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
@@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.
@@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
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?
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 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.)
@@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.
@@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.
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)
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.
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.
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).
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 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.
@@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.
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
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.
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!
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!
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.
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.
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.
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 :-)
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.
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
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?
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?
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.
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'.
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.
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.
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 😉
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)...
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.
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
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.
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?
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).
See brilliant.org/numberphile for Brilliant and 20% off their premium service & 30-day trial (episode sponsor)
tbf, Pascal's triangle mod 2 is actually a Sierpinski triangle for all n up to infinity.
Ayliean : you can timelapse this
Brady : don't tell me what to do
At what point do they have this conversation??
0:53
@@liambohlOh OK, the second part is in filming not in saying :-)
the Sierpinski triangle really just randomly jumpscares people when it feels like it
I thought it was pi that did that
"Pathological monsters!" cried the terrified mathematician...
@@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
I guess you could say, it likes to Sierpinscare people
My heart broke at 9:48 "until row 33".
For a moment, my brain heard "until rule 33" and I was like "don't you mean rule 34?"
@@RealCadde "If we mathematicians can write it down, it must exist."
And I was thinking Parker for some reason.
33 degrees in free masonry. Not a coincidence.
@@Testgeraeusch Calm down there, Max Tegmark.
Amazing. The moment you said, convert to binary, I saw it - but the effect not continuing forever, I didn't see.
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
@@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.
@@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
@@leif1075Binary is the smallest integral base. Binary is arguably the fundamental number system and everything else is arbitrary.
@@leif1075
Thanks for the comment, mate, but I don't think I'm that smart.
Hope your hangover's not too bad.
Cheers :)
Take Pascal's Triangle and dot out all the odd numbers - that also gives a Sierpinski triangle. Seashells can also produce Sierpinski-like patterns.
I love the way she can draw triangles with so much equilaterality.
very isosolece
@@johnjeffreys6440 - did you mean, isolesence?
But I got you.
@@jayluck8047 yust a yoke 🤤
@@jayluck8047 *isoscelescence
we are becominh angels of skill and grace
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?
That's what drew me in as well.
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!)
I remember that number because it's the maximum number of rows you could have in the old Excel format (.xls).
@@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.)
It might be weird, but, as a Pole, seeing a properly written polish name made me smile
What does the accent over the n do to the pronunciation?
@@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.
Grzegorz Brzeczyszczykiewicz
@@MichalGlowacz86 But n is already a nasal consonant...
@@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.
That is the straightest triangle i have ever seen.
To clarify I mean by hand not by any other means
Pretty much the straightest thing I've ever done.
💀
💀💀
💀 💀
💀💀💀💀
💀 💀
💀💀 💀💀
💀 💀 💀 💀
💀💀💀💀💀💀💀💀
@@bagelnine9 nice italic sierpinski you got there.
SKHULLLLEMOJIIIII
@@bagelnine9isnt this Wolfram automaton rule 90
@@thatonedynamitecuberno. No matt rose here...
Ayliean: timelapse this Brady
Brady: 👍
*awkward silence*
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.
Was looking for this comment. That's my favorite way to generate it.
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)
4:25 Looking at someone making a pentagon with compass and ruler is always so exciting :3
"Timelapse this."
"...No."
😄
I love the shell tattoo while talking about pretty math drawings 😂
Ah Ayliean coming back again with the amazing content! I love seeing her come back to the channel with her incredible mathematical story telling
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.
Gonna tease us like that and not offer photos? Maaaaaan...
yeah I wanna see that too!
It's always a delight to see Ayliean on Numberphile 💜
There’s always something so captivating about watching a person explain something that they’re truly interested in and excited about
I really appreciate keeping an acute over the letter N. Greetings from Poland ;)
I'm not even offered it. Just ñ.
Good to see a old style video, without the animations instead the very draws of our favorites mathematicians
Heck yeah, more triangles!!!!!
Isosceles!
5:08 and the length from that point to the edge of the circle is the *golden ratio*
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.
Come for the math, stay for the dazzling hair and makeup! :D
I know this video's about math but I just cannot get over the accent, it's just so unbelievably beautiful and lovely
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).
