動画数：214件
How is it possible that one of the points in the 3-cycle is outside of the black region? Wouldn’t the sequence from that point on diverge since it’s outside the set?
Idk about *your* YouTube Shorts Player, but *mine does have* a seekbar (this red/magenta line across the video's bottom edge that you can click to jump to a certain point in the video). Just hover your mouse across the bottom edge, and it should appear.
* *** Purpose of the video:* The video was created as a short explainer for an exhibit at the Computer History Museum on large language models (LLMs).
Wow, the museum piece at looks like the teletype machine I used in high school to learn BASIC.Now it's safely behind glass on display as an ancient relic. Wow.
👋
* *** Introduction to LLMs:* LLMs are sophisticated mathematical functions that predict the next word in a sequence of text by assigning probabilities to all possible words.
* *** Chatbot Functionality:* Chatbots utilize LLMs to generate responses by repeatedly predicting the next word based on the user's input and the ongoing conversation.
And that's why Mr.Chatgpt behave like that when answering.
You lost the nounce of tokens not being words (probably on purpose) and then used exactly when describing generating tokens .
- Does that mean there’s a web request back to the LLM after each word is produced? If it’s just one request why does each word seem to stream in as opposed to one block response?
Ah, yes, "Paris is a place... in Paris." Thank you, AI, very cool!
the wim Wenders Paris Texas reference ❤️
How important is the system prompt really? And is it even worth fine tuning a model when you could just alter this prompt and tada you created an expert on a specific topic? Which use cases is it worth to fine tune for?
- Извиняюсь за Каламбур и Рекурсию, интересно Можно ли понимать параметры В каком-то смысле Смыслом слов? Может всё не так-то уж и сложно и в то же время не так-то просто И у нас с машинами больше общего чем мы думаем. То есть аналогичным образом у нас в голове услышанные Или прочитанные слова Имеют какие-то параметры. А если говорить человеческим языком то какой-то смысл, Понятие. Таким образом мышление не такая уж и загадка. Мы тоже предсказываем следующее слово. У нас тоже параметры в зависимости от входных выходных данных могут менять значения. Так что не надо нас недооценивать. Кожаные мешки - это такие же алгоритмы работающие по тем же принципам только очень медленные. у нас тоже есть алгоритм Трансформера. Особенно когда смотрим или вспоминаем образы, а не слушаем. Наверное поэтому в некоторых эзотерических практиках и пытались выключить Внутренний диалог. Потому что речь - это примитивная машина тюринга ... А образы графика - это Параллельные вычисления. Поэтому лучше один раз увидеть Чем сто раз услышать. Что собственно здесь и продемонстрировано. Опять рекурсия.Но возвращаюсь к теме, думаю у нас тоже есть и рекуррентные нейронные Сети И всё остальное что присуще нейросетям.А значит это ещё один бонус в копилку того что нас можно оцифровать. Я имею в виду именно личность.А вообще этот канал отличная Находка! Спасибо алгоритмам Ютуба. Жаль нет времени, но подписался периодически когда буду есть смотреть. Тем более что в отличие от большинства видео здесь действительно видео. И есть что посмотреть. Особенно такое видео было бы полезно людям которые хотят понять как работают нейросети. Но именно тем кто хочет понять А не тем кто зациклена повторяет Что мы все не знаем как они работают.
* *** Training LLMs:* LLMs are trained on vast amounts of text data (e.g., from the internet) to learn patterns and relationships between words. This process involves adjusting billions of parameters within the model.
I love how OpenAI said they can't create tools like ChatGPT without stealing. Truly makes you wonder what the hell these companies even do
I could watch an animation like this for a while
It was the best of times, it was the blurst of times.
@ it was the best of times, it was the blurst of times??
* *** Backpropagation:* The training process uses backpropagation to refine the model's parameters, increasing the probability of predicting the correct next word in the training examples.
one billion addition and multiplication per second? easy. i've billions of neurons doing that every milisecond.
🤯 I’m somewhat comfortable with large numbers and to think we’re in the early stages! I’m excited to see how this evolves over the next decade(s).
it's incredible they've been working on large language models for over a hundred million years
* *** Reinforcement Learning with Human Feedback:* After pre-training on massive text datasets, LLMs undergo further training through reinforcement learning, where human feedback is used to improve the quality and helpfulness of their responses.
- Since this feat only took on the order of a year, the trainers must actually be doing quintillions (10^18) of operations per second🤯
million of years. mkay ()
"Workers" casually washes over the thousands of underpaid/enslaved people in exploited countries that these models depend on to perform the RLHF. If the museum doesn't address this elsewhere then it's a bad museum
afaik RLHF does not use human annotation for reinforcement learning on the base model. Instead, Human Feedback is used to align a reward model for the RL process on the base model.
"Workers flag unhelpful or *problematic* predictions ...making them more likely to give predictions that users *prefer*."A bit shocking that he says this with a straight face and seemingly takes no issue with the ethical ramifications of this practice.
this staggering amount of computation is also only made possible by an equally staggering amount of power and water consumption. AI training at this scale is exasperating climate change by rapidly increasing the amount of power big tech companies like Google are using.
at Why are the two “the” not associated to the same vector (numbers)?
* *** GPUs and Parallel Processing:* Training large language models requires immense computational power, which is made possible by GPUs that can perform many calculations in parallel.
