An Odd Property of the Sierpiński Triangle - Numberphile 

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

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

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

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

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

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 shell tattoo while talking about pretty math drawings 😂

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

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

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

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

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)

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

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

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).

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.

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

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.

"Timelapse this." "...No." 😄

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

Heck yeah, more triangles!!!!!

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

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

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!

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.

Patterns 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!!

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

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?

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.

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

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

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.

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 :-)

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?

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.

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

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

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

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?

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

Hey, cool golden ratio tattoo ❤

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

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

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

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

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)...

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.

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?

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.

Great video, fascinating! Thanks for sharing.

Fascinating.

Love the Sierpinski Triangle !

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

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

Really like her tattoos. She has a good artist.

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

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

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

FASCINATING

This was a really beautiful construction

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

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

ok that's very cool (and also heartbreaking 😂) but if we allow the use of non-prime Fermat numbers and their (distinct term) products, do we then get the full Sierpinski triangle? (sorry Poles, I don't have a grave n on my keyboard) /the mod2 Pascal's triangle

Better asmr than asmr

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

I could smell that Sharpie from here

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

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

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

Sierpinski ASMR

Thumbs up for the Keffiyeh!

9:11 isn’t that basically Pascal’s triangle but in binary (where 0 is even and 1 is odd)

Very interesting!

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

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.

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

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

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

Is Ayliean dating Tom Craawford? Match made in heaven.

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

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?

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

How do we know a certain polygon isn't constructable? The method we use for all the others doesn't work, sure, but could there be another approach that no one has thought of yet?

I wonder if there's a connection between Fermat Primes and Triperfect Numbers.

Ok but what would those binary numbers be if you kept going anyways? anything useful? maybe check for more fermat primes there

Huh, so thats how you triforce.

So there's only a handful of Fermat primes?

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

what we get is the mod 2 pascal triangle right?

How is constructing a nonagon not possible if a triangle is?

My favorite is pascals triangle color in the odd values

Ahh it's like last week's maths circle! 😉

Hey, that's a power of two Edit: one more than

How do you prove that line is one fifth of the great circle?

It almost looks like this binary sierpinksy construction is related to Pascal's (binary) triangle

That was neat

my mind is blown