Times Tables, Mandelbrot and the Heart of Mathematics


ok so to start off I’ve got 4 images for you. Quick question what do all of
them have in common. Well, I would be very surprised if you guess this one here. (Giuseppe) They are full of lines. (M) They are full of lines, they are in a circle. But actually they are pictures of times
tables. I better explain this. So to get these pictures what
you do is you start with a circle and then you pick a random number. And we just choose something nice to start with, a 10. Then we put 10 points on the on
the perimeter here equally spaced and then we label them
with 1,2,3,4,5,6,7. Actually, let’s start with 0–0123456789 10 10 we also want, so we kind of put it
right there again. So that’s also 10, that’s also 11 12 13 14 15 and so on. And then
again, and so on. So this guy here stands for 1, 11, 21 and so on. And now we’re going to do the
two times table and we’ll start with 0. So 2 times 0 is 0, same thing, so we don’t do anything. OK, two times 1 is 2. So we connect the 1 to the 2. Then 2 times 2 is 4, connect the 2 to the 4 and we keep on going, pretty obvious. Now 2 times 5 is 10 and
remember 10 is also over there so we connect that guy up. And then 2 times 6 is 12 which is also the 2 and we just keep on going like this okay and then at 9 you pretty much
gone all the way around now you could go on so for example could now do 2
times 10 and 2 times 11 and 2 times 12 and draw in those connections but actually the connections
are going to be exactly the ones that we’ve drawn before. Just to illustrate, let’s just go up to 2 times 12 is 24 which corresponds to
the 4 and that connection is already there. 2 times 13 is 26, and we’ve already got that. Of course we made a choice at the beginning that 10 If you change that 10 to anything
else well the picture changes but pretty much
everything I said stays true. So for example if you switch from 10 to
11 we get this guy here here. And then 12 13 14 15 and let’s just see what happens. (Giuseppe) Is there a reason why the diagrams have a horizontal symmetry axis. (M) yep yep yep yep yep there’s a reason and let us give this as homework. Guys figure it out in the comments, tell us why there is a horizontal symmetry axis.
Of course, what’s much more interesting is that curve that
somehow magically materializes when you up the number that we chose at the beginning and actually just highlight it a bit. This strange curve got a name and pops
up all over mathematics, it’s called the cardioid like in cardiology and means heart. So it’s like a mathematical heart curve. It comes up in all kinds of places. I’ll just show you a few. Ok so for example you can
have it as a rolling curve rule one circle around another circle and just
see what happens to one of the points on the boundary of the rolling circle and what it does is it traces the cardioid. Now where else can you see it? Well sometimes you can see it in
your coffee cup. If you’ve got a conical coffee cup like this and the Sun is in the
directions of one of those green lines in the coffee cup then we can also see it. Where else? Right
there in the Mandelbrot set so the biggest bulb in the Mandelbrot set is a cardioid and the second one is a perfect circle. Now this would be a good one to figure out why why is this guy
a cardioid. Just chase this down somewhere. Now just for those people who know something about the stuff, that’s the equation that we use to make up the Mandelbrot set
and there is a 2 in there. That’s important We just had the two times table. Let’s
keep going now (Giuseppe) One more thing about the cardioid shape. As a video producer I know that there is a microphone called cardioid microphoes and that is because they pick up the sound in that area, from that direction. (M) Okay maybe we do another video on this. Now instead of doing the two times table we can do the the three times table. What we get is this pattern here for initial our
choice 10 and then I just up up the numbers and what do we get? Another curve, another very famous
curve. This is called the nephroid which means kidney. It’s also a rolling curve, here the circle that is rolling is half the size of the other one. It’s also in a generalized Mandelbrot set. Remember before we head an exponent of 2. Now we’re
talking about the three times table we change the exponent to 3 and you actually
get the main bulb here being the nephroid. We also get it in the coffee cup. This time we need a straight straight coffee cup and the light
rays come in from one side. So they come in straight like this and
then they of bounce around inside the cup and and make up this curve.
Actually this way of making up a curve is called a catacaustic in mathematics. What else have we got. Four times you can see a pattern now. Four times gets you three petals. Before we had three times and so two petals and all these other things that we’ve highlighted they work too. Here the the exponent has gone up to to 4 in the generalized Mandelbrot set. Here we’ve got the rolling curve happening again. And then just for good measure
just one more guy here. That’s the 5. It gets four petals and we’ve got the Mandelbrot set here and we’ve got the rolling curve. All very neat. (Giuseppe) So the number of petals is n-1? Yep and it keeps on going like this. Ok so let’s start again with 2 times table. Now one one really nice thing to try is to actually
make increments not one … to 3 to 4 to 5 but actually do smaller increments actually do the transition here
continuously. I’ll just show you what it looks like when I do the multiplication times 2.1. What you get
would you get 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 3 and you can see how this cardioid turns into the nephroid. Now I just let it go. The numbers are going up and you can see the petals going more and more and more. You think well that’s probably what
it’s going to be. You can also look at this in the Mandelbrot set and you
can actually see exactly this sort of pattern happening in the middle. As we push the exponent to infinity the
whole thing kind of turns into one big circle. Okay and now you think that’s
the end of it but no, no, no. Let’s keep going here. Now the petals are pretty much going away but if you just look at what’s happening here
you see there’s a lot more structure somehow being hinted at. I mean I’m
moving very fast at the moment moving very fast but if you actually went
through this frame-by-frame you would see a lot of really really nice structure
happening and I’ve highlighted a bit here. Just a few of the frames that you get
you know some really amazing stuff happening here so if you stop at 33 you get this one. If you stop at 34 and there are a couple of other highlights here. There are a lot more but that is how
many I could fit here. Ok maybe let’s have a close look at some of them. So here we’ve got four. The number that all these correspond to is 200. What we’re really doing he has a
special name in mathematics So what we’re doing here is
the times table modulo 200. So it’s got a special name. It’s actually incredibly
important in all sorts of applications to do this sort of modular arithmetic.
But paper anyway for our purposes you know you understand how it’s built up
and if you know a little bit more you know a little bit more. So for the 2 times table we get the cardioid. For the 34 times table modulo 200 we get this guy here and then
some other interesting things happen. So, for example, if you multiply by 51 we get this guy here and if you multiply by 99 you
get that guy here. And obviously if you just look at the two numbers here below so 51 is pretty close to 1/4 of 200 so must have something to do is that. 99
is pretty close to one-half of 200, must have something to do with that. Now I
really want to explain something I really want to explain where this one
comes from. And well I mean the first time I saw it I was pretty surprised like
everybody else who sees it but then I remembered something I had seen the cardioid before in you know many many different guises and in one particular one that I was familiar
with I could see where the connection is. I just want to tell you about that one
and it’s got to do with these light rays bouncing around. It’s a little bit different
from what we had before. So basically we’ve got a circle here and then we’ve got a light
source right over on that side. And that circle reflects the light right, so we switch on
the light over there and then well what happens? Well light rays emanate in all kinds of directions and get reflected off the circle bounce around and you can see
this pattern emerging, the cardioid. So I knew this one I had actually calculated
this curve from this description at some point in university. And starting with this you can actually see very easily why the cardioid comes up in the two times table and I just want to show you.Let’s just highlight like one of those rays coming out here. It hits the wall here
and then it gets reflected at the same angle as it comes in, so this angle that you see here is the same as it comes in. So what does that tell us now? Well, if we if we put this line in the middle that means that this side really flips over
to that side, which means that this segment in particular is exactly as long
as that one here. If I travel from this light bulb to this point and then from
this point to that point it’s exactly the same distance, that’s going to be important. So let’s make this 0 let’s call this N. What’s 2N then? To get 2N we have to measure this distance here and then we have to measure it again and where do we get?
Right up there ! And so it’s pretty clear that when you do the 2 times thing you get that reflected ray, you know, in the picture. Just to finish off I’ll show you a little bit of a movie of all the stuff evolving a little bit slower than what
we had before.

