This is the absolute epitome of amazing RUclips educational content. Everything explained in this video is mind blowing. Keep up the great work mate, you're going to get millions of subscribers
Cool vid! Fractals and meta curves always fascinate me, no matter my age. Kinda sad you left out the trapezoidal meta curve which converges to sierspenki triangle, but it's still cool ❤
absolutely top 5 or even 3 most underrated youtubers. I damn well hope you are a teacher because you are so good at it. Learned more here without any tests than any math class.
My 4 yo son adores this video. We've been watching it every night before his bedtime for the last 3 weeks. His favourite bit is the Chaos Game creating the Triforce :)
How tf is this perfectly animated and demonstrated master piece gets less views man I'm glad I found this and subscribing since before this video great freaking video well said and well demonstrated love your video keep going on❤❤
Why does this channel make me so genuinely happy? Is it the accent? Is it the complexity of the subject explained in a simple way? I just want to watch this forever
Very interesting video, I love the animations! Though "Infinitely many points" can be slightly misleading for some people, if someone is not aware about certain intricacies. For example, it's not immediately obvious for many, that when you keep doubling numbers in every next approximation of Cantor's Set (8, 16, 32, 64... 2^n... points), then what you get at the end is actually uncountable infinity. Similarly with Hilbert's Curve going through the points with irrational coordinates 😉
Definitely amazing, but the one where you get Infinite points on a line remaining while having subtracted the whole line is not surprising at all. Its basically dividing an infinity by 2 and saying: see now we have two infinities! But this surely was too much fun! (Cantor ser example)
Very good video! There’s a bit of trouble with your definition of the Cantor set, unfortunately: See, the cool thing about the Cantor set isn’t just that it has infinite points and zero length, but that it has **as many points as the original interval does** and zero length. Proving this is probably more complicated than the level you wanted this video, so I understand why you wouldn’t want to do it. Anyways, the reason your “Cantor set” isn’t really a cantor set is that you only included the endpoints. Then you’ll end up with only numbers that look like (a whole number) divided by (a power of three), which is not nearly enough points to cover the whole interval. (Again, this can be a little tricky to prove.) To get the actual Cantor set, what you need is the infinite intersection of all the individual Cantor set stages. That is, you need to include any point that stays inside the intervals, no matter how small the intervals get. It’s hard to picture a point that isn’t an endpoint which does this, but such points actually make up (in a technical mathematical sense) 100% of the Cantor set!
If in your variation of the Chaos Game we choose a point slightly above the center of the triangle's base, then it would hit the restricted black zone.
If the starting point is in any restricted zone it never lands outside a restricted zone but into progressively smaller dead zones with every iteration, closer and closer to Sierpinski's Triangle but never landing on it. Although after very few iterations the dead zones it's landing in become so small they're essentially invisible. If, on the other hand, the starting point is on Sierpinski's Triangle, it must remain on it because every point in the dead zones can't be reached, every point twice the distance from a vertex (the points that would have jumped into the dead zone) are either outside the triangle or in other dead zones.
1:54 The triforce of courage, the triforce of power, and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers.