* *** Introduction to Transformers:* Transformers are a type of LLM that process text in parallel rather than sequentially, enabling them to handle larger datasets and learn more complex relationships.
And that's why ChatGPT doesn't know how many Rs are in "strawberry"
While the beginning of this video provides a basic overview of LLMs using the "prediction metaphor", it lacks depth in explaining how these models process language and generate text. It helps a bit when you reach but the way these numbers are generated is still "black box" magic. This means the connection between word-level processing, sentence structure, and overall meaning is not adequately addressed.
I would have also given more of an explanation of the “long list of numbers” —spend a few seconds describing how words are mapped out in vector space (the classic _man:woman::king:queen_ type of thing)—so they just don’t seem like random numbers. (I’ve seen videos about deciphering the language of other animals explaining this type of thing, i.e., for, arguably, an even _less_ technically-inclined audience, and the explanation doesn’t seem overly detailed.) Plus it’s pretty interesting.
* *** Attention Mechanism:* Transformers utilize an "attention" mechanism that allows different parts of the input text to interact and influence each other, enhancing the model's understanding of context.
"talk tuah one another". My brain is rotting, but great video
Cheque
* *** Feed-Forward Neural Networks:* In addition to attention, transformers also use feed-forward neural networks to further enhance their ability to capture patterns in language.
to
Awesome stuff! One piece of feedback: At , you use the word "vector", but up until now you've only been saying "lists of numbers". If this is for a general audience, I think throwing in new terminology right at the end without explaining it could be confusing.
Is it just me or does this bit get a lot louder on the audio mix?
* *** Emergent Behavior:* The specific behavior of LLMs is an emergent phenomenon arising from the interplay of billions of parameters tuned during training, making it difficult to fully understand their decision-making process.
you casually mention emergent behavior, have you been exposed to multilevel evolutionary selection? There appears to be great excitement about emergent behaviors in complex adaptive systems.I haven’t dug too deep into literature so the question might already be answered; what is the minimum complexity required for specific types of emergent behaviors?
"The words that it generates are uncannily fluent, fascinating and even useful."
* *** Where to learn more:* The video concludes by suggesting a visit to the Computer History Museum exhibit and recommending other resources (a deep learning series and a technical talk) for those interested in learning more about transformers and attention.
Never thought in a million years that I would ever see Kendrick Lamar in a 3B1B video
at he misses a tiny little bit on each side and each corner
leaves gaps in the “corners”
The projection part seems flawed to me. If you cast a shadow from a singular point, the area of the shadow will change as you move further. In this example light from Z axis is cast from a single point in top view but parallel ray from the side view. That’s why it seems like the with and height cancel out and the area stays similar. I’m not saying that it is incorrect, maybe the visualization is misleading or confusing.
He'd doing the trick vsauce made famous in the banach tarsky paradox video
It’s at . I have a screenshot of it
At there's quite a strange visual artifact for a few frames where an inverted and transparent copy of the display is visible, just letting you know
the question is begging to be asked. Two corners have a clear relationship between sums and products. What about the third???
i’m lost
It’s not just you the short is actually long. YouTube made shorts longer than one minute!
How the heck is this a long short?
It's actually
- - Intro
time), if you just look at the frame you are not seeing the whole video even though that frame has all the spatial dimensions of the video. I remember things in 4 dimensions, not all my memories overlap the same 3 dimensional space they exist distinctly in their own 4 dimensional spacetime. So in essence all real world "cubes" are in fact hypercubes they have coordinates in the time dimension that corresponds with when they were created and when they were no longer a hypercube. @3 dimensions and continued existing after it left the video's 3 dimensions. I tried to pay attention and I don't think at any time you mentioned that your video was only about spatial dimensions or that you were excluding temporal dimensions.
"At the end, you and I will CIRCLE them all together..."* moving circles wizardry showing *video just started and bro dropping the craziest bars
- - Twirling tiles
hexagons are the bestagons
What I'm told: Three rhombus shapes rotating,what i see: Stacked cubes, being added/removed, with a rotating fade effect
I'm just seeing cubes
the left one i can only see it as the interior sides of the cube the the one on the right looks more normal like it was a regular orthographical view of a cubeedit:interesting how i wrote this before watching the rest of the video and yet the exact thing i was seeing was exactly what 3b1b goes through in the video
I'm guessing it's to do with the decades of videogames, but when I look at that first example around I inherently see depth. I see a level a character could jump about on, like Q*Bert.
This tiling also corresponds to valid stack, it just so happens that the stack has infinite size (it is half space) and it is not possible to remove or add a cube and still have valid stack, which makes sence, because we cannot make a move on that tiling.
to me it looks like you are rotating cubes in and out of excistance
This would be such a great fidget toy. 😊
but the entire circle isnt covered D: u missed a spot on the lower right
"real analysis buffs will note that this circle is not completely covered by chords" :)Another excellent video, Grant !!
. if we try to mix some lines, not pararell, it will be just a waste of area cuz of crossing area will be just waste of this all. so the only way to make those uncross... so, the solution is thta what you said in , two. this is 2-dimensional solution.