98 Comments

  • whats the beat/ song at teh end?

  • жопу нарисовал по формуле

  • нихуя не понял, но очень интересно.

  • 3:43 heart?))) really, heart?)))

  • Matemáticamente seria un trasero perfecto.

  • That's the star wars republic symbol!!!!

  • Why the title and more is in portuguese?

  • Whats ment with the word Mandelbrot what does it mean?

  • Whenever I see a video mentioning mandelbrot, I need to watch it. I'm not a mathematician but a computer scientist, which is pretty nerdy too, I guess 😀

  • C'est une histoire d'amour entre le 1 et l'altérité que le 2 représente. Le 2 a le virus multiplicatif dans ses gênes (comme le 5, d'ailleurs)… Lorsque le 2 déploie sa force multiplicative, cela ressemble à un cul et le un, partant d'un centre mystérieux, s'allonge, comme un sexe masculin épris de désir. Dans l'alphabet arabe, c'est le dessin de la lettre ALIF, qui démarre d'un HAMZA primitif pour aller à la rencontre du BÂ horizontal… La suite de l'histoire est passionnante car la liaison est nécessaire…

  • Better than shrooms

  • This depends on you chosing a circle for your numbers…. suppose you decided to map them around a triangle??? or a square???… or if you did it in 3D on a toroid…..