Sierpiński triangle showing up in pascal’s triangle and through the chaos game can be understood quite intuitively. For the chaos game, consider only the biggest middle triangle. If we assume point was generated there through this method, we can construct three possible candidates for the point before it: we take the line from this point to each of the triangles tips and double them. All of these points land outside the triangle, so we can conclude that the middle section will stay empty, unless the process can also place points outside the triangle. The fact we can’t suddenly jump out of the triangle is easier to understand intuitively. The next smaller empty spaces can be understood by only considering the nearest corner, and the empty middle section: let’s say the second biggest void near the top of the triangle. A point in that space would imply the last point would either be well above the triangle (if the direction was either of the lower corners), or inside the middle void. And because we know the middle can’t generate any points, neither can these areas near corner. I don’t think this is a sufficient explanation to explain all the smaller triangles are also empty (as is), because the relationships gets more complex for the next levels. But having an intuitive understanding why at least the four biggest empty areas must form dispells much of the mystery. For Pascal’s triangle, the only two things we need to consider are that 1) for two numbers as input, the only possible way to add up to an odd number is adding exactly one odd and one even number, and 2) we have an infinite supply of odd numbers on the edges (as for the edges the parents always are 0 and 1, resulting a new 1). Say we had an arbitrarily large row of even numbers between the two ones add the end. Ignoring the zeros next to the ones, the next row would be one less wide. Because we chose that the row only contained even numbers between the ones, we can be sure the sumrow of them can at most have two odd numbers. Next we consider the each end of the row. In both cases we add a one to an arbitrary even number, so we must get an odd number. This locates all odd numbers in our sumrow. Our row now must be two odd numbers at the ends, and N-1 even numbers in the middle, where N was how many we chose to include in the first row. Ignoring the zeros and future ones that appear, the row we consider gets one shorter each time. So now we can repeat the same logic for our sumrow. It also has exactly two odd numbers, at each end, so the sum of it will be a row with odd numbers at the end, that is one shorter than it’s parent. This process forms the downward pointing triangle of even numbers inside Pascal’s triangle. Whenever there is a row of even numbers between two odd numbers (and we always will have at least the ones at the edge as odd numbers), it will taper down one number per row. The very last row will be just three wide, being odd-even-odd. Which results in a pair of odd numbers. This explanation doesn’t explain why more rows off even numbers appears, or why they are placed where they are, or why you eventually get a full row of even numbers and thus a big triangle after it. But for me at least, knowing that a row of even numbers must always taper down to nothing and form a triangle helps to explain the pattern.
That kind of question i had to solve myself with a cricle instead and never looked up if the answer of mine was right and now i know that it was thanks!
infinity means limitless/unlimited, but the shapes in the video have a clear boundry and therefore a limit to their area. what is limitless is the number out subdivisions you can make to the area of a hypothetical shape.
I think it would be good to show a 30s clip of calculating the integral of the triangle's missing area to subtract it from the big triangle. The reducing triangles have a really obvious and intuitive function driving their area, and it would be a great elementary intro that you could probably get most people to grasp if you don't "explain the reverse power rule" or other rabbit holes. Maybe not though, I've had some trouble even explaining how a functional average is like taking the diagonal through a square and turning into a "y=2" equation before, so maybe I'm just a terrible teacher.
Or does the lines between the corners guarantee that the irrationals between the corner coordinates are also touched somehow? Because it feels like the corner points should be countably infinite ( all the pairs of rational numbers) leaving an uncountable infinity of irrational points in between (all the possible pairs of irrational numbers)
Without watching the video: Assume the big triangle as a whole has an area of 1, the next smallest triangle would have an area of 1/4, the next smallest an area of 1/16 etc. With every iteration/size of triangles, there is 1 triangle out of 4 missing, so the area would be 1 - 1/4 - 1/16 - 1/64 … = 1 - Σ[n = 1; inf](1/4^n) = 2 - Σ[n = 0; inf](1/4^n), which is the geometric series, Σ[n = 0; inf](q^n) = 1 / (1 - q), |q| < 1. Hence A = 2 - 1/(1 - 1/4) = 2 - 4/3 = 2/3.
If I understand you right, then intuitively at some point there will be no longer any perfect squares to remove and some area will remain in the rectangle. Then the question to this is - what is the area of the final square removed from the triangle? You first must decide on the square you begin with AND the function you use to determine the area of the next square you remove. This function would be inherently recursive - meaning the ith input of the function is the output of the (i-1)th input unless you simply use a scalar. The outcome in both situations is the same. If you define a recursive function, then the area of the final square removed would be the supremum (read: limit as n -> infinity) of the function that produces the area of the next square. This would converge to some area since the square is contained within the rectangle. If you use a scalar, then you simply find the exact value such that the side length of the square equals the side length of the rectangle. Both will result in you being left with a rectangle missing a perfect square with side length equal to the vertical length of the rectangle. So the interesting thing is this: no matter what square you start with, the same result will occur. This is because of the obvious fact that a square has all equal side lengths! The area of the final square removed then, is R - S, where R is the area of the rectangle you start with, and S is the area of a square with side lengths equal to the shorter side length of the rectangle. So with a rectangle of area 15 that is 5 by 3, the area of this experiment would be R - S = (5 x 3) - (3 x 3) = 15 - 9 = 6. This is a brilliant question with a surprisingly elegant solution. Well done Josh.
