This vid is criminaly underviewed, its incredible the kind of deep math that can occur when just playing around with a seemingly simple operation. Will probably rewatch this again to try and understand it better
Not incredibility deep math. They are continued fractions which were even studied in Ancient Egypt. In fact, if you take the iteration and divide all terms by -k you end up with the cf [k;-1,....,-1,x]. Hence they are all the approximates of the same infinite continued fraction. It is clear that fixed points will be algebraic numbers and they will have a very specific relationship(because one is just inserting an extra -1 in the cf). The relationship can be easily calculated by looking at the two ways to approach the cf(from inside or outside).
@@MDNQ-ud1ty "Not incredibility deep math". Man, just let the guy declares as incredible what he thinks it is incredible. What an inconvenient reply you wrote.
There is a direct relationship between linear equations, angles of rotation and the trigonometric functions. If we look at the slope-intercept form of a line it is defined as y=mx+b where b is the y-intercept and m is the slope defined as m = (y2-y1)/(x2-x1) or dy/dx. Now consider the line y = x. We know that it has an intercept of 0 as it passes through the origin (0,0) and it also has a slope of 1. We also know that it bisects the XY plane within the 1st and 3rd quadrants. We know that the X and Y axes are orthogonal or perpendicular to each other as they create a 90 degree or PI/2 radian angle. We can easily see that a line with a slope of 1 has a 45 degree or PI/4 radian angle that is between the +x-axis and the line y=x. We know that there is 1 trig function that has an output of 1 when it's input argument is either 45 degrees or PI/4 and that is tan(t). We can substitute tan(t) in place of the slope formula y = tan(t)x + b. We also know through trig identities that tan(t) = sin(t)/cos(t). We can then apply this trig identity substitution to get y = (sin(t)/cos(t))x + b. We can see that sin(t) = dy and cos(t) = dx. This is also apparent in basic addition even from the simplest or first calculation you are taught 1+1 = 2. Without even realizing it, this simple addition which is a linear transformation also defines the unit circle located at the point (1,0). For each of the 1's we can treat them as a unit vector which would be radii to the unit circle and the output 2 or the addition of the vectors would be the circle's diameter. Since we are now dealing with vectors there is also a direct relationship between the cos of the angle between the two vectors and their dot product! This comes from the fact that the are also Pythagorean Identities within the trig functions and that the Pythagorean Theorem and the Equation of a Circle are basically the same thing! A^2 + B^2 = C^2 which pertains to Right Triangles and (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the origin of the circle and (x,y) is a point that lies on the circle's circumference! In elementary and high school they might teach you Algebra, Geometry, Trigonometry and even the basics or introduction to Calculus. Yet they may fail to teach these direct relationships as it will depend on your school's curriculum and the teacher(s) you may have.
I hate that my first time heating that was in a TH-cam comment but gotDAMN that's a relationship that woulda been nice to know years ago lol. Geometry is awesome
Thank you so much for this video! I started studying math at university this fall and last week I had to investigate exactly these functions of the form f(x) = (ax+b)/(cx+d) and I wanted to unterstand them a bit more and how you can interprete the parameters a, b, c and d. Your video couldn't have come at a better time :)
Outstanding video. Only one small point: you played Prelude in C twice when you could have played the Prelude then the Fugue! But seriously, thank you for this cool video.
6:58 I once realized that we don't see in 3D, but in 2.5D. We can perceive a full 2D plane + depth for each point. If I am given a 3D cube I can't perceive it all at once, I have to spin it just to see all 6 sides, then slice it into 2D sheets to understand its insides. Seeing 2.5D is powerful, but nowhere close to seeing 3D.
Continuous fractions, what are they all about? I always wondered. Then onto The Complex Plain and then Hyperbolic Space! It's mind blowing ... but here is one silly question: Is this why they invented the name 'Hyperbolic Functions'?
There is an application of this ideas in statitics? I mean, when we take log(P/Q) of two random variables we are take the geodesics in hyperbolic space?
That's a really interesting question! this sort of log probability seems to actually induce a metric with positive curvature, though I'm not exactly sure why. en.m.wikipedia.org/wiki/Fisher_information_metric
Yes, it's fine at times, and then the quality falls off a cliff. I'm not an expert in audio, but I think just having a consistent setup would help a lot.