Arguments from overlaps wasting area have a flaw. Wasting the area is a disadvantage, but you can gain some advantage by needing a narrower strip to cover a given area. At for example there is a triangle-like shape in the upper-right part of the circle, and you could cover that with a narrower strip if you placed it at a 45 degree angle compared to placing it horizontally. Obviously in this example that is very inefficient, but the point still stands - we cannot be certain whether the disadvantage always outweighs the advantage. Proving that the answer is 2 is to show that this claim is true.
I thought this was trivial but then he explained it at and I thought how someone couldn't immediately notice that? Aren't human brains designed to infer depth onto 2d images automatically?
I feel smart because I thought of the trick myself before it was mentioned.
Is there a study that claimed that a portion of humans can see the 3D cubes prior to needing to squint and then subsequently adding a 3 tone coloring in that example?Or was it simply to just simply avoid any possible confusion?
I don't even have to squint my eyes here: it's harder for me *not* to see it as a cube stack.
I was already thinking of that move as equivalent to remove a cube from a finite stack (It reminded me of the game Q*bert),and I was wondering if it could be done until you have no virtual cubes left
Us Minecraft buffs already see it as a stack of cubes!
Thank you for circling back: This is just "Q-bert" but now you get to build your own map and OF COURSE the original answer was "yes"
THATS EXACTLY WHAT I THOUGHT! I play minecraft and I tried remaking it once used these hexagons.
did u get a new mic in the middle in it why is the audio different compared to (2)
From the very beginning I saw this as a collection of cubes and at this timestamp this is made out to be THE trick to think about the puzzle?I'm really not sure if I just think naturally 3D, just thought ahead too much, or the video just assumes we're unable to make this a 3D problem from the get-go, but this realization alone stumped me more than the actual puzzle.
at this point you're catching up to me, that's how I automatically see those hexes filled with rombuses, my mind makes sees the cubes wether I want to or not. Lol
Edit: Ah, ok.
This is ONLY true if the stack of cubes fills up from the back. As soon as a cube obscures any kind of empty space the rhombus tiling fails.
did u get a new mic in the middle in it why is the audio different compared to (1)
"if you squint?" You mean it's possible to *not* see it as a 3D representation?
B at* : the curious viewers might enjoy taking a moment to pause and ponder and convince themselves it goes the other way around.*Me:* aww he called me curious <3 anyone else seeing me opening paint to draw rhombuses would call me weird
if we place only one cube the "closest" to our eye on the empty grid, then we do not get a proper tiling (there will be triangles appearing). I think we need to consider stacks such that each cube of the stack has cubes 'behind' it
Wait, what's the proof for the reverse direction? I.e. what's the proof that any tiling pattern has a corresponding stack-representation? What follows is not a proof - it already uses a stack-representation (to show that rotating a hexagon is equivalent to adding or removing a cube in stack-representation)! So couldn't there still be tiling patterns that do NOT have a corresponding stack-representation?
Can someone fill in the details of the question at ? I've spent a decent amount of time thinking about it but I do not have a satisfying answer. When I draw out a tiling I can see forcing consequences of choices, but nothing that I could write down nicely. Another route I went down was thinking about counting the number of tilings and stacks. As we have an injection from stacks to tilings, showing that the number of tilings and stacks are equal demonstrates the bijection we are looking for. For an a by b grid of squares the number of stacks is (a + b)! / (a! b!), but the problem seems harder in 3D, and I don't know how you would go about counting or generating the tilings. For anyone who is curious, some terrible python code gives the number of 4x4x4 stacks as 232848.
now, here's a fun part about representing 3D in 2D space like this! If you take the video and flip it 180° (that is turn your phone/computer screen upsidedown) then when he says he is adding a cube it appears he's removing one, and when he says he's removing a cube it appears he's adding one.
you could also think of the 'rotation' as pushing in the center corner of the cube to make an 'outie' an 'innie'.. lol
does any one else get a weird optical illusion when he removes the cube where the perspective shifts to being lit on the bottom and all cubes invert? that tricks my brain everytime I don"t know why.
😮 This realization blew my mind. I love this channel very very much.
What's nice about this is your brain can't NOT see the tile pattern as cubes. Like, the puzzle is solved intuitively without being actually proven solved, bc you just SEE the solution. Empty or full
I saw this in the first few seconds because I grew up playing marble madness and Q*bert
your trying to tell me you didnt think we would do that??? bruh all i dee are cubes and not hexagons
At I found myself locked in an Escherian optical illusion.
i keep optically illusioning myself and seeing the cube stack upside down
If someone thinks that algorith from proves that, then that is wrong, because that algoritm is not optimal. For example FULL -> FULL would require 2*N^3 moves with that algoritm, but the optimal amount is obviously 0.
I'm tripping or it's just being filled instead. I think my problem is the coffee I drink...
bro this is so trippy, as soon as you removed the last cube it turned from an empty room into a cube, that you can also kinda convince your mind to turn into an empty room, awesome
if you imagine that is full, it is so confusing
my brain somehow immediately started seeing this as an upside down cube instead of a hollow shape, which made really weird when more cubes were added.
this broke my brain. All of a sudden the cube is upside down.