Actually, with 6 lines of python code, we can say it's not infinity. a = int(input()) b=0 a2 = int(input()) for i in range(1,a): b = b+1/(a2**i) print(b) a: Repeations a2: Divide shape to a2 sides b: Colored area Example: a=69 a2=3 then b=0.5 (rounded because 0.499...83)
8:42 You know, it would be pretty nice if you explained what these four transformations do or *why* iterating a point's position by randomly picking them creates the Barnsley fern. If you made a deep dive into the subject explaining how this happens, maybe I'd be more fascinated by your content, but as such, this is just a shallow shovelful of factoids.
Doesn't the Hilbert Curve only reach points with rational coordinates? I get that going to infinity you approach everything because Q is dense in R. But you only get there by using the limit
using the thumbnail wouldn't a square minus the area of a fractal equilateral triangle equal 0.5 of the area of the square? just wondering cause at the triangle it's half there already, at the hyrule there goes a 4th of the triangle area so 3/8ths of a triangle in a square, but then the next one down 3/8 + 3/32 0.46875 or 15/32 just bearly half, do it again, 3/8 + 3/32 + 3/128 = 0.4921875 or 63/128 do it infinitly and you always get "just" under half
why does the chaos game work? ots infinite, so you could end up anywhere, and your point won't always be part of the triangle and make the triangle with 1 stray point.
this is the version of 1blue3brown that makes sense even if u only know simple math, underrated
3blue1brown
1blue3brown 💀
@@AdhrithMahesh ah crap your right
The hit channel by Sant Granderson
Skill issue tbh
This is the absolute epitome of amazing RUclips educational content. Everything explained in this video is mind blowing. Keep up the great work mate, you're going to get millions of subscribers
This channel is CRIMINALLY underrated
Geometry Dash Profile Picture spotted
based icon
lemoncake but blue
@@willm857 nuh uh
lemoncak3 is ass
I love gd colonge
9:40 he got us in the first half
Fr
8:19 that looked so satisfying...
Cool vid! Fractals and meta curves always fascinate me, no matter my age. Kinda sad you left out the trapezoidal meta curve which converges to sierspenki triangle, but it's still cool ❤
1:58 ah yes, the Triforce of Force, Courage and Wisd
underrated comment
When he was drawing the first shape, I was like "this looks 1/3 to me", was so satisfying when it turned out correct
very beautiful limits, it's even difficult to imagine what the result will be in the end👍
256+932=1000
2 3
23
10990
You have done a great job, getting an amazing result. Thanks for the video! This should have more views❤❤
absolutely top 5 or even 3 most underrated youtubers. I damn well hope you are a teacher because you are so good at it. Learned more here without any tests than any math class.
9:20that is the best leaf i have ever seen
this really is one of the most underrated channels as of right now
I love these kind of visual proofs.
Great content❤
She's in love with the concept.
Whoa finite object, with infite sides. Sounds amaizing
The perfect circle!
@@minotaurus4340 you could argue it has 0 sides easier than infinitee
how?@@Epicspacemacheene
Infinite is a cool thing 😊
Infinite Triforce is a cool thing 😊
My 4 yo son adores this video. We've been watching it every night before his bedtime for the last 3 weeks.
His favourite bit is the Chaos Game creating the Triforce :)
oh wow, cool that im not the only one that liked these types of videos
4 yr old son?!? you're raising a priodigy
@@Roperdo7 haha! :) I think he likes the pictures and saying the numbers he recognizes
@@AlanKey86 *I* think he's a prodigy
I mean how does he understand fractions?!?
How tf is this perfectly animated and demonstrated master piece gets less views man I'm glad I found this and subscribing since before this video great freaking video well said and well demonstrated love your video keep going on❤❤
less?
@@Fire_Axus i mean like he should've gotten 1million already lol
Why does this channel make me so genuinely happy? Is it the accent? Is it the complexity of the subject explained in a simple way? I just want to watch this forever
This is absolutely amazing and underrated
Congrats on blowing up
Very interesting video, I love the animations!
Though "Infinitely many points" can be slightly misleading for some people, if someone is not aware about certain intricacies. For example, it's not immediately obvious for many, that when you keep doubling numbers in every next approximation of Cantor's Set (8, 16, 32, 64... 2^n... points), then what you get at the end is actually uncountable infinity. Similarly with Hilbert's Curve going through the points with irrational coordinates 😉
In other words, you remove the entire length of the line segment, and there are still no fewer points than the original line segment.
