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Mathematical Coincidence
เข้าร่วมเมื่อ 9 ส.ค. 2021
Mathematical Coincidence is a channel which aims to explain mathematical concept with visuals and intuition.
In every video I try to show that one idea or equation or object, can appear naturally in many different fields. This is the idea that mathematics are a language of regularities and we only find links that exist anyway. But one may ask : Are those links just coincidence ? It depends, is it a coincidence that the difference between 2 consecutive square is always an odd number ? No because we know where this come from, it can be proven. But is it a coincidence that sin(666)=cos(6*6*6)=-(golden ratio)/2 ? It can still be proven but who would have expect that. So maybe a link become a coincidence when nobody was expecting it.
Twitter : MathCoincidence
In every video I try to show that one idea or equation or object, can appear naturally in many different fields. This is the idea that mathematics are a language of regularities and we only find links that exist anyway. But one may ask : Are those links just coincidence ? It depends, is it a coincidence that the difference between 2 consecutive square is always an odd number ? No because we know where this come from, it can be proven. But is it a coincidence that sin(666)=cos(6*6*6)=-(golden ratio)/2 ? It can still be proven but who would have expect that. So maybe a link become a coincidence when nobody was expecting it.
Twitter : MathCoincidence
Is there life on Cellular Automata ?
Cellular automata can reach complex structures, in this video we try to understand if a phenomenon such as life can occur in it and under which condition.
If you want to play with some of the cellular automata you have seen here are some resources :
Life Like : www.shadertoy.com/view/MfXyD8
PDE + LENIA + MNCA + NCA : www.shadertoy.com/view/Dl3yRH
Wolfram blog about CA and Second law of thermodynamics : writings.stephenwolfram.com/2023/02/computational-foundations-for-the-second-law-of-thermodynamics/
More detail about MNCA : slackermanz.com/
#SoMEpi #mathematics #maths #math #cellularautomata #entropy #emergence #life #chaos #selfreplication #gameoflife #originoflife
Music : soundcloud.com/user-938398110/ambiance-perceptron?
If you want to play with some of the cellular automata you have seen here are some resources :
Life Like : www.shadertoy.com/view/MfXyD8
PDE + LENIA + MNCA + NCA : www.shadertoy.com/view/Dl3yRH
Wolfram blog about CA and Second law of thermodynamics : writings.stephenwolfram.com/2023/02/computational-foundations-for-the-second-law-of-thermodynamics/
More detail about MNCA : slackermanz.com/
#SoMEpi #mathematics #maths #math #cellularautomata #entropy #emergence #life #chaos #selfreplication #gameoflife #originoflife
Music : soundcloud.com/user-938398110/ambiance-perceptron?
มุมมอง: 252
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Logical Perceptron
มุมมอง 6K2 ปีที่แล้ว
Exploration of the Logic of a Perceptron, the impact of its parameters, and how to interpret Neural Network in term of logic as well. #SoME2 #mathematics #maths #math #neuralnetworks #perceptron #neuron #deeplearning #machinelearning #logic #truthtable #digitalcircuits Music : soundcloud.com/user-938398110/ambiance-perceptron?
Diagonal Argument : Cantor, Turing, Tarski and Lawvere
มุมมอง 8K2 ปีที่แล้ว
Diagonal Argument with 3 theorems from Cantor, Turing and Tarski. I show how these theorems use the diagonal arguments to prove them, then i show how they are related to Lawvere theorem and how it can change your point of view about them. The paper on Lawvere theorem and self referential paradoxes from Noson S. Yanofsky : arxiv.org/pdf/math/0305282.pdf Update : There is a mistake at 5:50 on the...
Dynamical Billiard Series : 1 The Square
มุมมอง 1.1K2 ปีที่แล้ว
In this first video of the Dynamical Billard Series, I show you the trajectory resolution of a square billiard table. #maths #mathematics #math #billiards #trajectory #geometry #square #pooltable #systemdynamics #trigonometry #vector Music : soundcloud.com/user-938398110/ambiance-square-trajectory? TH-cam : th-cam.com/channels/H7SiBBxCSg03TwGeo5BTjw.html Twitter : MathCoincidence
Exploration of Lissajous Curves
มุมมอง 4.7K3 ปีที่แล้ว
Lissajous Curves through intuitive examples and some unexpected link. #SoME #Lissajous #mathematics #maths #math #firstvideo #curves #visuals #animation #numbertheory #fraction #rationalnumbers Video/Animation/Music : Benoit Arliaud Music : soundcloud.com/user-938398110/ambiance-lissajous? Channel : th-cam.com/channels/H7SiBBxCSg03TwGeo5BTjw.html
Information theory explains a lot of things.