Hello! I really enjoyed your video and I would like to recommend the works of Prof. N. J. Wildberger! He has his own channels: Insights into Mathematics and Wild Egg Maths. I hope these help out your studies!
The circle of inversion is the perfect model of our manifold. Space, event horizon, antimatter universe. We exist on one side of surface and antimatter is other. Time is the Planck second of all energy separating the two. Like a double layer and marangoni current. This is why chiralities. On this side outflow/convergence creates positive charge. This is why mass particle is proton. Gravity is convergence but in density. And a chirality on this side reversed when viewed from other side. The antimatter universe runs backwards in time but time is a thin sheet.
Especially once one looks at the numerical field density around the function. Just like gravity. And because the motion of the vacuum flux, the marangoni current, is absolutely tangential at every point and is moving at c, there is an intense negative pressure toward this surface/membrane. Gravity is this negative pressure in fact. If outflow along circular boundary layer then flow towards this. Inflow happens from the periphery.
This vid is criminaly underviewed, its incredible the kind of deep math that can occur when just playing around with a seemingly simple operation. Will probably rewatch this again to try and understand it better
Not incredibility deep math. They are continued fractions which were even studied in Ancient Egypt. In fact, if you take the iteration and divide all terms by -k you end up with the cf [k;-1,....,-1,x]. Hence they are all the approximates of the same infinite continued fraction. It is clear that fixed points will be algebraic numbers and they will have a very specific relationship(because one is just inserting an extra -1 in the cf). The relationship can be easily calculated by looking at the two ways to approach the cf(from inside or outside).
@@MDNQ-ud1ty "Not incredibility deep math". Man, just let the guy declares as incredible what he thinks it is incredible. What an inconvenient reply you wrote.
@@samueldeandrade8535 No, Fk of m@r@n.
That's fascinating.
I'm surprised this video wasn't labelled as being part of the Summer Of Math Exposition.
Exactly what I was thinking!
Yeah same. this seems like a perfect vid for that
There is a direct relationship between linear equations, angles of rotation and the trigonometric functions.
If we look at the slope-intercept form of a line it is defined as y=mx+b where b is the y-intercept and m is the slope defined as m = (y2-y1)/(x2-x1) or dy/dx.
Now consider the line y = x. We know that it has an intercept of 0 as it passes through the origin (0,0) and it also has a slope of 1. We also know that it bisects the XY plane within the 1st and 3rd quadrants. We know that the X and Y axes are orthogonal or perpendicular to each other as they create a 90 degree or PI/2 radian angle. We can easily see that a line with a slope of 1 has a 45 degree or PI/4 radian angle that is between the +x-axis and the line y=x.
We know that there is 1 trig function that has an output of 1 when it's input argument is either 45 degrees or PI/4 and that is tan(t). We can substitute tan(t) in place of the slope formula y = tan(t)x + b. We also know through trig identities that tan(t) = sin(t)/cos(t).
We can then apply this trig identity substitution to get y = (sin(t)/cos(t))x + b. We can see that sin(t) = dy and cos(t) = dx.
This is also apparent in basic addition even from the simplest or first calculation you are taught 1+1 = 2. Without even realizing it, this simple addition which is a linear transformation also defines the unit circle located at the point (1,0). For each of the 1's we can treat them as a unit vector which would be radii to the unit circle and the output 2 or the addition of the vectors would be the circle's diameter. Since we are now dealing with vectors there is also a direct relationship between the cos of the angle between the two vectors and their dot product!
This comes from the fact that the are also Pythagorean Identities within the trig functions and that the Pythagorean Theorem and the Equation of a Circle are basically the same thing! A^2 + B^2 = C^2 which pertains to Right Triangles and (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the origin of the circle and (x,y) is a point that lies on the circle's circumference!
In elementary and high school they might teach you Algebra, Geometry, Trigonometry and even the basics or introduction to Calculus. Yet they may fail to teach these direct relationships as it will depend on your school's curriculum and the teacher(s) you may have.
I hate that my first time heating that was in a TH-cam comment but gotDAMN that's a relationship that woulda been nice to know years ago lol. Geometry is awesome
One day, I will understand this all....
A modest tip: when using both "c" and "z", it helps to use the british way of saying "z" as "zed".