These shapes are so cool
Definitely amazing, but the one where you get Infinite points on a line remaining while having subtracted the whole line is not surprising at all. Its basically dividing an infinity by 2 and saying: see now we have two infinities! But this surely was too much fun! (Cantor ser example)
This channel is frickin awesome thanks to my recommendations for this
Very good video! There’s a bit of trouble with your definition of the Cantor set, unfortunately: See, the cool thing about the Cantor set isn’t just that it has infinite points and zero length, but that it has **as many points as the original interval does** and zero length. Proving this is probably more complicated than the level you wanted this video, so I understand why you wouldn’t want to do it.
Anyways, the reason your “Cantor set” isn’t really a cantor set is that you only included the endpoints. Then you’ll end up with only numbers that look like (a whole number) divided by (a power of three), which is not nearly enough points to cover the whole interval. (Again, this can be a little tricky to prove.)
To get the actual Cantor set, what you need is the infinite intersection of all the individual Cantor set stages. That is, you need to include any point that stays inside the intervals, no matter how small the intervals get. It’s hard to picture a point that isn’t an endpoint which does this, but such points actually make up (in a technical mathematical sense) 100% of the Cantor set!
Love your videos man!
How do you animate all this? With Manim?
yes, I use manim
Fascinating! 😮
3:33 I've never heard someone say "eighty-oneths" before but I love it
Amazing video. This is like 3blue1brown but understandable. Deserved like.
Btw what is the music you used at the last fractal?
What arrangements ar u using for presentation?? Can u say... pls because preparing in Power point it is hard .
9:27 I was waiting for the dragon curve since the start if the video
this video is really interesting, i just took a break from my homework and in my freetime, i just watched some more math lol. keep up the great work
THIS WHY MATH IS SO BEAUTIFUL, NICE WORK BRO
I saw the Triforce in the thumbnail and came out of a lesson in quantum physics.
That “Dragon’s Curve” is in the Mandelbrot Set
If in your variation of the Chaos Game we choose a point slightly above the center of the triangle's base, then it would hit the restricted black zone.
If the starting point is in any restricted zone it never lands outside a restricted zone but into progressively smaller dead zones with every iteration, closer and closer to Sierpinski's Triangle but never landing on it. Although after very few iterations the dead zones it's landing in become so small they're essentially invisible.
If, on the other hand, the starting point is on Sierpinski's Triangle, it must remain on it because every point in the dead zones can't be reached, every point twice the distance from a vertex (the points that would have jumped into the dead zone) are either outside the triangle or in other dead zones.
1:54
The triforce of courage, the triforce of power, and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers , and the triforce of Alzheimers.
the fern fractal blew me off
I don't even care about mathematics but this channel is so interesting that i watch everything fr
How do space-filling curves touch irrational points if rationals don't contain irrationals? Both get arbitrarily close
bruh this channel is so underrated fr fr fr.
8:33 what if I will put first dot in the middle of triangle, where we can see void?
It's kinda funny how that's exactly what I was learning in school last week
The formula
But i think it was a bit different looking
Sierpiński triangle showing up in pascal’s triangle and through the chaos game can be understood quite intuitively.
For the chaos game, consider only the biggest middle triangle. If we assume point was generated there through this method, we can construct three possible candidates for the point before it: we take the line from this point to each of the triangles tips and double them.
All of these points land outside the triangle, so we can conclude that the middle section will stay empty, unless the process can also place points outside the triangle. The fact we can’t suddenly jump out of the triangle is easier to understand intuitively.
The next smaller empty spaces can be understood by only considering the nearest corner, and the empty middle section: let’s say the second biggest void near the top of the triangle.
A point in that space would imply the last point would either be well above the triangle (if the direction was either of the lower corners), or inside the middle void. And because we know the middle can’t generate any points, neither can these areas near corner.
I don’t think this is a sufficient explanation to explain all the smaller triangles are also empty (as is), because the relationships gets more complex for the next levels. But having an intuitive understanding why at least the four biggest empty areas must form dispells much of the mystery.
For Pascal’s triangle, the only two things we need to consider are that 1) for two numbers as input, the only possible way to add up to an odd number is adding exactly one odd and one even number, and 2) we have an infinite supply of odd numbers on the edges (as for the edges the parents always are 0 and 1, resulting a new 1).