Even though the pattern stops, it's satisfying that it at least stops at the base of a full triangle!
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.
xd ur imagining it
@@bogdan_ostaficiuche's not
@@ulob how? can you please explain? i'm dumbfounded
@@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.
@@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.
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.
Ok, let's just name it The Parker Triangle.😂
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
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.
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!
"You're going to need a bigger paper".
I love to watch you draw shapes, and explain interesting stuff!
The Sierpinski Triangle is pretty wild, and that it shows up in so many weird places.
Triangles are my favorite shape. Three points where two lines meet. 😎
Patterns fool ya
How they fool ya…
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!!
Triforce from Zelda and mathematics- can't get much better combination of concepts.
My brain smoked a bit while keeping track, but hey, it makes sense 🙂Thanks for showing!
I admire the editor's dedication to not timelapse the video.
My favourite place where an unexpected Sierpinski triangle appears is the evolution of a long straight line in Conway's game of life.
That game got me through school without dying from boredome.
The Sierpinski triangle also appears from Wolfram elementary Cellular Automata Rule #90, and variants of the triangle appear in many other rules.
I remembered the Fermat Primes from that earlier video series on constructable polygons.
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!
I like how the triangle shows up in 1D cellular automata.
In the fractal limit you can make a sierpinski triangle using any shape, including fish
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.
One of my favourite shapes as well :D so cool! Thank you for sharing
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.
The 2nd way I remember from one of the books as "multiple-copy reducing xero machine".
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.
showing the hexadecimal representation of those numbers might be interesting too.
immediately my brain is wondering why that's exactly one more than 2^16
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 :-)
The "chaos game" method also constructs the Sierpinski triangle. Counterintuitive!
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.
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
Great video, fascinating! Thanks for sharing.
The sierpinski triangle is just the pascal triangle in GF(2), no? That's probavly a reason why it pops up a bunch.
"you can timelapse this"
Keeps showing it in real time
New prime was found!! looking forward to your next video about it :3
As soon as I saw the number in the thumbnail I knew it had something to do with powers of 2.
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?
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.
Going to have to print a whole new book for the new largest prime.
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?
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.
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'.
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.
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.
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 😉
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.
Really like her tattoos. She has a good artist.
7:45 The moment I heard binary I immediately saw where this is going...
Lmao. I felt like Brady was refusing to timelapse it just to make a point. 😁
I absolutely adore Ayliean. I love seeing her visual representations of the beauty of math(s).
Bonus: Those fingernails are sweet.
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)...
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.
Ooh! Two helpings of Ayliean in one day. What did we do to deserve this? 🎉
This was a really beautiful construction
Wait the digits of those binary numbers makes Pascal's triangle mod 2
Beautiful handwriting
When performing a Euclidean construction, you may not use a Rule. But you may use a Hyrule
Oh, that's brought back memories of CS classes.
65537 is also a common exponent used during RSA encryption and decryption.
Love the Sierpinski Triangle !
Gauss proved that Fermat's prime numbers as polygon sides are constructible, when he was around 16 years old
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
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.
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?
Please explain: How can one trisect an angle if one is allowed to pick up a length with a compass?
Wait why not? I thought that was one of the primary functions of the compass, to keep a set distance
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.
@@zmaj12321 but you literally cannot use a compass for its normal function of drawing an arc without it being able to hold its distance
Awesome episode.
I wonder what properties the binary sieprisnki numbers have
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. 😂
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.
someone could say it upgraded you
Fascinating.
You won't be surprised who discovered that that Fermat number isn't prime: Euler.
If you continued writing out the Fermat numbers in binary, including the composite ones, would that produce a Sierpinski triangle?
I could smell that Sharpie from here
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).
Princess Zelda finally explained the triForce.
If you spill Cantor Dust on your Sierpinski Carpet, you can clean it up with a Menger Sponge.
The strong law of small numbers strikes again!
Another way to generate a Sierpiński Triangle is with a Cellular Automaton. (Memories of BASIC and a C64 a long time ago.)
Hey, cool golden ratio tattoo ❤
I do think that a bit of stabilisation would benefit the watchability of this video.