since it's technically forming a one dimensional shape, it's pretty much impossible to not put the dot in one of the blank areas, since the points have zero length and width.
@@BakedAlaskaTM They're not one dimensional, just more like 1.3 dimensional or like that. You can understand that with areas, if you will try a square and a line, i can give you link to one video about that.
Long time ago, I was an 8th grade math teacher. At the end of the year, after all the state testing was done and we were counting down days until summer break, I did a unit on fractals. I spent a couple of lessons teaching my students how to program their TI-83s to graph the Serpinski Triangle. The program is about 12 lines long and uses the algorithm at the start of this video. The program is still available and works on all of the TI-84s.
Oh, that's nice. If you'd like to check out my code it is at the bottom of the description however keep in mind that im new to coding so feel free to share any tips.
Is "Patrick" a trick performed by your father? That would mean that "Matrick" is a trick performed by your mother. Then we have Brotrick and Sistrick (obviously) and, if you have a four-legged pet, we have dotrick.
Cool, A bit of research, putting the first point into proper position and it will become a rotation similar fractal. Right? Btw. I came from *3b1b featured creators #1* and I'm happy I have. Your videos are showing pure beauty of math!
I was surprised by that as well. I tried to check if I gave made any mistakes in my code but everything seemed fine. Let me know if you find the answer:)
Either there's a problem with your code, or your RNG isn't working properly. I did this same thing in Java and the generated shape had rotational symmetry.
I think “chaos” is interpreted incorrectly here, chaos essentially means *extremely sensitive to initial conditions*: change the initial randomly selected point marginally different from the first attempt and see a completely different pattern emerge. In this case it is more like “randomness”, and is opposite to chaos in that despite any random input conditions will always form a result with order.
After all that I have learnt about fractals and their formation; (fixed initial condition and a fixed algorithm put on loop) the video title intrigued me beyond words. This concept further pushes us from our understanding of fractals.
the rule was that the same vertex can't be chosen two times. Or the neighboring vertices can't be chosen. I can't remember from the top of my head right now.
@@ThinkTwiceLtu I raised from one dollar to two is not much but I can't afford a lot, is just you, Veritasium, 3Blue1Brown, Science Asylum and Eugene Khutoryansky. So you are one of my top five! I'm a physics teacher in Portugal.
@@jlpsinde No worries, I appreciate any kind of support whether it's monetary or not. I am happy that you enjoy my videos it keeps me even more motivated to create more high-quality content. Thanks a lot:)
It's interesting that if you ran infinite cuts, there would basically be straight lines from the corners to the center, except for an infinitely small point in the very corner of the triangle. The middle points on the edges of the blue triangle create the midpoint on the edges of the cut out triangles, as the midpoint of the edge midpoint and vertices create triangles.
What you are describing here is an Iterative Function System, more specifically the "fractal flame algorithm", if you google that there is a paper as the second result. The first triangle algorithm you describe is the first one described in the paper. The paper goes on to describe techniques on how to improve the fidelity of the images generated (which can become very vivid and intricate!). These are used by programs such as Apophysis(7x) or Fractorium (which is a gpu based implementation, which runs many many times faster on basically any computer with a graphics card). For people interested in this, I highly recommend reading the paper and then trying the programs out, maybe looking at more informal tutorials as well.
mathematicians: *put together an elegant closed system that works beautifully in creating an image that infinitely repeats itself over and over, evolving the world of mathematics and our understanding of it* me at 2:20 : hehe patrick
For those wondering why a fractal is produced, think of it like this: For the large middle black triangle, there is no random vertex and random point in the triangle you could pick such that the midpoint lies in that triangle. So, the random movement of the point will never land there. For the three smaller triangles, the only way for the midpoint to land there is for the starting point to be in the middle black triangle; however, as said before, the point never lands there, so the point never lands in these three triangles either. And then for the nine yet smaller triangles, their starting point would have to be in one of those three smaller triangles, and so on.
ohh omg huge thanks for the explanation ! so for the midpoint to land in the middle black triangle initially, a random point must be outside the triangle (and that's not the case here) and this would lead to the approach that our 3-sided triangle shown is adjacent to 3 larger black triangles outside on the screen
Nice! Have you tried to use an initial point that lies inside one of the empty areas in your fractal? What will happen? Will that be the only point outside the fractal? Or maybe the fractal only appears for certain initial points?
