An Odd Property of the Sierpiński Triangle - Numberphile

[@numberphile]

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[@FrankHarwald]

tbf, Pascal's triangle mod 2 is actually a Sierpinski triangle for all n up to infinity.

[@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 :-)

[@betoneiracromadarebaixada8187]

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

[@jimi02468]

I thought it was pi that did that

[@kwanarchive]

"Pathological monsters!" cried the terrified mathematician...

[@betoneiracromadarebaixada8187]

@@jimi02468 yeah, pi does it too. The Fibonacci numbers also pop up from time to time, there are some math things that are just kinda omnipresent

[@JustAnotherCommenter]

I guess you could say, it likes to Sierpinscare people

[@tremapar]

Just look in the mirror and say Sierpinski 3 times

[@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.

[@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.

[@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

[@iidoyila_live_]

we are becominh angels of skill and grace

[@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

[@drenzine]

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

[@MichalGlowacz86]

@@drenzine 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.

[@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 :)

[@NatiNugasu]

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

[@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.

[@therealax6]

All 2^2^n numbers come up in low-level computing all the time, of course. Up until n = 6 they're all very recognizable numbers. (Beyond that, we don't have the CPUs for those yet!)

[@TheAngelsHaveThePhoneBox]

I remember that number because it's the maximum number of rows you could have in the old Excel format (.xls).

[@therealax6]

@@TheAngelsHaveThePhoneBox That used to be 2^16, aka 65,536. (Now it's 2^20 = 1,048,576. Likewise, the number of columns went up from 2^8 = 256 to 2^12 = 4,096.)

[@psymar]

Yep! And the first number in the triangle sequence that *doesn't* work is 2^32+1, which 2^32 comes up in computer science quite often too!

[@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...

[@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)

[@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.

[@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

[@oliverfalco7060]

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

[@JohnRunyon]

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

[@Nawakooo0]

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

[@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!

[@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.

[@jaymanier7286]

"Timelapse this." "...No." 😄

[@ggb3147]

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

[@xinpingdonohoe3978]

I'm not even offered it. Just ñ.

[@WAMTAT]

Heck yeah, more triangles!!!!!

[@johnjeffreys6440]

Isosceles!

[@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).

[@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.

[@user-hr8fj5ve3s]

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

[@Buzk_4]

Patterns fool ya

[@Rubrickety]

How they fool ya…

[@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.

[@Kestrel1971]

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

[@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

[@TheSabian321]

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

[@thirstyCactus]

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

[@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'. Edit: it's been a few... decades... since I learned this stuff. Likely a better term to search is 'constructible numbers'.

[@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.

[@RobinDSaunders]

About "finding another Fermat prime", there's a probabilistic argument that there aren't any. The prime number theorem says: take a number near N, then its probability of being prime is p(N) ≈ 1 / log N. If Fₙ is the nth Fermat number, Fₙ ≈ 2^2^n, so log Fₙ ≈ 2^n log 2, and p(Fₙ) ≈ 1 / (2^n log2), which is tiny relative to n. The probability that *any* Fermat numbers beyond Fₙ are prime is then bounded by the sum of the individual tiny probabilities, Σ p(Fₖ), for k > n. This sum is ≈ Σ 1 / (2^k log2) for k > n, a geometric series which sums to the (still very small) value 1 / (2^n log2). So the probability that there are *any* Fermat primes beyond Fₙ is bounded by this very small value. We can rule out the first few Fₙ by checking them with a computer, which lets us start with a bigger n, so the probability becomes even smaller.

[@HoSza1]

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

[@xenontesla122]

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

[@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

[@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.

[@likebot.]

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

[@jacejunk]

Triforce from Zelda and mathematics- can't get much better combination of concepts.

[@lornasmith5571]

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

[@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.

[@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 :-)

[@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. 😂

[@tom7]

When performing a Euclidean construction, you may not use a Rule. But you may use a Hyrule

[@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!!

[@waltercisneros9535]

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

[@dogsareawesome9197]

Ive always been a _little_ into math, but otherwise didnt really care for it. THIS has got me hooked.

[@myamacke4159]

Another way of constructing a sierpinski triangle (kind of) is with a l-system tree with 3 branches, no trunk, and 120 degree angles! It's not a true sierpinski triangle because of the branches going through the voids, but it's still really neat! It was one of my first forays with l-systems before i started making art using stochastic l-systems constructed to model plants from my photography!

[@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.

[@nazokashii]

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

[@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.

[@herbpowell343]

Ironically, as you said, "You must be thinking, 'We have strayed far from the Sierpinski path here'" as I was thinking, "x, x+2, x(x+2), x(x+2)+2, x(x(x+2)+2), x(x(x+2)+2)+2(x(x+2)+2), etc." If it makes you feel better, I remain irate that pi~10^0.5 rather than pi=10^0.5. I console myself with the knowledge that, while (7^2*10-7+1)/7^2=/=10, 22/7=/=pi in the first place, so it could never be an exact match anyway. It does make the average of 7/5 and10/7 VERY close to 2^0.5 though, or at least closer than the tables they gave us in our high school math books. And I am trying VERY hard not to think about the fact that 10^0.5=(2phi+1)2^0.5. Ultimately, there is only Unity and Void, so binary is all there is, hence the whole 0/0=???? thing: What would you like it to be... ?

[@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

[@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

[@N7492]

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

[@losveratos]

Really like her tattoos. She has a good artist.