  • The universe is a fractal and it is both infinitely large and infinitely small.

  • Fellas, that Cardioid is thicc af

  • Озвучьте ролик, смотреть нужно его а я читаю

  • Its strange how Youtube recommended this video to me today, since just a few hours ago I tilted my water glass and was intrigued by the shape the light was making on the table…the Cardioid.

  • Haha when you imagine this guy is looking at green screen, and he thinks he is seeing something

  • EXCELLENT!!! but um, no yellow on lines anymore please, especially when its high density, hard to see 😀

  • Ну и хули сссука, где перевод то ёбана нахуй в рот вас всех…

  • I am pretty sure this can be called a mathematical acid trip

  • This reminds me of Louis Slotin's death.

  • 2:59 …1.2.4.8.….???? ведь умножать стали с 1,2,4,6… подмена понятия визуализацией, нет?

  • Please, made the video of the mic patterns!

  • You can watch this in 4k
    https://www.youtube.com/watch?v=f7IAvLoRPCU

  • таким образом получается что жопа это правильная геометрическая фигура

  • Имеет горизонтальную симметрию, потому что синус нечетный. Прямая связь видна, если глянуть в полярных координатах.

  • the spirograph game reaches new high for sales in universities!

  • play the last bit at .25 speed for a profoundly different experience

  • Is there a connection whith the works of the famous Lituano-Spanish-« Syrian »« mathematicien » of the sixteenth century Cordovero?

  • en español hijo de puta, NO SOLO LOS GRINGOS quieren saber. voy a leer o voy ver las imágenes. que alguien lo boble

  • Thumbnail looks just like Mercury's retrograde movement on a ptolemaic flat earth. Funny.

  • How does Fibonacci correlate to Mandelbrot and times tables?

  • Warum ist der Titel mit deutschen Wörtern aber der Spricht english

  • 800points implementation
    https://editor.p5js.org/pkoch/present/qInvm-nMw

  • thumbnail lied to me, this isnt a coffee latte art

  • I noticed that every so often, but regularly it forms a spiral. Is there a name for that?

  • The projection through a magnifying glass, too.🔎

  • What software is used?

  • It looks like a vortex in middle

  • I can't help but look at the evolving patterns and think of them as a 3D pattern evolving even though there is no third dimension. What if you made a 3 dimensional version where you space points on a sphere?

  • the force is strong with ^2

  • There is a God, and He speaks math.

  • Take me to church! Loved every second of this video. Subbed and belled. Again, Thank you for this /

  • Yo veo el trasero desnudo de una mujer. Que triste es mi historia.

  • If you put water under high frequency it would do the same thing

  • Maths are real bc NOW i can see them XD

  • wow

  • Awesome! Congrats!

  • This should be taught in every high school geometry class. We're so fixated on base 10, this is the way to address that in math ed. Everyone hated times tables, this is the gateway to showing how math is cool and past learning wasn't worthless.

  • The blue pattern on a peacocks feather is the same shape.

  • Hey, great video! When you multiply numers(dots) by 2, can that be connected with complex solutions of a polynomial of n-th degree. (e^(i*alpha_k))^2 , where k=1 to n gives that line which connects 2 dots on the circle.

  • I’m in shock

  • I’m in shock

  • Ну теперь то вы поняли, что мандалы это математика?
    Ну и стоит подумать, что именно так нужно проектировать термоядерные реакторы.

  • Why Mandelbrot?  See: https://youtu.be/OW8XMIlhFyI

  • Bist'n Deutscher?

  • Que maravilla¡!

  • I saw Chakra

  • That looks exactly like a DMT trip man.

  • 변태;

  • Would like to be able to purchase a high resolution of this for a wall display (monitor with memory) as a dynamic art display that cycles repeatedly.

  • I'm just surprised when that plane becomes part of mandelbort set…

  • How did I fall into this rabbit hole?
    This stuff is an absolute trip.

  • Now I know how the star wars "rebel" logo was created.

  • I'm very surprised at the comments.?

  • So why is it symmetrical?

  • My guess is:
    He uses a projector, shows it to an imaginary auditorium with one person who is asking him questions.

    Then he gets rid of the projector image and overlays an accurate digital image.

    My question is how does he get rid of the original projector image?
    What would it take me to do something like this using Matlab and Davinci Resolve?
    I want a video on how he does it.

  • you can see the cardioid when a plumber bends over to work.

  • if i had known about this kind of math when i was in school i would have done a lot better, for some reason seeing the geometry makes better sense to me than trying to interpret math as a language. it just clicks.

