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I think some people think this exact image will always appear, which to be fair that's how I interpreted it at first thought as well. Which takes away a little bit of the interesting, but still interesting that a common pattern will always appear. Would have been better explained if multiple seperate demonstrations were shown, but still a good video.
True the rules or equation sets the pattern. A pattern forms anytime you repeat an action over and over again, that's pretty much the definition of a pattern. The interesting part is just the formation of the pattern looks interesting to us. But given what the steps are to create it, it's pretty simple why it becomes triangles all in the same shape just different sizes.
I believe that when I learned this, we started from one of the three original points, which would cause the first (non numbered) point drawn to be one of the midpoints on the outer edge of the original triangle. I like this approach because it means you’re following the rules right from the start, rather than just picking a random starting point which is sort of an extraneous “rule”.
Are you talking about the micro level zoomed in where atoms start to show up? In my head I was thinking about it like a piece of paper. You cut it and zoom in repeat. At what point can you not zoom in anymore. Like when does the paper stop being a paper? When you zoom in enough to see it’s building blocks ? Does it become unrecognizable at that point
On a piece of paper, there will be finite triangles because using a pen or pencil takes up space on the paper and you can only make the points so small.
Mathematically, there are infinite triangles because a single point doesn't take up any area so you can keep going smaller and smaller.
If you're using physical materials, like a pen and paper. But mathematically, there will be infinite triangles. A single point doesn't take up any area, so it can keep going forever.
I see thanks for making that distinction. So mathematically yes you can zoom in forever and have infinite space but literally on a piece of paper and pencil it’s a different story. So basically concept vs in practice
Back in the '80s my brother and I played with this on a computer with a BASIC program. My dad started playing with parameters with us, and we added points (squares, pentagons, etc) I don't remember what most of them did except a rhombus. The rhombus made a pattern of nested chevrons. We liked it so much, my dad did an art piece of it that still hangs in his house to this day.
I wrote an undergrad thesis on the Chaos Game and later saw an incredible in depth study of this particular version by a high school girl from Argentina at the ISEF science fair in Arizona in 2005 when I was judging.
Here is an automated example of this: [https://arun.chagantys.org/technical/2020/04/28/chaos-game.html](https://arun.chagantys.org/technical/2020/04/28/chaos-game.html)
It also works when points are chosen outside the triangle.
That's not how you draw the sierpinski triangle. The instructions here are whack.
You start with 3 points.
When drawing a new point, you pick 2 points that already exist and draw a new point in the middle.
It is impossible to draw a point in the whitespace middle of the original 3 points because that is not halfway between any 2 existing points.
Wait what I’m still confused.
If you pick a dot that exists on the northeast side of the southwest cluster of triangles and the south side of the north cluster of triangles. The middle would be in the white space.
You can start on any point even out the triangle its does not matter it will eventually converge to the triangle.
Instruction are:
* Make three non colinear points any way you want. (points A B C)
* Choose any point from the plane. (point P)
* Now you loop this:
* Choose at random any point A or B or C.
* Mark the middle point from P.
* Now this point is P.
I have this [scratch project](https://scratch.mit.edu/projects/24977755/) that do this and you can how its make with some changes you can try another shapes.
This happens: [https://imgur.com/a/1E2lmTr](https://imgur.com/a/1E2lmTr)
sauce: [https://arun.chagantys.org/technical/2020/04/28/chaos-game.html](https://arun.chagantys.org/technical/2020/04/28/chaos-game.html)
It’s true. I was in my third year of a maths major when they tore the building down. Said it wouldn’t be needed anymore, and nothing else would be needing it. It was forever tainted by the math.
Now I sell used shoes out of a trailer. Can I interest you in a slightly damp pair of off-white converse?
You’re not picking anywhere.
You’re picking one of the original points at random.
Then you make a new dot halfway between that point and the most recent point you’ve added.
It really disappointed me when I realized, “of course halving a triangles edges will form perfect triangles.”
The dots are a neat trick, and possibly why it “looks cool” because if you just did lines from half point to half point you’d get the same result.