When all the cubes get removed and we are left with on big hexagon, I start feeling nauseated. I can’t tell if I’m looking down on an empty shell or up onto the bottom of a cube. It’s keeps flipping and it almost feels like I’m moving from viewpoint to viewpoint very fast. It’s like inverse of being sick because your body (specifically head) moved too fast for your eyes to keep up.
the moment you “remove” the last cube, my mental representation of the space into which the little cubes are “placed” switches from looking like a depression into the screen and instead looks like a projection out of the screen. Then, once cubes are “added” back in, they look like they are sitting above the projection until a critical number are added (about 12) at which point the whole thing snaps back to looking like a depression into the screen with little cubes inside.
You have shown that EMPTY -> FULL stack can be done in N^3 moves and that it cannot be done in less. But you have never proven that every pair of positions can be done in N^3 moves or less.
what about 37 steps? we don't actually need to put blocks that at the end will be invisible. This is 3d projection, not 3d world. we can add some floating stuff ;)
— but you can’t place a cube only at the very top in the closest “corner” to us.
why is the adding and removing of the cubes or the rotation of the rhombuses making me feel strange… this is freaky
At can the puzzle not be optimized when adding cubes to only care about the closest cubes to the perspective, making a hollow object but greatly reducing the number of rotations to get to the desired end state/perspective?
re can you not get from the empty configuration to the full configuration (insofar as the 2D scenario goes) in just n^2 + (n)(n-1) + (n-1)^2 moves, which is strictly less than n^3 for all n > 1? What you're doing is building a hollow shell around the empty n^3 "alcove": n^2 moves builds one of the three walls you'll need, (n)(n-1) more moves builds out the rest of a second wall, and (n-1)^2 moves builds out the rest of the third wall. In the 3D scenario you'll then still have (n-1)^3 missing cubes behind those walls, but in the 2D scenario you won't be able to tell the difference.
wouldn't going from the empty configuration to an "apparently full" configuration take only something like N*(N-1)+1 "cubes"?When I say "apparently full" configuration, I'm talking about a configuration which only needs to have cubes on all of the "slots" of the 3 visible faces from our POV.
just saying, you can't see the innermost cubes nor the cubes that are not on the surface, so shouldn't the solution be 3*N^2 (representing the three sides visible) - 3N (representing the are covered twice) or smth?
- For the first problem, I was kind of surprised that N^3 (64) was the answer rather than something which was less than that, just because I was imagining a configuration with a full stack of blocks would just require the 3 sides to be full, and could actually be "hollow" with the inner/unseen 3x3 cube being absent. I see that it really is 64 because when you start empty, there's only one place (the origin) where the edges meet as hexagon, and you need to build from inside-out, rather than just skipping to the outside.
Why do you have to fill the full volume with n^3 steps? Why not leave it hollow, which only needs n^2 + n×(n-1) + (n-1)(n-1).
because we're talking about 2d projection the amount of "cubes" needed to make it full is less than n**3.
Actually, it does come from a bias because our species created this puzzle, and our species is three dimensional, therefore it was created by a three dimensional creature and the bias is baked into the puzzle. Thank you.
But isn't the empty configuration equivalent to the full configuration up to rotation? Are those really two "different" states?
I swear I saw all of this, until when you just counted the number of cubes it takes to fill the whole thing, it feels wrong because there are hidden cubes aren't there?!
what if the cube isn’t filled and just a shell of smaller cubes? How would you know if there was an internal void or a break in the pattern on the other three surfaces?
- - Tarski Plank Problem
My god, into the video, and this is gourmet for my mind.
Tarski plank problem : Answer: min sum is 2* sqrt(2) = 2,828... It is 2 strips perpendicular in the middle or the circle, each having the width of min sqrt(2) and max 2. (if they less then sqrt(2) they do not cover the whole circle, if they have over 2 then they are not strips inside the circle, when is is exactly 2, it does not makes sense to me that 2 points are parallel, i only count strips inside the circle.. no tangency's).
Couldn't the second puzzle ( ) be solved by saying that the minimum sum of the widths of the strips is 2 because if you want to cover the circle by strips whose sum of the areas is equal to the area of the circle (minimal possible area if you want to cover the whole circle), therefore the strips must be parallel. If you would cover the circle elsewise, there are going to be overlaps. This will result in sum of the areas of the strips being greater than the area of the circle, therefore the sum of the widths of the strips being greater than 2.
I'm really thinking "this is a tensor algebra problem" (each strip as a vector, a collection of strips makes a tensor)not sure how that helps there though…I mean, it would be interesting to look at what the right quotient of the tensor algebra would be…
This is actually quite intuitive. Parallel strips have the greatest efficiency of coverage because none of them overlap with each other, they only fill UNFILLED space in the circle, but any other arrangement of strips REQUIRES that two or more strips intersect in the circle, meaning some of their total area is being wasted covering parts of the circle that are already being covered, and that wasted area is area that COULD have been used to cover NEW area instead, which now needs to be substituted by adding yet more strips, which bumps the total area of strips up, so 2 must be the minimum.
- мне кажется, что тут не пример "сложной" задачи, а пример манипулятивного формулирования задачи.
when i heard this I was like "oh, it's all coming together"
I managed to solve that first puzzle (but i didnt really make a solution) just before you hinted at the 3D because thats something i noticed but never bothered to usenow i have no idea how to do this second one but then i saw the napkin ring problem and im like aaaaaaaaaaaaai honestly expected 3D rhombic dodecahedrons to pop out at some point
isnt dπ the area of a strip on a CYLINDER?
out of the brown too, but only drawn ¼ out of the brown
nd problem at . Also if 2 strips overlap each other, then aren’t we double counting the area so the area shouldn’t necessarily be sum of pi into d.
into pi into radius into width come from. What area is that exactly and how are we considering the circumference of the circle for that strip part because that strips length isn’t the circumference of that whole circle? I’m confused can somebody pls help me understand this. It’s around @ minutes
But if you can project a sphere onto a silinder why can't we make a flat rectangle map of earth thats completly accurate?