  • It's looked like inside of the eye to me …

  • Spirograph in the 70’s. Wasn’t that’s what the drawing kit w spiral rings etc was called? 🤘🏼

  • Cycloids are everywhere.

  • what happens when you use a sphere instead of a circle?

  • oh cute, if I leave a comment I'm gonna get stalked…lol

  • great thank you

  • does anyone know how I can reach the speaker?

  • Naow days sf movie just like a really ( basic on mathematics )

  • Mandelbrot is a matter detection functions. Mostly used in how structures are formed. Mathematics of.relative development. Even in developmental biology.

  • the fibonacci numbers appear in the center if you slow the last sequence to half the speed amazing. Really a proof that math is somehow gods favorite tool.

  • A stupid question….what is the utility of this, what is the application in life? I am sure , there is, its just that I would like to know what is this knowledge applied in.

  • I am very happy to see all this stuff you to visualize the meaning of number..

    Which application you used to develop this animation or simulation…

  • is that 86 or 88 the one which look like flower?

  • https://github.com/honeydatax/marina-os very good video about can you help me to make a video about my new operation system builder

  • hello

  • Please make a t-shirt or a coffe mug with these patterns!!!

  • Aeh … shouldnt the rays of the sun be parallell ??

  • I am trying to understand the movements of a market chart (candles) with Mandelbrot or fibonacci, but I still can't make it, some of you have already done it to help me please, thank you !!

  • wait this was actually that easy???

  • what program did you use to visualize?

  • YES! #jungEinstein reiteration, perhaps? K, thanks Doctor.

  • fyi, heads up on the penrose numbers for mentoring me: cant give you the muskrat's function but I can give you my pow(x,y) = x^y function, where x and y can be lots of great things like dimensionalNumbers, topologicalNumbers, x is usually time or position. Thank you , Master Jedi. bows . ps, I call it pow for a reason…ever play Mario and watched him scale lol?!

  • But why?

  • I know this doesn't have anything to do with the video, but I would like to share these questions and this message with as many people as possible.

    1. What do you think happens to someone after they die?

    2. Do you think there is an afterlife?

    3. If heaven exists, how do you get there?

    4. Would you be good enough to get into heaven?

    Here are a few questions to see if you are a good person:

    1. How many lies have you told in your whole life?

    2. Have you ever stolen?

    3. Have you ever used God's name in vain?

    4. Jesus said that "whoever looks at a woman to lust for her has already committed adultery with her in his heart" (Matthew 5:28). Have you ever looked with lust?

    5. If you have broken any of these commandments, would you be innocent or guilty on Judgment Day?

    6. Would you go to Heaven or Hell?

    The Bible says that "all liars shall have their part in the lake which burns with fire and brimstone" (Revelation 21:8) and "Those who indulge in sexual sin, or who worship idols, or commit adultery, or are male prostitutes, or practice homosexuality, or are thieves, or greedy people, or drunkards, or are abusive, or cheat people — none of these will inherit the Kingdom of God" (1 Corinthians 6:9-10). So, if you are guilty of any of the ten commandments, then you are in trouble when Judgment Day comes! Do you know what God did for guilty sinners so that we wouldn't have to go to Hell?

    2,000 years ago, God became a human being, Jesus of Nazareth, who suffered and died on the cross for our sins and rose from the dead on the third day, defeating death. The punishment that we deserve for breaking God's law, Jesus took in our place. He paid the fine that we could never pay so that God can legally set us free by forgiving our sins. He is just, by punishing evil through Jesus’ death on the cross, while being rich in mercy towards us.

    What you have to do is repent, or turn from your sin, and trust alone in Jesus. Don't trust your own goodness, because, as Jesus said, “No one is good — except God alone” (Mark 10:18). Trust alone in the savior. The moment you do that, God will forgive every sin you have ever committed and grant you eternal life, and He will give you a new heart with new desires, so that you will love to do what is right. You must repent and trust in Jesus.

    When are you going to do that?

    Here is a conversation between a Christian and an atheist on this subject: https://www.youtube.com/watch?v=QRZlYg08Za4

  • Fascinating. The very foundations of creation are based upon math, geometry, light frequency and the frequency of vibration.

  • It feels like there is a concept really important about how the universe works behind this but I feel too dumb to see it

  • Hi. would you be interested in a 3d version of the pattern at the end?

  • Also is there a spherical representation of this rather than a circle?

  • Be interesting to track the path of the dynamic centre as it moves towards 200: sometimes the movement appears continuous and sometimes it appears to jump in a discrete way.

  • Who do I contact to teach y'all motherfuckers something?

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