Yet again the one didn't fail to impress me This time my mouth was open for 2 mins. Legendary!!! Just wait for me to get a job and I will pay half my salary💚
I have done this simulation and created Sierpinski triangle images. I've also created a Sierpinski triangle using another algorithm, and I know a third algorithm. These are cool.
The first triangle is very interesting. If you look at the pascall triangle. And always draw a dot for an odd number than you will get the exact same triangle pattern. Math ist beautifull
The math with the triangles came out perfectly matching what our universe probably would have done. The list of dots per size range from 1XXL to 3XL to 9L to 27M to 81S. To see how 9 plays a major part in our universe, check out Marko Rodin; or take the 1-9 multiplication table and reduce all the numbers through addition, for example 10=1+0=1, 12=1+2=3, 14=1+4=5 ect. This produces what I'm dubbing the Layman's Fingerprint of God because you can see how 9 exerts its dominance over it all. If you add it all together you get 459 or 4+5+9=18=1+8=9 and how it is surrounded by 9 and has 9 running though it. Making 9 very similar to our description of God being one who is equal to it all, yet outside of it all, yet running through it all.
i believe this concept sparked a notion in my mind, that... this is one of the ways the Cosmos has to create reality, like a rule* that particles and atoms arrangement , a building block of the physical realm can never be in the same spot twice... a quick example, the snow crystals, but reality's matrix is 6 triangles facing each other or hexagonally based... 🤔
Awesome! I think I saw this on numberphile, but this video took it further. Thanks! Btw, it's cool to see stuff done in processing. I've used it a few times, but mainly I just stick to python.
It's rare to find youtube channels that produce the "style" of videos(which I enjoy) that you produce, not too long neither too short, a good explanation of what are you going to do/happening and I particularly liked the fact that you showed that the rule that we use at the triangle doesn't work with the square and showed how to correct it! And as always you choose a good song to follow the video with (:, for example I think that a piano song(Erik Satie for istance) wouldn't go as good as this song that you choosed went. Edit: I forgot to mention that that last few videos that you made, we can actually try it by ourselfs (and not only see a different/cool proof, say that's nice and close the video/investigate about it), one example is "Cutting a Möbius strip in half" ahahha. Sorry for my mediocre and redundant English.
Thanks:) It's good to hear from people like you that they enjoy watching my videos. I don't think I would be making a lot more videos if not for feedback like yours. Glad you like the music choice as well. Tbh I've been really struggling in finding songs that sound good for the background of my videos. Most of the royalty free songs don't sound that well, so most of my videos get copyrighted for the songs I use. If you have any songs in mind that you think would be suitable for my future videos feel free to share haha. Thanks again for watching:)
Nuno Mateus You can always email me at thinktwiceask@gmail.com , but you don't really have to look for the songs, dont want you to feel like you're being forced to do that. Thanks again:)
Numberphile made a video on the same topic too. All those who liked this video should check that too. Just type in "Chaos Game Numberphile" on your RUclips search and you should find it.
Reality cannot be whatever you want because other people have different ideas of what is the center . The relation between your Center and other centers is what makes "reality"
Im calling you out on that triangle. If you drew a midpoint from a point to any random vertex it would pretty unlikely to be so close to the middle of an edge unless that point is in the corner
Would you always get the same fractal given any point, or does the initial point matter? It seems like getting a perfect fractal would be impossible if the middle was selected.
The first few points will not be on the fractal, but all future points will get exponentially closer to the fractal. So after 100 iterations, the point will only be less than 1/(2^100) of a unit away from the fractal.
You hit on a really interesting point there btw: these images are technically just very very close approximations of the fractals they describe. They are what is called strange attractors. Rulesets that somehow make points chosen at random "attract" into a shape. If the shape is a fractal, the attractor is called strange.
I'll just note that the systems shown in the video are not chaotic: even though we might casually understand chaos as "random" (for example, in the context of a mob), mathematically speaking, that view is completely incorrect. Chaotic systems are fundmentally deterministic: if you know exactly how a system evolves and know the exact state of the system, you will be able to compute all past and future states of the system. Chaotic behaviour refers to how small changes in initial conditions of the system can produce vastly different behavious as time goes on. By contrast, with the systems shown in this video the next state is chosen at random: you can't predict with 100% certainty what the future states will be. That makes this video a demonstration of how stochastic processes - systems which evolve randomly - can sometimes form structures which are also interesting for other reasons.