You should watch Wolframs New Kind Of Science series he has up on YT. He makes the connection between Turing universality and the behavior of CA’s under this notion of computational equivalence: that all rules are equivalent (to each other) and to a Turing machine.
I've read Wolfram readings for the thermodynamic part, i didn't know there was YT series, i will have a look thank you !
I don't really think that Cellular Automata are an alternative to computer programing. Automata theory which CA is a part of is the basis of comp scince.
Of course, i agree with you CA is part of Automata theory. But I didn't say computer programming i said programming language. It's easy to think that computer science is equal to programming language - since it is a big part of it - but it is not. There are other ways of doing computation without no language like CA.
There is a lateral jump-rope movement pattern called the "dragon roll" where the rope's midline follows the first Lissajous curve described @1:55 in this video: x(t)=sin(t) and y(t)=sin(2t). The rope travels laterally from left to right both in front of and behind the body. It's actually a 3D pattern, because the rope also moves up and down. The rope criss-crosses over the head. The entire 3D curve looks like Viviani's Curve (best shown on the Wolfram MathWorld website). That description of Viv's curve shows the parametric equations for x, y, and z of the motion. The "dragon roll" movement is shown live at th-cam.com/video/J25_41nPFLo/w-d-xo.html -- a demonstration by its inventor. Interestingly, our CNS is able to figure out the movement far faster than my cognitive brain was able to dissect its mathematics. Our nervous system knows how to copy/imitate quite well, but that doesn't explain how Weck was able to imagine/invent the movement in the first place. Brilliant! There's a device called a Harmonograph to plot a Lissajous curve with two decoupled pendulums moving pen or dispensing sand. Beautiful harmonograph-created curves are shown in the book "Harmonograph: A Visual Guide to the Mathematics of Music" (2003). Apparently, this was a popular device in the 1800s for after-dinner entertainment well before the days of TH-cam. Actually, the harmonograph pre-dated essentially all electronics (including vacuum tubes).
Hidden gem this channel
Very nice!
Cantor was a genius who discovered uncountable real numbers cannot be one to one with the natural numbers, so he created two dimensional array of numbers, now the problem isn't that of counting but one of accommodation, in which even a new room can be created out of infinite rooms.
I found this while looking for mvc iron man infinite
What a BRILLIANT video and critical research! Well done! AMAZING and Hats Off!!!
Thank you so much !!
Why does diagonal argument not work on computable real numbers?
Because computable real numbers are countable, there exist a bijection with Natural numbers. The diagonal argument can be applied only on an uncountable set, we suppose that it is countable and it leads to a contradiction which means that it must be uncountable. (Cantor : Suppose Real numbers are countable, create a diagonal numbers that is not in that set, contradiction)
@@mathematicalcoincidence5906 no it is not as simple. If i list the computable numbers and the apply the diagonal argument, then effectively i have created a procedure to compute that diagonal number. Hence this diagonal number is computable and by the diagonal argument it was not in the list. And yet, the computable numbers are countable. You see the problematic is deeper.
@@mimzim7141 Ok sorry i've missed your point. If we apply Diagonal Argument to a countable set, the diagonal numbers become a procedure to create a number outside of that set. For Rationnal Q, if you compute the diagonal numbers you end with an irrationnal numbers. Because in the construction your number has always something different from all others rationnal, (different from all periodic decimals sequence that exists which means it is irrationnal). In the case of computable real numbers, the diagonal numbers give you an uncountable real numbers, because the diagonal numbers will be different from all decimals of other computable real numbers.
@@mimzim7141 The fundamental reason why this doesn't work is: you can enumerate all computable real numbers. (Meaning: the set of all real numbers whose decimal expansion can be generated by an algorithm. And that's true because there are countably many algorithms - an algorithm is a finite object, under one of the several equivalent definitions of algorithmic computation. For example, there are countably many Turing machines.) But this enumeration itself can't be evaluated by an algorithm. There's no such algorithm which, given natural numbers a and b (under a suitable encoding) returns the b-th digit of the a-th computable number, such that this covers all computable numbers (and such that this algorithm is guaranteed to halt on every valid input). If you apply the diagonal procedure to the sequence of all computable numbers, you get a real number which differs from all computable numbers, and therefore the resulting number is not computable. What if I replace this with numbers definable with a formula? The problem with this is that the notion of "a formula defines a real number" can't be formalized within the theory itself (e.g. in set theory). You can represent formulae as natural numbers (this is the Gödel numbering). But the notion of "formula represented by number n is true" (within the current model of set theory) can't be expressed as a formula. This is Tarski's theorem on non-definability of truth.
15:41 why is d bar defined in this way? it should be the reverse...