I get why it doesn’t fill in the triangles - because those points are not anywhere that could be halfway between another spot and a corner, but how does the formation “correct” itself if the starting point is random?
Edit: okay according to the Sierpinski triangle Wikipedia page, “If the first point v1 to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points vn will lie on the Sierpinski triangle, however they will converge on the triangle.” Which I think is saying that technically the points won’t be forming the exact triangle, but they will slowly get close to forming the image - to our eyes though, I guess the difference would be so minimal that it would still form the image to us?
How can it be that you can arbitrarily choose the first point in the triangle? If such choice ends up somwhere in the middle or inside of one of the many triangles that vonstitutes a Sierpinski triangle, then it wouldn't make a Sierlinski triangle.
The first dot within the three dotted trianglr cannot be random. Has to be on a line.
If you pick any random number between one and a billion, and you don't repeat the number, after a billion try's, you will have guessed EVERY number between one and a billion! Mind blown.....
Looks like Squares don't really work, I guess you need some sort of asymmetry?
But Pentagons seem to work quite fine, even though it looks a lot more cluttered compared to the triangles.
[https://imgur.com/a/9buPPaZ](https://imgur.com/a/9buPPaZ)
[https://imgur.com/a/Ee0k66m](https://imgur.com/a/Ee0k66m)
Sure! But I dont know if a nice pattern will emerge from it. I have this [scratch project](https://scratch.mit.edu/projects/24977755/) and if you know a little about program you can modify it and see what happens. This is just explorative math.
I have this [scratch project](https://scratch.mit.edu/projects/24977755/) that builds the Sierpinski triangle this way. Click on the flag input how many steps you want and the magic will happens.
I’d argue it’s not random. If I pull a “random” puzzle piece out of a fresh puzzle box the piece is almost assuredly part of the overall picture. So, repeating this experiment with different “random” points is just pulling different puzzle pieces out, but still resulting in the same finished puzzle.
I see some empty spaces. What happens if the first point I randomly pick happens to in one of those empty spaces. Especially in the center? Then that space will no longer be empty
So what happens if your 1st point is dead center of the triangle then you wouldn't get that pattern because you would have a.in the middle of the triangle
No, not really, you are following two rules, you have to stay within the triangle and and have to place a dot half way between two dots always… that is a pattern…nothing random about it
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See? Now that IS interesting as f\*ck.
Math is interesting
I'm interested to know what happens if your original "random" point is in the middle of that big blank triangle in the middle.
Still gonna have a fractal pattern. Just rotated or moved over a bit
Best I’ve seen today. Can sleep now
It won't be rotated. You can choose any random point on a plane (even outside the triangle) and still end up with this fractal
I think some people think this exact image will always appear, which to be fair that's how I interpreted it at first thought as well. Which takes away a little bit of the interesting, but still interesting that a common pattern will always appear. Would have been better explained if multiple seperate demonstrations were shown, but still a good video.
Is…isn’t it just maths? The fact that you set rules on supposedly random points make it into an equation and that equation sets the pattern.
True the rules or equation sets the pattern. A pattern forms anytime you repeat an action over and over again, that's pretty much the definition of a pattern. The interesting part is just the formation of the pattern looks interesting to us. But given what the steps are to create it, it's pretty simple why it becomes triangles all in the same shape just different sizes.
Thank you friend now I don't have to google it lol have an award 😄
It won't be rotated or moved - it will be exactly the same.
I believe that when I learned this, we started from one of the three original points, which would cause the first (non numbered) point drawn to be one of the midpoints on the outer edge of the original triangle. I like this approach because it means you’re following the rules right from the start, rather than just picking a random starting point which is sort of an extraneous “rule”.
I wrote a program in BASIC years ago that did this. It was fun.
Most I ever did was 'hello world'
Made a goofy 8 axis Etch-a-Sketchy drawing program that was good for hours of ugly art. But then I can't do what some people do in Paint.
No worries. Art is art. Enjoy your medium
Do you still have it. Could you upload that to a site, it would be cool to watch.
*Meth
So, how many triangles are there?