I'm still in pretty much the same condition as a month ago, I just got to wait so I don't really know how long it will take to get better. Thanks for asking though, glad you liked the video:)
![BLACK BAG - Official Trailer [HD] - Only in Theaters March 14](http://i.ytimg.com/vi/Du0Xp8WX_7I/mqdefault.jpg)
Me putting a point directly in the center: *Reality can be whatever I want*
Okay, but seriously
What happens if you put a point in the middle?
@Sai Sasank thanks, just watched it, I actually understand the phenomenon alot better now
since it's technically forming a one dimensional shape, it's pretty much impossible to not put the dot in one of the blank areas, since the points have zero length and width.
@Sai Sasank link?
@@BakedAlaskaTM They're not one dimensional, just more like 1.3 dimensional or like that. You can understand that with areas, if you will try a square and a line, i can give you link to one video about that.
The Legend of Fractal - Breath of the Chaos
ha
underrated comment since the last 2 years
Lol
triforcw
@@123cityperson now 4
Long time ago, I was an 8th grade math teacher. At the end of the year, after all the state testing was done and we were counting down days until summer break, I did a unit on fractals. I spent a couple of lessons teaching my students how to program their TI-83s to graph the Serpinski Triangle. The program is about 12 lines long and uses the algorithm at the start of this video. The program is still available and works on all of the TI-84s.
Oh, that's nice. If you'd like to check out my code it is at the bottom of the description however keep in mind that im new to coding so feel free to share any tips.
Hey! I didn't find your code mentioned in the description...
Maybe you standardized video descriptions and overwrote it?
*Sierpinski
@kurz1000 I bet that in English it is „Sierpinski Triangle", while in Polish "trójkąt Sierpińskiego".
Yes but why?
"Is this a fractal?"
"No, this is Patrick"
Lol
Patrick
Is "Patrick" a trick performed by your father? That would mean that "Matrick" is a trick performed by your mother. Then we have Brotrick and Sistrick (obviously) and, if you have a four-legged pet, we have dotrick.
@@simonmultiverse6349 dotrick = two tricks in 1
tritrick =
Woof.
Furthermore, *WOOOOOF* also bark, bark!
"Oh, just random points, it can't be that intresti-- Wait what?"
Can we now say that math is art?
Who ever said the two weren't linked?
It's always been art
Depends on your definition of art. Math is related to art? Yes. It IS art? I don't know. As I said, it depends on your definition of art.
Cool,
A bit of research, putting the first point into proper position and it will become a rotation similar fractal. Right?
Btw. I came from *3b1b featured creators #1* and I'm happy I have. Your videos are showing pure beauty of math!
MIMIK 100th like
It’s an error in the code
I have a Guide Voodo, and imma throw it to the lava
@@ragnarock1008 Andrew the Guide was slain.
Wall of Flesh has awoken!
@@haxel96 *MIMIK the guide
I love how it pulls a 64 bits 32 bits 16 bits 8 bits 4 bits 2 BITS 1 BIT HALF BIT *QUARTER BIT* at 1:07
Yeah, cool animation
Why doesn't the last one have rotational symmetry?
I was surprised by that as well. I tried to check if I gave made any mistakes in my code but everything seemed fine. Let me know if you find the answer:)
Either there's a problem with your code, or your RNG isn't working properly. I did this same thing in Java and the generated shape had rotational symmetry.
yes I guess there was a problem with the code. My bad:(
It was a great video nonetheless, and your new videos are also great! Please continue making more!
thanks for nice words:) I will try to upload whenever I can. Hope this community will keep growing.
We love you Think twice! Thank you for creating this iteration animation!
I think “chaos” is interpreted incorrectly here, chaos essentially means *extremely sensitive to initial conditions*: change the initial randomly selected point marginally different from the first attempt and see a completely different pattern emerge. In this case it is more like “randomness”, and is opposite to chaos in that despite any random input conditions will always form a result with order.
0:58 Zelda logo complicated
1:45 snowflake
2:20 summoning a demon
You missed out on the fern one
Its really beautiful, a perfect fern just from chaos
Param Rathour ya i was gonna make a separate video about it
Think Twice I would be waiting for that.