[@machevellian79]

Great video, fascinating! Thanks for sharing.

[@stephendemone]

Triangles are my favorite shape. Three points where two lines meet. 😎

[@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.

[@lombre9149]

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

[@PawelKraszewski]

Another way to make a (skewed) Sierpiński triangle is to take points of nonnegative integer coordinates. You paint the point black if (x AND y) == 0 (that is bitwise logical AND) and white otherwise. That's the fastest way to generate the triangle.

[@ant0n1o13]

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

[@DustinRodriguez1_0]

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

[@DukeBG]

Heuristically there should be a finite amount of Fermat Primes. So just those five. F12 is the smallest fermat number that is not fully factored. For mathematicians factoring it would be more exciting & bigger news than discovering the next largest mersenne prime & stuff like that. And a message with a factor would be smaller than this comment. F20 is the smallest fermat number where we don't have any factors at all (but we know it's composite). Finding a factor here would also be big news. F33 is the smallest fermat number for which we don't know the status (composite or prime) because it's too large to test (2.6 billion decimal digits). Actually, already for F31 we have some factors, but we don't know the status of the remaining cofactor (646 million decimal digits).

[@nosy-cat]

The problem of constructible regular polygons is a deep one, and very interesting. The channel "Another Roof" has a two part series on this, and I can't recommend it enough!

[@Pheonix1328]

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

[@jamesyoungquist6923]

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

[@timetraveler1234-m3q]

Hey, cool golden ratio tattoo ❤

[@platypi_otbs]

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

[@zathrasyes1287]

Beautiful handwriting

[@PrimordialOracleOfManyWorlds]

in the Sierpinski triangle fractal, i noticed the binary ones made up the upright triangles and the binary zeros made up the inverted triangles. suppose you do the fractal, fill binary ones in the upright triangles and binary zeros in the inverted triangles, then assemble the binary numbers, and convert to base 10 numbers. What numbers do you get?

[@WK-5775]

Fill the entire fractal? You'll get nothing at all because there are infinitely many layers of numbers. Instead, if one stops at some level, (and if I understand your question correctly), one will get exactly the products of finitely many different Fermat numbers.

[@Ny0s]

This was a really beautiful construction

[@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.

[@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)...

[@martingallagher1780]

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

[@joysanghavi13]

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

[@mrsillytacos]

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

[@Hambonillo]

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

[@FFVison]

I remember my TI-82 (my first semi-serious introduction to programming) had a test program you could enter into your calculator and it would plot out points to generate a sort of Sierpinski Triangle. It was not perfect, for one, the pixels wouldn't align exactly (every even row should have 2 values which doesn't line up with the odd number of values on the odd rows) and it would also have some stray pixels as it did use some RNG algorithm. Granted, this is from memory from like 30 years ago, so it's a hazy recollection of a program that I didn't fully understand how it worked that I copied and pasted into my calculator to generate a neato graphic.

[@Sylocat]

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

[@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

[@keir92]

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

[@soulsand4287]

7:45 The moment I heard binary I immediately saw where this is going...

[@psymar]

sierpinski gasket is the inductive step to the triforce

[@Tobi9012]

9:10 You can also create the shape by starting with the three 1s on the tip and and calculate the row below, like in the Pascale Triangle, but use XOR instead of addition as an operation. But it works beyond row 33 😉

[@David_Last_Name]

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

[@coulie27]

Love the Sierpinski Triangle !

[@definitelynotadj]

Can anyone explain how she constructed the midpoint of the line at around 4:50? I feel like i understood how to make the perpendicular line but i couldn't follow the midpoint.

[@WK-5775]

To construct the midpoint of a line segment, draw (segments of) circles with identical (and sufficiently large) radii around both end points of that line segment. The line through the two intersection points of these circles is the perpendicular bisector of the given line segment, and its intersection with it is the midpoint.

[@definitelynotadj]

Thank you!

[@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.

[@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?

[@bigsarge2085]

Fascinating.

[@bengoodwin2141]

I think it wasn't super clear, is that list the only odd constructable polygons? So there's a finite number of them?

[@Milan_Openfeint]

Depends on the number of Fermat primes. They get big fast, so we didn't check many, and I guess there's no proof either way. Statistically, the chance of Fermat number being prime is 1:2^n while size is 2^2^n, so I'd guess maybe there's one more somewhere and that's it.

[@bengoodwin2141]

@@Milan_Openfeint the statistical argument doesn't seem very sound. Nothing involved is truly random, but we'd need some kind of breakthrough on prime numbers to understand better. If the series is finite, it would be surprising if there are any more. If it is infinite, then they must just get more and more spaced apart, like the primes.

[@Milan_Openfeint]

@@bengoodwin2141 I was thinking like 1/2+1/4+1/8... is finite, you'd only get 2 primes ever if these were the chances, and the chances are actually lower.

[@exodusfivesixfivesix8050]

Princess Zelda finally explained the triForce.

[@JordynPi]

The strong law of small numbers strikes again!

[@DeathlyTired]

If you increase the tools to {compass, straight edge, can fold/unfold the paper (plane)} are all polygons then constrctible?

[@vapormermaid]

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

[@NoNeedForRandomNumbers]

Better asmr than asmr

[@MrBmarcika]

what we get is the mod 2 pascal triangle right?

[@MrScottev]

Why did you use 1 as a starting point for constructible polygons?