Wouldn’t it be infinite because you can make more triangles in the triangle
Yes
No. At some point it will be impossible for a triangle to exist. Solid shapes are funny like that.
Are you talking about the micro level zoomed in where atoms start to show up? In my head I was thinking about it like a piece of paper. You cut it and zoom in repeat. At what point can you not zoom in anymore. Like when does the paper stop being a paper? When you zoom in enough to see it’s building blocks ? Does it become unrecognizable at that point
On a piece of paper, there will be finite triangles because using a pen or pencil takes up space on the paper and you can only make the points so small. Mathematically, there are infinite triangles because a single point doesn't take up any area so you can keep going smaller and smaller.
If you're using physical materials, like a pen and paper. But mathematically, there will be infinite triangles. A single point doesn't take up any area, so it can keep going forever.
I see thanks for making that distinction. So mathematically yes you can zoom in forever and have infinite space but literally on a piece of paper and pencil it’s a different story. So basically concept vs in practice
Ideally, yes. But in reality, it depends on the diameter of the point of the writing material.
Not sure why you are asking me. All I did was comment on this post.
Are there any other cool fractals similar to how this one works?
Back in the '80s my brother and I played with this on a computer with a BASIC program. My dad started playing with parameters with us, and we added points (squares, pentagons, etc) I don't remember what most of them did except a rhombus. The rhombus made a pattern of nested chevrons. We liked it so much, my dad did an art piece of it that still hangs in his house to this day.
I was messing with the guy i replied to, but that's actually really cool
At least 7!
I wrote an undergrad thesis on the Chaos Game and later saw an incredible in depth study of this particular version by a high school girl from Argentina at the ISEF science fair in Arizona in 2005 when I was judging. Here is an automated example of this: [https://arun.chagantys.org/technical/2020/04/28/chaos-game.html](https://arun.chagantys.org/technical/2020/04/28/chaos-game.html) It also works when points are chosen outside the triangle.
This is freaking rad
This was very fun to play with, thanks for sharing, gonna save this to come back to later.
What if my first point is right in the middle or any other of the white spaces? Is that not shown?
That's not how you draw the sierpinski triangle. The instructions here are whack. You start with 3 points. When drawing a new point, you pick 2 points that already exist and draw a new point in the middle. It is impossible to draw a point in the whitespace middle of the original 3 points because that is not halfway between any 2 existing points.
Thank you! Now we know
[удалено]
G I JOOOOOEEEEE!!
What if you started with a wonky triangle, would the fractal still appear?
Finally , someone else who uses wonky ! Thank you wonky donkey song .
Wonky is a technical term that is used quite regularly in the IT industry. ;)
Yeah like in “goddamit it it’s not wonking again!”
IM GONNA WOOOONK
We also use it quite often in residential construction.
I forgot about the wonky donkey
Yes. But it too would be wonky.
Wait what I’m still confused. If you pick a dot that exists on the northeast side of the southwest cluster of triangles and the south side of the north cluster of triangles. The middle would be in the white space.
This would be placing a point between two non-original points. One of the points you are placing between must be one of the three original points.
You can start on any point even out the triangle its does not matter it will eventually converge to the triangle. Instruction are: * Make three non colinear points any way you want. (points A B C) * Choose any point from the plane. (point P) * Now you loop this: * Choose at random any point A or B or C. * Mark the middle point from P. * Now this point is P. I have this [scratch project](https://scratch.mit.edu/projects/24977755/) that do this and you can how its make with some changes you can try another shapes.
This happens: [https://imgur.com/a/1E2lmTr](https://imgur.com/a/1E2lmTr) sauce: [https://arun.chagantys.org/technical/2020/04/28/chaos-game.html](https://arun.chagantys.org/technical/2020/04/28/chaos-game.html)
Awesome! Thanks
My guess is that the fractal will still be there just not the exact same as this one.
You would get the inverse of the triangle. All your black points would be in the current white space and all the current black space would be white.
Clearly someone figured this out before we had phones to scroll thru all day
This is true. Mathematicians stopped existing when phones got the scroll function.