I like your way of explaining visually.
After all that I have learnt about fractals and their formation; (fixed initial condition and a fixed algorithm put on loop) the video title intrigued me beyond words. This concept further pushes us from our understanding of fractals.
The title deserved a long ponder and a like STRAIGHTAWAY
2:26 is this a fractal?
No,this is Patrick
I just woke up and I am not ready for this. I will come back when I am actually awake
legends say he's still waking up to this day
why pentagon does not generate a symmetrical figure ? initial condition ?
jr tao i messed up the code, sorry
its strange ! github.com/appleminis/PolygonDreamcatcher
the rule was that the same vertex can't be chosen two times. Or the neighboring vertices can't be chosen. I can't remember from the top of my head right now.
random walk what iam looking hehe
mine don't break symmetry, i will try to understand something, thank you twice ;)
Amazing, this man deserves more people supporting on Patreon!
Thank you, really appreciate your support.
@@ThinkTwiceLtu I raised from one dollar to two is not much but I can't afford a lot, is just you, Veritasium, 3Blue1Brown, Science Asylum and Eugene Khutoryansky.
So you are one of my top five!
I'm a physics teacher in Portugal.
@@jlpsinde No worries, I appreciate any kind of support whether it's monetary or not. I am happy that you enjoy my videos it keeps me even more motivated to create more high-quality content. Thanks a lot:)
Looooove your choice of music. Will be looking forward to hear your own music as background, would be cool ❤️
thanks Lucy :) I'll try to write my own song
hope you feel better soon x
on the triangle, what if you put the first point in the center of the triangle? would that ruin the fractal?
no
No it's somehow manage to get its shape
It's interesting that if you ran infinite cuts, there would basically be straight lines from the corners to the center, except for an infinitely small point in the very corner of the triangle. The middle points on the edges of the blue triangle create the midpoint on the edges of the cut out triangles, as the midpoint of the edge midpoint and vertices create triangles.
@@aidenwallin3523 fddd ddo dddo did did didnt 😮😂do Z
What you are describing here is an Iterative Function System, more specifically the "fractal flame algorithm", if you google that there is a paper as the second result. The first triangle algorithm you describe is the first one described in the paper. The paper goes on to describe techniques on how to improve the fidelity of the images generated (which can become very vivid and intricate!).
These are used by programs such as Apophysis(7x) or Fractorium (which is a gpu based implementation, which runs many many times faster on basically any computer with a graphics card). For people interested in this, I highly recommend reading the paper and then trying the programs out, maybe looking at more informal tutorials as well.
ty!
This place NEEDS more views, you are awesome!
When you are just having fun with shapes but end up summoning a demon
THIS CONCEPT ITSELF IS TOO AWESOME SO AS TO DESERVE A MILLION VIEWS.
This is one of the best vídeos that i ever seen
No way! It's absolutely fascinating! Thanks for sharing.
Your work is brilliant. It's so good that I even let the ads finish just to support you. And, yes, I'll hit up Patreon
Good work.
Really appreciate your support~
Watching a one hour ad on a 2 minute video lets gooooooooooooooo
mathematicians: *put together an elegant closed system that works beautifully in creating an image that infinitely repeats itself over and over, evolving the world of mathematics and our understanding of it*
me at 2:20 : hehe patrick
This is the scariest video of 2018! :o
If i were an artist, a musician I would buy the rights to the animation of the triangle and use it as a cover, its soooooo satisfying!!
Absolutely magnificent I've been working on this Serpinski triangle and I never imagined such a thing
You need more Subs!!!!
appreciate your support:)
I completely agree with this!
I am thoroughly terrified yet incredibly shocked at the same time
For those wondering why a fractal is produced, think of it like this:
For the large middle black triangle, there is no random vertex and random point in the triangle you could pick such that the midpoint lies in that triangle. So, the random movement of the point will never land there.
For the three smaller triangles, the only way for the midpoint to land there is for the starting point to be in the middle black triangle; however, as said before, the point never lands there, so the point never lands in these three triangles either.