Thank you i didn't notice i made a mistake, So it should be {1 if d(i)=0, Undefined if d(i)=1} which means it should stop(1) if the program loop(0) and loop (undefined) if the program stop(1).
That square is not a perfect square so is a rectangle, count the little triangles.
10:34 It is instructive to look at the contraposition as well. If there is a surjective function f: A -> Y^A for some A, then all α: Y -> Y have a fixpoint. These are classically but not constructively equivalent but the proof of this version is very nice and constructs the fixpoint explicitly. To prove this, let α: Y - > Y be any function. Consider now g: A -> Y, a -> α(f(a)(a)). Let a‘ in A be a preimage of g under the surjective function f. Then α(f(a‘)(a‘)) = g(a‘) = f(a‘)(a‘), so f(a‘)(a‘) is a fixpoint of α.
That's just a roundabout way of proving that a surjective function from A to Y^A (Y^A meaning the set of functions from A to Y) cannot exist, as long as Y has at least two elements. (If a and b are elements of Y, then it's trivial to create a function on Y without a fixed point: f: if x=a, then f(x)=b, otherwise f(x)=a.)
Great talk, nice job!
The intro is very cool, the video as well of course. :)
Thank you, Glad you like it!
Great video, especially with the visualizations, e.g. with the hyperplane cut by a line. Interesting extension of this idea: If you consider that *multiple-input* Boolean logical gates/functions are non-linear, then you can recover the full range of possible logical gates/functions by including the non-linear terms as additional inputs to the perceptron. For example, for binary Boolean gates, you need not only an input for x1 and for x2, but *also* another input for the 'product' x1x2, where the 'product' (think like multiplication) is exactly equivalent to Boolean AND. So, if your perceptron now takes a third parameter, let's call it x12 (meaning the AND value of x1 ^ x2), then you can have a third weight w12 on this input, and with these *three* inputs, plus the bias input, you can recreate XOR like so: y(x1, x2, x12) = b + w1x1 + w2x2 + w12x12 With b=-1, w1=2, w2=3, w12=-6. (x1, x2, x12, forumula, y) 0 0 0 -1 0 0 1 0 2 1 1 0 0 3 1 1 1 1 -2 0 I only used those values of weights to show that perceptron can have a variety of weights that generate the same XOR output. But, more clean and simple values for weights would be b=0, w1=1, w2=1, w12=-2. This gives exact output for XOR, no thresholding needed. So, the point is that binary Boolean gates/functions are not fully described by simple linear operation on two Boolean variable inputs. To make them behave *like* a linear operation, also need to supply the product (or conjunction, or join) of the two Boolean variables as a third Boolean input. This is interesting and perhaps useful, but it quickly runs into a problem. In order to handle Boolean functions of *more* variables, say 3 or 4 or more Boolean variables, then you need not only the conjunctions of x1 and x2, but all *pairs* of basic variables, and *also* all triplets of variables, and all *quadruplets* for 4 or more variables. For 2 'basic variables', you need a total of 4 weights (bias, x1, x2, x12). For 3 basic variables, you need 8 weights (bias, x1, x2, x3, x12, x13, x23, x123). For 4 variables, you need 16 weights, etc. For n basic variables, you need 2^n inputs! In some cases, with few Boolean variables, you might be able to handle this. E.g. with 8 variables, you could give each perceptron 2^8 = 256 inputs and weights, but for 16 inputs, you would need 65,536 inputs and weights per perceptron! That's a bit crazy, and it just gets worse the more basic variables you want a single perceptron to handle. However, one beauty of perceptrons is that they themselves are non-linear! And, in particular, a single perceptron can take in two inputs, x1, x2, and produce the conjunction (product / join) of those two, i.e. what we earlier called x12. And so, if you start with two basic variables, then the *first* layer of perceptrons cannot recreate XOR, but if you have a *second* layer of perceptrons, one which produces x1, one produces x2, and one produces x12, then you can combine those three to create XOR in the final output layer! There are multiple ways to do this, and you can use more than one internal/hidden layer, but a simple way is to have: 2 inputs: (x1), (x2) 2 hidden perceptrons: (x1 . not x2), (not x1 . x2); call these as h1 and h2 as inputs for next layer) 1 output perceptron: (h1 + h2) = XOR! (This version does not require any perceptron to take in more than 2 inputs. Of course, can achieve the same thing with a three-input perceptron at the end with inputs x1, x2, x12.) So, my point is that a single layer of perceptron would require more than two inputs, and they would have to be non-linear inputs (e.g. additional conjunctive inputs such as x12), in order to make XOR (or EQU). But you can also get the same non-linear effect by having multiple layers of perceptrons. (In the latter case, however, the extra layers of perceptrons would have to learn to generate the correct non-linear functions such as x1.x2 or whatever. So, there may still be good cases to supply the extra conjunctive inputs directly to the first layer.) Anyway, it's an interesting exploration!