Say we had an arbitrarily large row of even numbers between the two ones add the end. Ignoring the zeros next to the ones, the next row would be one less wide. Because we chose that the row only contained even numbers between the ones, we can be sure the sumrow of them can at most have two odd numbers. Next we consider the each end of the row. In both cases we add a one to an arbitrary even number, so we must get an odd number. This locates all odd numbers in our sumrow. Our row now must be two odd numbers at the ends, and N-1 even numbers in the middle, where N was how many we chose to include in the first row.
Ignoring the zeros and future ones that appear, the row we consider gets one shorter each time. So now we can repeat the same logic for our sumrow. It also has exactly two odd numbers, at each end, so the sum of it will be a row with odd numbers at the end, that is one shorter than it’s parent.
This process forms the downward pointing triangle of even numbers inside Pascal’s triangle. Whenever there is a row of even numbers between two odd numbers (and we always will have at least the ones at the edge as odd numbers), it will taper down one number per row. The very last row will be just three wide, being odd-even-odd. Which results in a pair of odd numbers.
This explanation doesn’t explain why more rows off even numbers appears, or why they are placed where they are, or why you eventually get a full row of even numbers and thus a big triangle after it. But for me at least, knowing that a row of even numbers must always taper down to nothing and form a triangle helps to explain the pattern.
Thank you for this.
Great work mate!
That kind of question i had to solve myself with a cricle instead and never looked up if the answer of mine was right and now i know that it was thanks!
damn, scrolling down, I thought this was a zelda video
infinity means limitless/unlimited, but the shapes in the video have a clear boundry and therefore a limit to their area. what is limitless is the number out subdivisions you can make to the area of a hypothetical shape.
How do you edit your videos???
I think it would be good to show a 30s clip of calculating the integral of the triangle's missing area to subtract it from the big triangle. The reducing triangles have a really obvious and intuitive function driving their area, and it would be a great elementary intro that you could probably get most people to grasp if you don't "explain the reverse power rule" or other rabbit holes. Maybe not though, I've had some trouble even explaining how a functional average is like taking the diagonal through a square and turning into a "y=2" equation before, so maybe I'm just a terrible teacher.
Nice!
Wow Digital Genius I saw a video of math is art well I just wanted to thank you for that video it's so fascinating how did you figure that out ❤
Does the space filling curve touch every rational coordinate inside the original open box shape, but not any irrational coordinate?
Or does the lines between the corners guarantee that the irrationals between the corner coordinates are also touched somehow?
Because it feels like the corner points should be countably infinite ( all the pairs of rational numbers) leaving an uncountable infinity of irrational points in between (all the possible pairs of irrational numbers)
Without watching the video:
Assume the big triangle as a whole has an area of 1, the next smallest triangle would have an area of 1/4, the next smallest an area of 1/16 etc. With every iteration/size of triangles, there is 1 triangle out of 4 missing, so the area would be 1 - 1/4 - 1/16 - 1/64 … = 1 - Σ[n = 1; inf](1/4^n) = 2 - Σ[n = 0; inf](1/4^n), which is the geometric series, Σ[n = 0; inf](q^n) = 1 / (1 - q), |q| < 1. Hence A = 2 - 1/(1 - 1/4) = 2 - 4/3 = 2/3.
This channel is actually 3b1b lite ❤❤❤
Absolutely amazing
I saw the triforce in the thumbnail and "geometry" in the title and clicked
the fern one blew my mind
good short summary
Am I the only one that clicked on the video because the thumbnail looks like the triforce?
Wait…that thumbnail is a nintendo refrence!!!
I was expecting a funny shape video, instead I got educated. I feel disapointed in myself
We love to see it 😮
Bro since when do math channels start talking about the TLOZ triforce 💀
When you start seeing art in mathematical patterns.
What if we take a rectangle, remove an exact square from the middle, and repeat the process?
If I understand you right, then intuitively at some point there will be no longer any perfect squares to remove and some area will remain in the rectangle. Then the question to this is - what is the area of the final square removed from the triangle? You first must decide on the square you begin with AND the function you use to determine the area of the next square you remove. This function would be inherently recursive - meaning the ith input of the function is the output of the (i-1)th input unless you simply use a scalar. The outcome in both situations is the same. If you define a recursive function, then the area of the final square removed would be the supremum (read: limit as n -> infinity) of the function that produces the area of the next square. This would converge to some area since the square is contained within the rectangle. If you use a scalar, then you simply find the exact value such that the side length of the square equals the side length of the rectangle. Both will result in you being left with a rectangle missing a perfect square with side length equal to the vertical length of the rectangle.