And then for the nine yet smaller triangles, their starting point would have to be in one of those three smaller triangles, and so on.
ohh omg huge thanks for the explanation !
so for the midpoint to land in the middle black triangle initially, a random point must be outside the triangle (and that's not the case here)
and this would lead to the approach that our 3-sided triangle shown is adjacent to 3 larger black triangles outside on the screen
this is the best thing to watch when sleep deprived
Nice! Have you tried to use an initial point that lies inside one of the empty areas in your fractal? What will happen? Will that be the only point outside the fractal? Or maybe the fractal only appears for certain initial points?
Why is the pentagon one not rotational symmetrical? Why is the bottom side the only one that fills?
Yet again the one didn't fail to impress me
This time my mouth was open for 2 mins.
Legendary!!! Just wait for me to get a job and I will pay half my salary💚
really appreciate your comments. thank you for the support:)
Your videos are amazingly exciting for me.
I could never imagine Maths could be so fun.
Thank you.
Cool. Mathematically makes sense, but awesome to see it in action
I have done this simulation and created Sierpinski triangle images. I've also created a Sierpinski triangle using another algorithm, and I know a third algorithm.
These are cool.
2:20 I don't know what that fractal is called so I'm going to call it the serpenski rocket
How do you make this animations? It’s awesome :)
Julio Toboso García he said in the description that he used a programming environnement called Processing, so I gess he programmed it himself
Processing is a really powerful visualization language.
Look for The Coding Train on RUclips, you will find a lot of tutorials.
@@mathieugouttenoire9665 please remind me in a yearrr
processing.org
@@nahfid2003 it’s been 7 months, why did you want reminding in a year?
Really Really good vids! As a sience teacher this is gold! Thank you!!
Btw Also came through 3B1B...
Everybody gangsta until he makes Mandelbrot's set with this
"Total chaos" lmaoooo. The amount of misconception in the layman is just absurd.
Woah nice Ultra Triforce!
I was obsessed with fractals a few years ago
I love this chaos game because I can't code, but these are so simple operations you could run it in MS Excel or any other spreadsheet app.
nice alternative
Desmos too?
I love your animations!!!
thank you:)
The Pentagon one looks like an Evil Version of Patrick star
I absolutely love your videos!
glad to hear that! :)
Link's been hallucinating.
doing this on 3d, in a tetrahedron, will actually result in the 3d version of the sierpinski triangle which is the sierpinski tetrahedron
The first triangle is very interesting.
If you look at the pascall triangle. And always draw a dot for an odd number than you will get the exact same triangle pattern.
Math ist beautifull
Excelente video! Good job, keep up with the good work, please!
Thanks a lot:)
The first one is called sierpínski’s triangle and it’s very cool
Incredible! thank you! please keep it up!
thanks a lot~
The math with the triangles came out perfectly matching what our universe probably would have done. The list of dots per size range from 1XXL to 3XL to 9L to 27M to 81S. To see how 9 plays a major part in our universe, check out Marko Rodin; or take the 1-9 multiplication table and reduce all the numbers through addition, for example 10=1+0=1, 12=1+2=3, 14=1+4=5 ect. This produces what I'm dubbing the Layman's Fingerprint of God because you can see how 9 exerts its dominance over it all. If you add it all together you get 459 or 4+5+9=18=1+8=9 and how it is surrounded by 9 and has 9 running though it. Making 9 very similar to our description of God being one who is equal to it all, yet outside of it all, yet running through it all.
i believe this concept sparked a notion in my mind, that... this is one of the ways the Cosmos has to create reality, like a rule* that particles and atoms arrangement , a building block of the physical realm can never be in the same spot twice... a quick example, the snow crystals, but reality's matrix is 6 triangles facing each other or hexagonally based... 🤔
Nice change of music!
cheers
OH MY GOD WHAT?!
I *never* knew that
Good Stuff Man!
Nightmare patric wants a hug
Awesome! I think I saw this on numberphile, but this video took it further. Thanks!
Btw, it's cool to see stuff done in processing. I've used it a few times, but mainly I just stick to python.
Thank you:) I haven't really tried coding in python so I don't really know the difference.
I did it in p5.js so it was easier to share with others and edit on any PC
Randomness is the name we give to the ocean of unknown.
Interesting!
Wonderful!
Love the video, hope you're feeling alright
thank you~
beautiful
It's rare to find youtube channels that produce the "style" of videos(which I enjoy) that you produce, not too long neither too short, a good explanation of what are you going to do/happening and I particularly liked the fact that you showed that the rule that we use at the triangle doesn't work with the square and showed how to correct it! And as always you choose a good song to follow the video with (:, for example I think that a piano song(Erik Satie for istance) wouldn't go as good as this song that you choosed went.