Great video - NOTE to all math video creators. We don't need music to make it "more interesting". It just makes it harder to hear.
Fascinating and BEAUTIFUL! Good presentation! THANK YOU!
This video is amazing. Thanks for putting it together.
this is criminally underrated in the math community. I hope to see your channel blow up soon, this is incredible stuff
Thank you very much!
p
Thank you so much for this video! I started studying math at university this fall and last week I had to investigate exactly these functions of the form f(x) = (ax+b)/(cx+d) and I wanted to unterstand them a bit more and how you can interprete the parameters a, b, c and d. Your video couldn't have come at a better time :)
many functions are periodic. mostly in basic root-finding methods.
I learned this same math but in electrical engineering, it's nice to see how this relates in an abstract math sense.
Man this is some awesome content, keep up the great work!
Such a great video
I was very surprised by the pattern shown at the beginning 😮
Outstanding video. Only one small point: you played Prelude in C twice when you could have played the Prelude then the Fugue!
But seriously, thank you for this cool video.
Such a good question!
thank you so much, i find this topic so compelling
6:58 I once realized that we don't see in 3D, but in 2.5D.
We can perceive a full 2D plane + depth for each point.
If I am given a 3D cube I can't perceive it all at once, I have to spin it just to see all 6 sides, then slice it into 2D sheets to understand its insides.
Seeing 2.5D is powerful, but nowhere close to seeing 3D.
Mind blowing xd I have no idea what's going on anymore :p nice video guys
this goes to the golden ratio :D
.... u dont need 4d to visualizze a hyperbolic Plane. just need some hyperbolic crochet- look it up! it looks very cool
That pygame animation looks slick!
We levelled up from powerpoint for a second there
great .... wonderfull
Continuous fractions, what are they all about? I always wondered. Then onto The Complex Plain and then Hyperbolic Space! It's mind blowing ... but here is one silly question: Is this why they invented the name 'Hyperbolic Functions'?
Great video, can u explain group theory in easy way in next video.
I'm feeling there is some group theory connection here, I immediately thought symmetric group :p
What is the k_n value for a periodicity of 2? What about 1 (if there even is one)?
"identity"
If you solve k-k/x=x or k-k/(k-k/x)=x for k, the solutions depend on x, so there is no fixed k that works.
There is an application of this ideas in statitics? I mean, when we take log(P/Q) of two random variables we are take the geodesics in hyperbolic space?
That's a really interesting question! this sort of log probability seems to actually induce a metric with positive curvature, though I'm not exactly sure why. en.m.wikipedia.org/wiki/Fisher_information_metric
And all that started after my physics exam.
#some2
Great video! But the audio is a little too low, eager for the next video!
Yes, it's fine at times, and then the quality falls off a cliff. I'm not an expert in audio, but I think just having a consistent setup would help a lot.
Hello! I really enjoyed your video and I would like to recommend the works of Prof. N. J. Wildberger! He has his own channels: Insights into Mathematics and Wild Egg Maths. I hope these help out your studies!
3:00
you could probably use k7 as an approximation of pi
Why would you? Looks like coincidence that k_7 is somewhat close to pi, and approximations like 22/7 are easier to compute and closer.
The circle of inversion is the perfect model of our manifold. Space, event horizon, antimatter universe.
We exist on one side of surface and antimatter is other. Time is the Planck second of all energy separating the two. Like a double layer and marangoni current. This is why chiralities. On this side outflow/convergence creates positive charge. This is why mass particle is proton. Gravity is convergence but in density. And a chirality on this side reversed when viewed from other side. The antimatter universe runs backwards in time but time is a thin sheet.
Especially once one looks at the numerical field density around the function. Just like gravity. And because the motion of the vacuum flux, the marangoni current, is absolutely tangential at every point and is moving at c, there is an intense negative pressure toward this surface/membrane. Gravity is this negative pressure in fact. If outflow along circular boundary layer then flow towards this. Inflow happens from the periphery.
8:00 flat earth society :p
(not trying to be rude )the word for three times is thrice
it's not a word anymore though
@@blackcat5771 what
TL;DR it’s all about the Möbius.
Perhaps then f(z) with |S(f(z))| |S(φ)|
First
You can't see 3 dimensions! Everyone keeps saying that! You can only see 2! 😡