Great work, hoping for more videos
Thank you !! ☺️
Very good, I like how personal the style is. I had encountered all of the single topics on their own but the way you brought them together is really intriguing. Well done!
Thank you very much! ☺️
It appears to be number factorization and then triangulation, with the hidden layer similar to what may be compared to the mathematics referred to as the vacuum space. Gates are Maxwell mathematically.
Thank you for your comment, can you explain a bit more please, I don't get the link between Maxwell, gates and vacuum space
@@mathematicalcoincidence5906 you are the Educator. I am a dumbfounded person.
@@mathematicalcoincidence5906 I am like a person working on a backroad training to recognize a car. 😂
@@mathematicalcoincidence5906 The triangulation I spend time with is: 1000564416 times 1000799917 with four products. I enjoyed your channel presentation.
That's a good video :D I especially liked the conclusion at the end, as it made it 'click' for me on why NeuronalNetworks are not performing well on certain tasks. That said, I've heard quite a bit that it's not the new hotness to have NeuronalNetworks that do not just work on a given input, but also "concider" previously processed input(s); that for example beeing extremly usefull in object tracking on video and similar applications where a sequence of data points are (more or less) steadily connected and thus their existing connection can be used in processing them instead of relying on extracting all information just from a single point/frame/picture. In essence what I just described above should be some kind of im plementing "a clock" to a NeuronalNetwork so it can also use the information of the previous input / timestep / cycle. How exactly that is done is the next intersting question :D Follow up video? ;)
Thank you ! :) You're right, actually neural network have evolved to resolve this problem like you said in a way that process multiple times the input data, like residual connection, recurrent neural network, stack of encoder decoder in transformers,.. And it is a trade off to keep optimization and have access to complex recursive behavior (that simple feed forward NN might miss) But it is not possible to have a clock and a statistical optimization mechanism (because it would be stuck in any loop of the NN) so one should rethink NN architecture from scratch like maybe isolate the statistical part from the looping and recursive part. I'm still thinking on this so maybe not the next video but one day... ;)
Very Interesting Great video!
Thank you!!
What I found interesting by myself is that we can imagine both a cylinder which rotates left to right (or right to left) *and* a cylinder which rotates top to bottom (or otherwise)! For some frequency ratios it’s easier than for others. And that’s because we can have both cylinders with a wave drawn on them as different projections of one wave drawn on a single 4d torus {(x, y, z, w) : x^2 + y^2 = z^2 + w^2 = 1} where (x, y) = (cos(ω1 t + ϕ1), sin(ω1 t + ϕ1)) trace the first circular motion with angular velocity ω1 (and initial phase ϕ1), and (z, w) likewise trace the second one with ω2, ϕ2. Then we can discard either w or y coordinate to get differently oriented cylinders in x-z-y or x-z-w space, then discard the remaining y or w to get a Lissajous curve in the x-z plane.
Interesting! I read there is an other way to find the torus projection : lissajous Curves are also billiard trajectory, and if you unfold its square shape in x and y direction so that it represent the mirror trajectory, you end up with 4 square that ensure continuity on up & down border and right&left, and these limit condition referred as a torus surface
This was an amazingly produced video! Really well structured and animated, please make some more! You deserve far more attention..
Thank you very much ! That is so motivating ! Don't worry new content is on the way ;)
Beautiful video
Absolutely amazing video!
Wow ! Thank you !
You should look at the equation (arg(z)+pi)^(i*abs(z))
I don't see how it is related to the subject, Can you tell me more about it?
Looks like it could be a novel pairing function
You make really high quality content ! You should have participated in the Summer of Mathematical Exposition challenge. It would surely have given your channel more visibility.
Thank you very much ! I'm so happy to hear this ! Yeah i did it but there were also many content at the same time
Fascinating video. Really enjoy maths applied to music. This video should have more views!
Thank you so much! 🙏☺️
Meta
Didn't get it immediately, that's right!!
Man, this was so good!
Thank you, it is so motivating! 😄
Please continue this series
Thank you! Don't worry it is planned
i've seen chaotic systems as described as pool tables with some form of curved obstruction somewhere on the board... and now i'm watching this and wondering if some wave (or maybe noise, in fact i fear that it would have to be noise 😨) can define a curve that best reflects the chaotic motion of such systems...
You should speak louder
Nice stuff, thanks for breaking this down! I now have a greater appreciation for these curves :)
Thank you ! I'm glad you enjoy it !
good video!
thank you !