Edit: I forgot to mention that that last few videos that you made, we can actually try it by ourselfs (and not only see a different/cool proof, say that's nice and close the video/investigate about it), one example is "Cutting a Möbius strip in half" ahahha.
Sorry for my mediocre and redundant English.
Thanks:) It's good to hear from people like you that they enjoy watching my videos. I don't think I would be making a lot more videos if not for feedback like yours. Glad you like the music choice as well. Tbh I've been really struggling in finding songs that sound good for the background of my videos. Most of the royalty free songs don't sound that well, so most of my videos get copyrighted for the songs I use. If you have any songs in mind that you think would be suitable for my future videos feel free to share haha. Thanks again for watching:)
I should be the one who thanks for posting the videos! I will try to finds the songs, but how can I "share" them?
Nuno Mateus You can always email me at thinktwiceask@gmail.com , but you don't really have to look for the songs, dont want you to feel like you're being forced to do that. Thanks again:)
Inspired to program these examples in Mathematica CDF, thx
Numberphile made a video on the same topic too. All those who liked this video should check that too. Just type in "Chaos Game Numberphile" on your RUclips search and you should find it.
The first was sum good Zelda's lore shit
What's the rule for the pentagon? It's pretty suspicious that it wasn't rotationally symmetric.
this channel still doesn’t have 10billion subscribers? tf
Reality cannot be whatever you want because other people have different ideas of what is the center . The relation between your Center and other centers is what makes "reality"
it's not exactly emerging from total chaos, since there are rules guiding where dots get placed, but nonetheless this is fascinating.
Im calling you out on that triangle. If you drew a midpoint from a point to any random vertex it would pretty unlikely to be so close to the middle of an edge unless that point is in the corner
Pretty cool
Would you always get the same fractal given any point, or does the initial point matter? It seems like getting a perfect fractal would be impossible if the middle was selected.
Out of chaos...
Comes order.
Can you give us a link to simulate this ourselves with different shapes and dot placements?
Wait, but what if I choose a part that is NOT part of the fractal in the first place??
Gabriel's Epic Life of Goals you will still get the same result
The first few points will not be on the fractal, but all future points will get exponentially closer to the fractal. So after 100 iterations, the point will only be less than 1/(2^100) of a unit away from the fractal.
You hit on a really interesting point there btw: these images are technically just very very close approximations of the fractals they describe. They are what is called strange attractors. Rulesets that somehow make points chosen at random "attract" into a shape. If the shape is a fractal, the attractor is called strange.
I'll just note that the systems shown in the video are not chaotic: even though we might casually understand chaos as "random" (for example, in the context of a mob), mathematically speaking, that view is completely incorrect. Chaotic systems are fundmentally deterministic: if you know exactly how a system evolves and know the exact state of the system, you will be able to compute all past and future states of the system. Chaotic behaviour refers to how small changes in initial conditions of the system can produce vastly different behavious as time goes on. By contrast, with the systems shown in this video the next state is chosen at random: you can't predict with 100% certainty what the future states will be. That makes this video a demonstration of how stochastic processes - systems which evolve randomly - can sometimes form structures which are also interesting for other reasons.
Hey i absolutely loved this ... and how do you make a computer do so ?
2:22 fancy patrick
Order is chaos at its most chaotic point.
Great video as always! Any update on your health? I keep asking every video but I really hope you get better !
I'm still in pretty much the same condition as a month ago, I just got to wait so I don't really know how long it will take to get better. Thanks for asking though, glad you liked the video:)
Did the doctors give you any assurance of when are you going to be "fine"?
no, the doctors cant really do much at this point, they just told me to wait. It's quite frustrating but I can't really do anything about it.
would love to see a more finely-grained simulation, but this is still very cool nonetheless
I'll upload it to patreon just for you :)
Why does that happen?
Ok, but what happens if the initial point is in the empty part of the triangle fractal? Or if you get the dead center as a starting point even?
Very good
The star-like shape is supposed to be symmetrical
0:53 wow look it’s the sirpenskiri triangle
poop
amazing
See if someone told me in middleschool that math is secretly wizardry
that's cool
*THE TRIFORCEEEEEEEES*