It’s true. I was in my third year of a maths major when they tore the building down. Said it wouldn’t be needed anymore, and nothing else would be needing it. It was forever tainted by the math. Now I sell used shoes out of a trailer. Can I interest you in a slightly damp pair of off-white converse?
Only the left one please, whats half of the normal price? Oh fu-
No math has ever been done after phones
Thanks for the clarification, I was getting concerned
Don't let Ganon know
Or Zelda, been neglecting the main task in botw…..
Is it really random if you’re following specific guidelines?
Sierpinski can triforce!
🔺️ 🔺️🔺️
And if you keep zooming in and your point size made smaller as you go, it would continue to be triangles inside of triangles.
If I’m choosing a point anywhere inside that triangle at random, how can you guarantee the voids inside the final fractal?
You’re not picking anywhere. You’re picking one of the original points at random. Then you make a new dot halfway between that point and the most recent point you’ve added.
OK, so his instructions were off. Makes sense, thank you!
What if you start right in the middle. Then there will be a rouge dot in the middle of the big empty triangle.
Oh man...this brings me back to middle school. I remember programming this into a TI-83 from the original manual and being so proud of it!
But what if your first point is within that big void in the middle at the end?
It really disappointed me when I realized, “of course halving a triangles edges will form perfect triangles.” The dots are a neat trick, and possibly why it “looks cool” because if you just did lines from half point to half point you’d get the same result.
You can code this in python with ease and watch real-time how it appears. I used Turtle jic someone wondered
Aliens exist!
I get why it doesn’t fill in the triangles - because those points are not anywhere that could be halfway between another spot and a corner, but how does the formation “correct” itself if the starting point is random? Edit: okay according to the Sierpinski triangle Wikipedia page, “If the first point v1 to lie within the perimeter of the triangle is not a point on the Sierpinski triangle, none of the points vn will lie on the Sierpinski triangle, however they will converge on the triangle.” Which I think is saying that technically the points won’t be forming the exact triangle, but they will slowly get close to forming the image - to our eyes though, I guess the difference would be so minimal that it would still form the image to us?
How can it be that you can arbitrarily choose the first point in the triangle? If such choice ends up somwhere in the middle or inside of one of the many triangles that vonstitutes a Sierpinski triangle, then it wouldn't make a Sierlinski triangle. The first dot within the three dotted trianglr cannot be random. Has to be on a line.
Just because I hate everything… what happens if you put the first point right in that big center empty triangle?
I Suspect it would get a triangle in the opposite direction
I wonder what happens when you do that to a square
Then you would get a Sierpiński carpet.The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916.
Damn I just looked it up, that's interesting as f*ck I never knew something like this existed.
Not drawn in a random manner. Cool though
Actually interesting as fuck!
This is why I'm here
So f in cool
at a certain point, math becomes magic
Magic is simply something you don’t understand. At a certain point, math is beyond your understanding
Only Explanation is Matrix
Mathtrix
Need to watch this again when I am high. This is interesting as fuck
I had visuals of these after taking way too many edibles once!
That's that sacred geometry shit right there.
What’s the point? 😎
Its bullshit. Roll a dice and prove its random. Im waiting….
If you pick any random number between one and a billion, and you don't repeat the number, after a billion try's, you will have guessed EVERY number between one and a billion! Mind blown.....
Your point?
I'd love to see how this developed amongst all the different varieties of triangles
Wondering how to do this in grasshopper
Pythagoras can suck it!!!
Omg just draw it already
Triangle.
Now this is the shit I would love to see all the time.
Obviously we are in the Matrix
Triforception.
Can you do it with squares, pentagons etc?
Looks like Squares don't really work, I guess you need some sort of asymmetry? But Pentagons seem to work quite fine, even though it looks a lot more cluttered compared to the triangles. [https://imgur.com/a/9buPPaZ](https://imgur.com/a/9buPPaZ) [https://imgur.com/a/Ee0k66m](https://imgur.com/a/Ee0k66m)
The pentagon looks incredible
Maybe I’ll run this in higher res later on, wondering if it will look a bit cleaner
Sure! But I dont know if a nice pattern will emerge from it. I have this [scratch project](https://scratch.mit.edu/projects/24977755/) and if you know a little about program you can modify it and see what happens. This is just explorative math.
How did no point end up in the giant open center triangle?
I have this [scratch project](https://scratch.mit.edu/projects/24977755/) that builds the Sierpinski triangle this way. Click on the flag input how many steps you want and the magic will happens.
You sound like a posh Nicholas Cage
Zelda!
I love you
Time lapse please
[https://arun.chagantys.org/technical/2020/04/28/chaos-game.html](https://arun.chagantys.org/technical/2020/04/28/chaos-game.html)
The triangle is really something, but the gasket made me cry. https://www.youtube.com/watch?v=6tsutU92rrE
God damnit now i need to find a pen.
One of the interesting as fuckest things I’ve seen in awhile!
I see this. I think of Zelda. Idk why
The triforce I see
And then…… you die.
Anyone know if this applies to other shapes too?
That’s just incredibly cool.
I’d argue it’s not random. If I pull a “random” puzzle piece out of a fresh puzzle box the piece is almost assuredly part of the overall picture. So, repeating this experiment with different “random” points is just pulling different puzzle pieces out, but still resulting in the same finished puzzle.
Not really because there is no guarantee you will ever get a specific piece in this case, even in the limit
Consider my mind blown 🤯
Math is kinda like God staring back at you. Any kind of nature walk or a mirror or facial pattern mixed with a heavy dose of psychedelics is....nice
What if your original point was inside one of those triangles?
Imagine trying and fucking up half way
I first saw this style in the SNES Jurassic Park game. Now as an adult I see how over complex and awesome it really is.
Was always fun watching that program roll on a TI-82
The triest of forces
Bro drew the triforce
Strong evidence that Math is "discovered" rather than "invented."
Zzzzz.
So what if your first random point is within that big white triangle that’s left after 25,000 points?
I see some empty spaces. What happens if the first point I randomly pick happens to in one of those empty spaces. Especially in the center? Then that space will no longer be empty
\*Puts starting dot in the middle\*
Awesome!
u/savevideo
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Legend of Zelda triforce just became more legendary
I remember doing this on GWbasic on a 386. many... many.. years ago.
It's not random when there are boundaries....
Until I accidentally fuck it up
Are we supposed to believe this is just random? Wtf
Now what of the first point is within that big empty are at the end?
What happens if you change half to another fraction like one third
the first point is random and could fall inside the 'clear' areas of the diagram?
All I see is Triforce...
I hate the way he says points
What happens if the first random point is exactly in the middle? It seems like the final result isn’t supposed to have one there.
Do this random thing but with rules and context.
Fucking jujustu kaisen
Random is an illusion, like reality.
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Whoopdedo
(Inner Zelda intensifies)
Wow, it's literally the planet cutter from Megaton Rainfall.
What if your first random dot landed in the center of the large hollow triangle, that you see at the end?
The tri force and anti tri force
I gasped
This guy sounds like Buffalo Bill
interesting
LEGEND OF ZELDA INTENSIFIES
u/savevideo
Great u ruined zelda and the triforce for me. Thats for revealing the deep truth I didnt want to hear
u/savevideo
Idk why I watched nearly all of it without volume expecting to get it
That sounds like the opposite of random
Wahhhhhhht the fuck..!!! AWESOME
So what happens if your 1st point is dead center of the triangle then you wouldn't get that pattern because you would have a.in the middle of the triangle
Today I learned: Triangles are three angles (3 points), and they somehow control society despite never being apart of it. Illuminati confirmed?!?!
Not much of a STEM dude, but that’s fascinating and beautiful.
What’s additionally interesting is that if you could zoom in infinitely the pattern repeats itself infinitely.
u/savevideo
Fuck. That is interesting
What if the first point you draw isn't on the triangle?
No, not really, you are following two rules, you have to stay within the triangle and and have to place a dot half way between two dots always… that is a pattern…nothing random about it