I had this exact intuition, but without thinking 3-dimensionally.I began with the area covered with non-overlapping strips, and reasoned that if any of the strips overlaps any of the others, that means one of two things: that we either increased the length of a strip, which is not what we want, or that we rotated it, but rotating it would leave gaps for which the strip's length would have to increase, therefore the non-overlapping model is the one with the smallest area.
area of hemisphere is 2πr^2 not 2π right.and i didn't get how πd for area of strip
- - Monge’s Theorem
why cant you just use a whole sphere and wrap a single strip around it? There was a guy that made a soccer ball like this froma double J shape. So the strip as thin as you want because its a sphericon.
alg 1 has got the most goated snack setup of all time
I am writing this midway, while viewing the video. Might edit it later, might add on in the replies.
does Grant climb?
Not sure if you remember me, but I was the was the contestant who mentioned Monge's theorem at ! (although I had no idea how to salvage the proof lol)
that's a bit harsh, i think he looks great!
the small green dot in the corner at looks exactly like the signal a samsung phone gives you to tell you either the mic or the camera is on. you spooked me lol
- the three circle and tangent crossings subjct forming a line...if you think about for example the marked green circle to move along the tangents towards the red circle and the center created from themPLUS making the circle size to adapt to the tangents so that the common tangent shape for them stays unalteredTHEN you will have to watch where the green-to-cyan center dot moves along at the same time.it will move...first towards the cyan-red center point (where the green circle matches the red circle by position and diameter)and second towards the "fixed" green-red center point (where the green circle matches the green-red center point - again by position and diameter, which is zero here).during that green circle movement the center and the contact points are just moving linearily and at totally equal factors.and this is not only true for the green-red but also for the green-blue circle.so as anything is just going linearly, the result movement for any tangent meeting center point (at least for the two moving ones) goes linearly as well.the result is a linear trace for both of those moving center points.we do have three center points and at the green-to-red circle identity case we suddenly only have two such center points.also on the green-to-center identity we only have two such center points.for each of those case its valid: two points are able to define a straight line.and as the moving center points (in that ruleset of the moveable idea from just the beginning)are shown to be at some time on either of these other two (and of course all three) initial center locationsPLUS them only moving along on a straight linethey have the two common points and the straightness as their attributeswhich in the end just defines them as an identity "curve" to the straight line that we have just around by the other two center points (and by any center point anywehre).the same effect will happen if you have any other combination from two fixed and one moveable circle.thus as with degree of freedom is in place, all modification in positions and diameters in a coulpled way will be reachable by transforming the base deisgnwith the proof of all center points living on the same line still intact.and as its true for any such combination, an equivalent proof is possible for any (ignore very special cases) such 3-circle setup,even when diameters are set different and even if their (relative) positions do change.in effect we will probably alter the slope and position of the common line between the centers when either moving a circle or scaling a circle,except for at least the case where movement and scaling take place in a tangent and diameter coupled way as described before.(doing multiple such moves is subject on the circles one after the other is expected to fully preserve that center connecting line.)PS:on linearity:triangle congruency gives you a first hint why all is just going at linear scale.(think about it a classical copy machine for e.g. engraving numbers in a mill or some children's toy...it just scales up or down something - whith orientation, directions and angles typically all preserved.)
Don't know if it's relevant, but I noticed that for any 3 circles, they can be arranged such that all 3 external tangent pairs intercept at the same point
I looked at this and imidiatelly thought that the 3d-way of thinking about it would be to think of this image as a picture of 3 balls lying on a plane, and the "horizon" line of that plane behind them (the line where the 3 points lie)And I think I actually found a way to prove a theorem with this - all 3 balls have a center point, and these points must form a plane, and this plane must have the same "horizon" line as the one they lie on - because if you assume that all 3 balls are the same size, then the tangent lines between each two of them represent the line of perspective - sort of a tube between two balls, the axis of which is a line between two center points, hence it must be on the plane, and as this line goes to infinity it should fall on the horizon line, and the two other tubes must fall on the same horizon line because they are all on the same planeI thought this was gonna be the solution, but when the solution went in a completelly different direction I felt so smart by discovering this, and it still feels like the most beautiful thing I've ever discovered
You can also imagine the circles as spheres of the same size, which we only see as bigger or smaller due to perspective. The tangent intersection points would then be vanishing points, which all lie on a horizon line, belonging to the plane the three spheres lie on.
was he referencing möbius transformations here?
for the prev problem i did indeed watch that video but i entirely forgot about it LOLi think i can do this third one with a bit of algebra6 relevant values (not bothered to use the fact that all 3 circles are in a triangle to reduce that to 5 values)r1 r2 r3 d12 d23 d13nevermind this is impossible without knowledge of the coordinate values!!r1 r2 r3 p1 p2 p3ok im so done im not bothered
a three legged table can rest flat on any surface. So if you imagine the top of the table as the plane and extend it....?
- no waaay!What's amazing is the video are this point showed me the reason why they are all on the same line in 2D space, moments before you explaining it in words. But I had already just conceptualised the scenario before you got a chance to say the words, because of your brilliant video content. And then, you said the words! Which put into English the confirmation of my anticipatory understanding. You "collapsed the wave function" of my understanding, why going to a 3D projection is the mechanism to prove a 2D conjecture, in my head in a moment 🙂This approach is revelatory!! Amazing...Thank you!I had already solved this problem in my head tho, of minimising the widths of pairs of parallel chord strips, by a reasoning in 2D space. For me, the solution had to be the base case of a single pair of chords that are tangents to the circle, having a width of two. Any other 'random' combination of parallel chord-pairs involves duplication of the area inside the circle, due to the necessary overlaying the areas in any non-parallel chord arrangements.In 2D, if you want to minimise a quantity like area, you have to minimise the "amount of area" covered by the strips. You can't afford to have overlapping strips if you want to minimise the strip widths - it wouldn't make sense. So, to minimise the total area of the circle covered by the strips can only hold when all strips are parallel - which in the simplest case is the two tangents to the circle, meaning the answer to the total stip width that covers the circle has to be 2, for the unit circle.But - that feels nearly like a qualitative answer, a logical reasoning over the 2D projection of the broader problem. "Going 3D" gives the Mathematical proof.... Wild!
In Monge's theorem, my inner "intuition" (AKA: "Specialty on not paying attention and going off the subject by noticing something extremely dumb") made me see a triangle between the vectors put.Has that something to do, or was that a coincidence?
- это просто ерунда. Разумеется, что в 3Д пространстве, существует прямая для всех положений разно-размерных кругов расположенных в любом положении относительно друг друга. А в 2Д - есть ограничения. Как в 3Д пространстве появляется ограничение для касательных (а не вершин), которое перестаёт быть ограничением для 4Д пространства.
THE CONES of course! And here I forgot about the essence of the game. Ben would have been ashamed.. lol
For the cones, you made sure they have the same angle at the point for a center of similarity among them all, but wouldn't that still work with spheres since they are also similar shapes?
how are u using cones? what is the height for each of those cones?
this problem immediately made me thought of how people draw 3d things in perspective on 2d paper, as projection using a (or multiple) horizon line. When the circles were presented I immediately thought oh the projections gonna make a line with those 3 points since they are bound by each others' sizes. It's great to see the mathematical proof of it right after
why cant you take the 3 poles of the spheres as the reference point?
About the origin proof from , a similar idea is used in the proof of Casey's theorem in Math Olympiad Dark Arts - Goucher (2012), so it might be worth looking into Casey's theorem.
I find it easier to follow if yo imagine connecting the tips of the cone to a triangle, because that's the crucial part of the plane. Pretty much anything aside fromt the triangle is unimportant to the viewer. This would also make it easier to understand that the dots must form a line when intersecting with the floor. Triangles in 3d space are always planar so if you shoot rays alon their edges until they hit the floor, it's obvious why they must form a line, no matter the orientation or size of the cones.
when I first encountered this theorem, I was told that it is called the "three cone-hats theorem" (теорема о трех колпаках), and the fact that it is called "Monge's theorem" was unknown to me until now
@ I just had a thought and wondered , could you some then think of these as the pyramids in Egypt , we in this video are one of the stars reference points , by changing our view reference point in the video , we are changing which star we are looking from , onto a this 3 body object , I wonder if it means this what we see is a video explaining a reference point of something that will project a hologram. Enjoying the video so far !!! nice🥳🥳🥳🥳
As soon as you used the centers of similarity and moving the circles around @ I started seeing tunnels and due to perspective, all should meet on the horizon, which is a line.
These immediatly look like perspective lines! As you move the circles around my brain interpret them as such.
- huh, that also looks like a bunch if identical objects in a perspective drawing, all with vanishing points on the horizon.
If three circles lay on the same centre of similarity it all breaks. We get only 1 point centre of similarity and in 3D projection 3 points that are tips of cons form a line and therefore do not uniquily determine a plane. Maybe I am missing something and there is a condition that disallows it. But just wanted to point it out.
a cone with a pi based shaped is called a piramid, apparently
I know this is a month late but couldn’t you use this Monge’s theorem as a way to prove shape similarity? Because in u say “how any shape is possible as long as they’re similar in shape” the Theorem is true.
At , when you say that these 3 centers of similarity always must fall on a line, I suddenly remembered my drawing classes from high school : This line is actually the horizon line, when drawing 3D objects in perspective.🙃
This looks like an architectural perspective with a horizon point.
did you say- A PIramid shape?
missed opportunity for a pi-ramid
Though it’s still fair to call these generalized cones. We’ll need to rewrite the condition of them having the same vertex angle but in the end it would be what we actually use in the proof: that the heights are similar the same way as the bases are.
This is still unnecessarily obfuscatory and misses important insights.The only relevant fact about the shapes is their size. We can this factor out the shape and be left with only three positions in the plane, each with a value. This is by definition 3D in each point. We can visualise the solution by imagining the three shapes as three heights over the plane, on which we place the plane. This top plane obviously intersects the ground plane in a line.
- - 3D Volume, 4D answer
I happen to be using the theorem relating to the mystery question for a math essay. I know it’s in relation to the shoelace formula, but I don’t exactly know how to explain or “prove” why parts of the formula just works. Some explain things in a complicated way, or explain how but not why it works. I’m not the biggest fan and probably because I’m not good at math, but I like to watch these random math videos recommended to me, I think it’s interesting to learn about random math stuff that is connected to our world even if I don’t fully understand it
I was pondering on this problem and came up with my own polyhedron with coordinates a(0,0,0), b(0,4,4), c(3,3,0), d(1,2,0) and e(3,0,1). Can someone double check if I got the volume right V=11+1/12?
please could you explain more about from to
now this puzzle becomes really easy for us 12th grade students in India as we are taught to find vlomue of parallelopiped and tetrahedron in vectors so basically if you have all four points of a Tetrahedron take one of the four points as A and describe 3 vectors along AB , AC AD where B C and D are the remaining three points now just take cross product of any two of the three vectors and take the dot product to the cross product and find the magnitude you get is 6 times the volume of a tetrahedron which is the volume of a parallelopiped we also described this as [ AB AC AD] which is (AB×AC)•AD !! Which gives us the volume required now The problem is our professors were not really able to visualize us the proof so I will be waiting for you to help us out thank you!!!
"i'm not going to show you the full answer to this puzzle" i am the pi guy on the left
..... where can i find the answer.... thanks a lot!
I know this one because my college thesis was about decomposing a k-cube into k! equal volume simplices (triangles, tetrahedra, etc.). So I needed to take all sets of k+1 vertices of a k-cube and find their volume and eliminate the ones that are coplanar or are too big.
I am not being factitious this is just the only way I can think of to do this.Step 1: Translate the triangle so one of the 3 points lies on the origin, let this point be (x1,y1) reflect the triangle as necessary so all 3 points are in the positive quarter of the X,Y plane.Step 2: let sides A and B connect to the origin, let B be defined as the line segment with the greater slope. Let (x2,y2) be the second point of B. let line segment C connect (x2,y2) to (x3,y3)Step 3: calculate the slope of A let it be m, B let it be n, and C, let it be oStep 4: f(A) = mx, f(B) = nx, f(C) = oxStep 5: Use a definite integral from the origin to coordinate (x2,y2) calculate the areas under B as b and A as a, subtract a from b, let this value be T1Step 6: Use a definite integral from (x2,y2) to (x3,y3) to calculate areas under A as a and C as c, subtract a from c let this value be T2Step 7: add T1 to T2.
hmmmmmm, I'm foreseeing a new addition to the Essence of linear algebra series. I hope it comes true, linear algebra is really beautiful
can't wait for it
- - The hypercube stack
Seriously, is no one gonna mention the fact that Grant is finally making a video about all n dimensional determinants for the linear algebra playlist.I've been waiting for such a video for years now (4-5 years now).Go on Grant, I'll be waiting patiently and excitedly for that video
ive forgotten totally what the video is about and finally am brought back
"a little mental gymnastics keeps the mind flexible" be careful you're going to give ammo to facists and and science deniers 😂👌👌
a cube exists and continues to exist until around @
. Presumably it actually existed before entering the video's
That projection resembles the diagrams you get when showing all of the underlying morphisms of the compositions of natural transformations. I think that would be an interesting connection to explore since anyone programming for a living today makes use of that category theoretic concept whenever they use libraries like LINQ, languages like SQL, or really anything involving functors. At least that’s my understanding.
Swoosh... That went way over my head
YES IT'S THE RHOMBIC DODECAHEDRON LET'S GOOOOO
Have you ever played with a shashibo toy? The name is just the first letters of shape shifting box. Each toy is a cube that you can unfold and invert to a rhombic dodecahedron. You can use these, and their intermediate shapes, to make many other 3d tilings, which can make some of these puzzles easier to visualize! I recommend getting some power of two of them so you can make more combinations. I've made larger cubes and dodecahedrons without using a cube or dodecahedron, as well as a shape that only followed the vertices and edges of a dodecahedron.
I was so excited to see the rhombic dodecahedron. It's my favorite non-platonic 3D shape! A really nice aspect of it is that if you use it to make dice, they are just as fair as platonic solids. I never liked that a d4 is hard to roll, and alternatives to a tetrahedron typically are either also hard to roll, or are just another existing die shape relabeled making them hard to pick out from a bunch. But the rhombic dodecahedron is perfection, you can label is 1-4 three times producing a fair d4 that rolls very well and doesn't look like any other die. I actually 3D printed some of these, and they're super nice.
I'm so happy you mentioned this, I watched it like 3 times
one of my oral exam questions for SNS admission was to prove that rhombic dodecahedra tile space. The professors first asked me to consider the shape obtained by a cube by gluing regular pyramids of square base to each face such that their heights are half the cube side length. They first asked me to count the number or faces. Then they asked me to prove that these shapes tile space, which was quite simple at this point.
This might not be true! In fact quasicrystals come from projecting/cutting higher dimensional tilings to lower dimensions. For example a specific angle of a 5D cube to 2D will produce regular pentagonal "quasi-symmetry", which is isomorphic to Penrose tilings.
so 6 square sides = 6 sides in 2d but the rhombic dodecahedron is a dodecahedron with 12 sides, where does that number come from? 24 faces?
What's the name of the talk shown at ? The link in the description doesn't work.
d ( ), out popped the bonding angles for carbon, which was a fun surprise (although maybe for symmetry reasons it shouldn't have been?).
hey it's a tetrahedron
a rhombus is a squished square
so this rhombic dodecahedron can be formed by taking the three vectors of x,y and z from the project of the cube into a rhombus and then these projection vectors of the projection of x,y and z vectors forms this rhombus hexagon projection.Now this rhombus projection can also be split into four vectors that lie on the same plane and if we create a new projection for these vectors all the new projection go to another plane of existence and create this new rhombic dodecahedron that if projected as a 3d shape will form a 4d cube.So summarising what you have said here:If we take a cube and project it's as a projection vector formed by 3 projection vectors formed by xî,yj and zk.Now we can take one out of the projection vector formed by x,y and z vectors a project this cube as a 2d rhombic hexagon.Now this rhombic hexagon can be projected as a result of 3 projection vectors from the remaining 2 projection vectors and now if we take a projection of those vectors we get a rhombic deco tetrahedron which is projected with respect to x,y and z (initial vectors) will be projected as a 4d cube that has infinite reflections on all directions of this normal cube.So this 4d cube forms a tesseract, right?
D cube projections ( ish) have the same effect as adding extra vanishing points!
- I think another way to describe this is to rotate it by 180° around the axis that runs through the blue piece, and then invert it by that axis.
- I like the visualization of "inverting" a hypercube projection. I always like seeing animations of rotating a hypercube, particularly ones where it cycles through the projection of having one cube inside another. Similar to the 2D version's 60 degree rotation version, are you able to show what the hypercube rotation version looks like of getting from the original to the inverted form?
Bold of you to assume such a nerdy cocktail party would have people that don't watch 3b1b
the faces of the rhombic dodecahedron are not 60°-120° rhombi. they have different angles
could you make some of these cubes transparent?
4D creature: "Just squint your eyes and you'll see it's basically a stack of hypercubes"
- - The sadness of higher dimensions
"not a lot of direct utility". Your statement is incomplete. We just haven't found the utility yet.
This quote by Hamish Todd is kinda bizzare. Like, 3x3 matrices are 9 dimensional and that is very valuable for thinking about them.
Why that error correction code works and is unique, is nicely explained in Another Roof's video "Why Do Sporadic Groups Exist?" I'm somewhat surprised it's related to sphere packing as well, although perfect correction codes are somewhat related to spheres, so maybe it shouldn't be so surprising.
Aaah, I actually had a revelation about that pair of random vectors last week! Though, when I imagined it, it was actually the most likely position one point would appear on a sphere relative to another point if they were placed randomly.. even though they have an equal chance of appearing anywhere on the sphere, they are most likely to be 90° perpendicular to one another.. seeing that brought up at the end of this video made me so happy, because that exact thing is much easier to prove in 3 dimensions!
what do we mean by random vector?
- очень странное открытие. Увеличение размерности это буквально увеличение перпендикулярных состояний случайной пары векторов.
In which episode on neural networks is it mentioned?
There's a crossover with neuroscience and neurodivergent brains here (I write as someone with ADHD, so please no one take offense if anything seems to mis-characterize you or be pejorative - it definitely is not the latter): stereotypical "autistic" thinking is often characterized as "robotic" or "rigid", or "computer-like", in there sense of experiencing very strongly defined categories/ontologies. Stuff very manifestly *is* or *is not* something, in this mode of thinking. Rules are very rigorously adhered to. It's also been documented that autistic brains have higher density of synapses than non-autistic brains. In this sense, autistic brains are configured with a "higher number of synaptic dimensions". So given any particular pair of stimuli, there is higher probability that they will output approximately orthogonal vectors. Or put differently, the likelihood of a false positive for the test "are these two things the same" is much lower, but correspondingly that can result in more frequent false negatives, i.e. difficulty with the notion "these two things are fungible because they represent the same category".
"But what if" is what Mathematics is all about.
At , it is not necessarily requiring minds eye to handle, play around, or do something with this concept. As I have aphantasia (a pretty high degree of, at most only a vague shadow of things can be seen), I do not have trouble making sense of it or even using it though I do not see it in my mind.
Wow, I wonder what the manim code looks like in !
Who drew the painting at , pi creature in the woods?
- Part of me wonders... When you play games that involve 4D mechanics, you begin to get a "feel" or intuition for 4D. As we grow as a species, I wonder if spending more time in "faux" higher-dimentional space will allow us to access some level of intuition that might shed light on these sorts of problems. Can a human, being raised playing in 4D, 5D, or even higher projection environments, come to "see" these things in their mind?
to
I disagree. The brain is a very capable machine such as neural networks, the only thing is missing from us is to be able to input another dimension to our brain which we technically can, we could create simulations where we encode a third dimension as audio for example and the brain would be able to "see it". Or maybe in a future we can find a more direct way of sending those extra dimensionalities directly for the brain to process. I mean at the end of the day the eye just encodes 2D, light intensity and a very tiny fraction (fovea) of RGB (cones), and with that plus another eye our brain is able to construct a 3D scene (note that we dont get 3D input, the brain is the one that makes sense out of our 2D inputs)
Speak for yourself! :D
Lambda GPU Workstation?