So the interesting thing is this: no matter what square you start with, the same result will occur. This is because of the obvious fact that a square has all equal side lengths! The area of the final square removed then, is R - S, where R is the area of the rectangle you start with, and S is the area of a square with side lengths equal to the shorter side length of the rectangle. So with a rectangle of area 15 that is 5 by 3, the area of this experiment would be R - S = (5 x 3) - (3 x 3) = 15 - 9 = 6.
This is a brilliant question with a surprisingly elegant solution. Well done Josh.
You gained my respect at 2:14
That's really cool
Actually, with 6 lines of python code, we can say it's not infinity.
a = int(input())
b=0
a2 = int(input())
for i in range(1,a):
b = b+1/(a2**i)
print(b)
a: Repeations
a2: Divide shape to a2 sides
b: Colored area
Example: a=69 a2=3 then b=0.5 (rounded because 0.499...83)
One might conclude from the chaos game that within infinite chaos, there is order.
Fractals are the best, not only because they are beautiful (Even non self-similar shapes). but because they are a rebellion against calculus
At 6:33 the denominator should be 729 instead of 243
Thanks for confirming that I am right 😅
The chaos game won't work with all random first dots tho? What will happen if the first dot is somewhere in the empty middle of the triangle
Hello how can i connect with you for tips and guidance? If it's paid i can pay tooo
You lost me a bit when explaining some of the equations but shit this is so educational and so crazy my mind kept getting blown🤣
8:42 You know, it would be pretty nice if you explained what these four transformations do or *why* iterating a point's position by randomly picking them creates the Barnsley fern. If you made a deep dive into the subject explaining how this happens, maybe I'd be more fascinated by your content, but as such, this is just a shallow shovelful of factoids.
Havent watched the video yet but i assume the shape in the thumbnail converges to a limit and not to infinity right?
The second I saw the thumbnail i immediately thought it was about the legend of zelda
Doesn't the Hilbert Curve only reach points with rational coordinates? I get that going to infinity you approach everything because Q is dense in R. But you only get there by using the limit
The Hilbert curve is the limiting curve, so it works out.
That square one I think was called Sierpínski's _gasket_ , not carpet. But, very nice to see all these gems together in one video!
The square one is called Sierpiński Carpet, but sometimes the Sierpiński Triangle is called Sierpiński Gasket
@@digitalgenius111 By Jove, your're right. I must stop doing these things from memory!
I had it at the start 1 att without knowing any of it! I love my eyes
so cool!
Even better with chimie and white powder to see this in the walls
This happens in music too, you have a note that is worth 4, then 2, then 1, then half, 1/4, 1/8, 1/16...
using the thumbnail wouldn't a square minus the area of a fractal equilateral triangle equal 0.5 of the area of the square? just wondering
cause at the triangle it's half there already, at the hyrule there goes a 4th of the triangle area so 3/8ths of a triangle in a square, but then the next one down 3/8 + 3/32 0.46875 or 15/32 just bearly half, do it again, 3/8 + 3/32 + 3/128 = 0.4921875 or 63/128 do it infinitly and you always get "just" under half
Wasn’t the triangle spilt in 4 just the triforse?
Doesn’t one of the Julia Sets resemble the dragon curve? I am not sure though, I might be mixing something else up.
Prequel to fractals video
6:34 correct me if I wrong but can anyone explain why is it 64/243 instead of 64/729?
This video explains where the Triforce gets it's power.
That's fractal!
damn this got me studying maths
why does the chaos game work? ots infinite, so you could end up anywhere, and your point won't always be part of the triangle and make the triangle with 1 stray point.
If anyone wants to know the first two are sierpenski shapes
Oh the nightmarish flashbacks of my 3** intro to mathematical proofs writing course 😅
The Line Fractal Is Actually The Mandelbrot Julia Set 5:13
"Or can it?"
*vsauce theme starts playing*
This looks almost as response video for mathologer latest clip
when the shapes are both finite and infinite at the same time: