polynomial equations only applies for powers n=1,2,3,4 and that's it. Fermat primes only for n=0,1,2,3,4. I recall at least in the first two cases they are related, but no-one knows if this also applies to Fermat primes or is just a coincidence.

[@MooImABunny]

what is it with the number 5 that keeps breaking these sequences 😂 (also there's no proof currently that there is no other Fermat primes at all. but it does seem pretty likely

[@marwynthemage]

Interesting. However, my favorite method of constructing the Sierpiński triangle will always be using recursive quad trees: draw the upper right quadrant black, and the other quadrants as the original quad tree (with the upper right quadrants black, recursively). You obviously need to stop rendering after a while, otherwise the entire image will be black :-)

[@Kestrel1971]

The Sierpinski triangle also appears from Wolfram elementary Cellular Automata Rule #90, and variants of the triangle appear in many other rules.

[@ronny332]

My brain smoked a bit while keeping track, but hey, it makes sense 🙂Thanks for showing!

[@KarolKarasiewicz]

Wow! Two things: 1. Miss, You're great at drawing, triangles drawn by hand, double wow. 2. So mamy theorems You just mentioned by the way, just like toystory... Triple wow! Thank You, that was great.

[@bkuker]

Any chance you'll talk about why there is this relationship between odd constructible polygons and fermat primes? Is it proven, or just coincidental? Would finding another fermat number mean finding more (large) odd constructible polygons? Does the relationship tell us anything about how we can construct them?

[@stephenbeck7222]

I believe the connection is proven in Gauss’ seminal work on arithmetic (number theory), in the same book he demonstrated the construction of the 17 sided polygon. I would guess the proof is beyond the scope of this channel.

[@tomkerruish2982]

It's too much to fit in this comment (appropriate for something Fermat-related), but it boils down to algebra. A straightedge and compass allow us to add, subtract, multiply, divide, and take square roots. (This is why we can't duplicate the cube since that would require a cube root.) Constructing a polygon with Fermat-prime-many sides can be done by performing a sequence of such computations. For further details, look up 'splitting polynomial'.

[@jamesknapp64]

Another Roof did a breakdown on the proof this. And yes this is proven that construcble odd factor distrinct odd fermat primes Yes finding another Fermat prime would mean there is an incredably large number of sides constructable polygon. Currently the smallest Fermat number that we don't know if its Prime or Composite is F_33 or 2^2^33 + 1 which is about *2.5 Billion DIGITS* long. However most number theorists believe that there are only 5 Fermat Primes. Yes Being a Fermat Prime tells you how to construct 17, 257 and 65537 sided polygons.

[@Sylocat]

I remembered the Fermat Primes from that earlier video series on constructable polygons.

[@janTasita]

My favourite place where an unexpected Sierpinski triangle appears is the evolution of a long straight line in Conway's game of life.

[@soilnrock1979]

That game got me through school without dying from boredome.

[@u2befake149]

thank you for wearing the keffiyeh

[@OneTrueBadShoe]

I absolutely adore Ayliean. I love seeing her visual representations of the beauty of math(s). Bonus: Those fingernails are sweet.

[@yoshi-cs6ib]

The sierpinski triangle is just the pascal triangle in GF(2), no? That's probavly a reason why it pops up a bunch.

[@N7492]

The "chaos game" method also constructs the Sierpinski triangle. Counterintuitive!

[@lombre9149]

New prime was found!! looking forward to your next video about it :3

[@jamesyoungquist6923]

In the fractal limit you can make a sierpinski triangle using any shape, including fish

[@nate8334]

My favorite Fractal. The blood type compatability chart is also a sierpinski triangle. I thought it was interesting that information about us could be Fractal in addition to the physical shapes of things like blood vessels.

[@cesarmontes7235]

First I was sad at that "until row 33", but then I immediately remembered Gaudí and this makes it somehow more magical and intriguing. Is there something more to this?

[@zathrasyes1287]

Beautiful handwriting

[@Pheonix1328]

I like how the triangle shows up in 1D cellular automata.

[@nazokashii]

One of my favourite shapes as well :D so cool! Thank you for sharing

[@David_Last_Name]

Lmao. I felt like Brady was refusing to timelapse it just to make a point. 😁

[@ant0n1o13]

"you can timelapse this" Keeps showing it in real time

[@Hambonillo]

showing the hexadecimal representation of those numbers might be interesting too.

[@likebot.]

"You're going to need a bigger paper".

[@mrsillytacos]

Going to have to print a whole new book for the new largest prime.

[@HunterJE]

Noticed that at least as far as it goes that triangle of odd constructible primes in binary is the same as if you make Pascal's triangle by the "add the two terms above each position" method but do the addition mod 2 (or, equivalently, XOR the terms above each position)...

[@MrMctastics]

With the 1's and 0's serpinski triangle, I thinks its called Glaisher's Theorem which implies that the sum of each row constructed this way must be a power of two. This kind of builds off the discussion in the comments about pascal's triangle since the nth row is 2^n

[@RealCadde]

Start of video. All i know is, the number in the thumbnail is 2 to the power 16, plus 1. Dealing with powers of 2 all my life has damaged me.

[@esajpsasipes2822]

someone could say it upgraded you

[@losveratos]

Really like her tattoos. She has a good artist.

[@jonprudhomme7694]

This is a fun construction. I am a big fan of we using binomial expansion coefficients to do the same thing, doesn't break down at any tho.

[@winnablebtw459]

Strictly speaking, at 3:00, you can't pick up lengths with a compass in construction problems. Doing so would allow you to trisect an angle which is famously impossible.

[@PaulFisher]

Can’t you transfer a distance between two arbitrary points by constructing a parallelogram with one edge being the distance you want to transfer and the second being the line from the source to the destination point?

[@WK-5775]

Please explain: How can one trisect an angle if one is allowed to pick up a length with a compass?

[@dingus42]

Wait why not? I thought that was one of the primary functions of the compass, to keep a set distance

[@zmaj12321]

I'm pretty sure one of the first proofs in Euclid's Elements is how to transfer a distance without being able to "store" distances on the compass.

[@dingus42]

@@zmaj12321 but you literally cannot use a compass for its normal function of drawing an arc without it being able to hold its distance

[@hedlund]

Oh, that's brought back memories of CS classes.

[@machevellian79]

Great video, fascinating! Thanks for sharing.

[@timetraveler1234-m3q]

Hey, cool golden ratio tattoo ❤

[@Ny0s]

This was a really beautiful construction

[@flamencoprof]

Another way to generate a Sierpiński Triangle is with a Cellular Automaton. (Memories of BASIC and a C64 a long time ago.)

[@PhilBagels]

If you spill Cantor Dust on your Sierpinski Carpet, you can clean it up with a Menger Sponge.

[@joysanghavi13]

Gauss proved that Fermat's prime numbers as polygon sides are constructible, when he was around 16 years old

[@rosiefay7283]

5:20 Your line cut the little circle in two points and you used the further one. An alternative would be to pick the *closer* of those two points. Your third circle would have then been smaller and perhaps easier to draw?

[@keir92]

immediately my brain is wondering why that's exactly one more than 2^16

[@TheSabian321]

I admire the editor's dedication to not timelapse the video.

[@vapormermaid]

As soon as I saw the number in the thumbnail I knew it had something to do with powers of 2.

[@tlhIngan]

65537 is also a common exponent used during RSA encryption and decryption.

[@Doktor_Vem]

I know this video's about math but I just cannot get over the accent, it's just so unbelievably beautiful and lovely

[@idrisbalavakos]

I could smell that Sharpie from here

[@coulie27]

Love the Sierpinski Triangle !

[@bigsarge2085]

Fascinating.

[@barutjeh]

You won't be surprised who discovered that that Fermat number isn't prime: Euler.

[@ex59neo53]

I used to find fractals beautiful ,then I had to study them 30 years ago ,before Internet ,and learned to hate the name Hausdorff :)

[@EastBurningRed]

just learned about haushorff in topology, what made you hate him?

[@mathphysicsnerd]

_"What do you MEAN represent a set of points with transcendental metric definition?!"_

[@iamalsometao]

Sierpinski ASMR

[@CHAYITO-ii5pt]

FASCINATING

[@eliotjacobson]

I don't see what the fermat numbers have to do with being Prime in the creation of the triangle. Surely if you just continue using those numbers in the indicated fashion you would continue to generate more levels for the triangle irrespective of them being Primes

[@dead-claudia]

that 65537 is bugging me way too much as a programmer. it's not 65535 or 65536. it feels off by one in the wrong direction. 😂

[@karlwaugh30]

Awesome episode. I wonder what properties the binary sieprisnki numbers have

[@NoNeedForRandomNumbers]

Better asmr than asmr

[@oliverwashbrook8834]

Was this recorded at KCL? An amazing link between number theory and geometry

[@JohnPretty1]

Is Ayliean dating Tom Craawford? Match made in heaven.

[@xethlorien4736]

well i wasn't expecting all of that. :D

[@allwaysareup]

Came for the maths, but staying for the surprise Ayliean.

[@acaryadasa]

IMO, using the center hole of the compass to find the center of a circle is kind of illegal according to the rules of Euclidian constructability. It doesn't really matter for the sake of this great video explanation, but strictly speaking one should/could construct a circle, draw a chord, construct a perpendicular bisector of the cord to construct a diameter, then create a perpendicular bisector of the diameter to find the center.

[@rmsgrey]

How do you construct the circle in the first place without starting with the center?