The Brachistochrone, with Steven Strogatz
ฝัง
- เผยแพร่เมื่อ 28 พ.ค. 2024
- Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist.
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with TH-cam, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
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About your challenge... I do not love to be dunned and teased by foreigners.
haha...that was a good 1
Hadn't Newton already solved it when he said that?
Gorky
Well, then he must be Newton.
@Gorky OOOOOOOOOOHHHHHHHH SNAP
And also the claw thing
Do you challenge me because you think you've got an unusually clever solution, or because you think I'm Newton?
+Larzsolice No need to be Newton to do math ;)
+Larzsolice lol
+Gna Nay Depends on the math
None of these both. And that is why you can do better.
haha! Favorite comment.
i have an excellent proof for this but this comment section is too small to contain it
the math geek in me laughed out loud :D genius!
Fermat's Last Theorem :)
Rafa Martínez-Avial find your owns copycat
That made my day XD
lol who dragged Fermat into this
I rarely ever comment on TH-cam, but I wanted to say that your content is amazing and most importantly thought provoking!
Keep up the good work and thank you for the time you put into your videos!
+MegaAproductions Thanks so much!
+3Blue1Brown superb video but one thing i still confused about is why you replace the speed of light in different refractive index with square root of y?because i think this only apply if the particle initial velocity is zero and was accelerated only with gravity whereas light initial velocity in vacuum is about 300,000km/s
@@3blue1brown hello
its still 1 comment on this channel. they werent lying
@@cringium shit yeah
Just wanna say that all your videos are beautiful, well edited and animated, insightful, and a joy to watch. Thanks for the awesome content, 3Blue1Brown.
Really neat! I wish I had this video when I took physics to learn the connections between different concepts.
did you figured that out?
MindYourDecisions hello fresh tall walker
Shit message
hahahahaha!!
Who has the cleverer community? You or 3Blue1Brown?
The best math in youtube by far
Baby you have to google the legendary problem number 6
This solution might interest the skateboarding community! Does any skateboard park exploit the solution to the Brachistochrone problem, I wonder?
or rollercoasters for that matter!
Yeah,now that I think about it skateboard tracks do look like a cycloid
omg I totally thought that while watching the video :)
It would definitely be a very smooth, efficient transition. It would give the greatest speed for a given ramp height.
@@brainmind4070 It gives the shortest time, but the speed is independent of the path (being proportional to the square root of the distance dropped).
The answer is "no"
no, I can't :)
N4w4k You have to *believe*
If I made it, so can you !! 👊💪 Hahahaha
why not 'Yes'? "Yes, I'll marry you, Math' xD
If you have a paper, and you try to use the point C like 3b1b does in the video, you will get an equation like v=2cosθ√gR and if you are correctly using point C, you could get that point C is moving in constant speed and thus the whole circle is moving at constant speed.
"Newton stayed up all night....solved it..."
Meanwhile me: *_"I stay up all night and create problems out of thin air"_*
Lol
Honestly, if the legend is true, then this was absolutely badass.
He was SIR ISAAC NEWTON
Leibniz did it as well.
Solution to the challenge:
Consider an infinitesimally small time interval dt, when the velocity of the pi-creature is v at angle theta with the vertical line.
In this time interval, the pi-creature descends a height of dy = v cos(theta) dt (1).
By conservation of kinetic energy, v^2 = 2gy (y being the total height the pi-creature has descended).
By Snell's law, sin(theta) = Cv (C is a constant).
Take the derivative of both equations above:
2v dv = 2g dy (2)
cos(theta) d(theta) = Cdv (3)
Putting (1) (2) (3) together, we have d(theta) = Cg dt, so the theta-t function is a straight line :D
Nice one!!
A good formal prove. But the teaser asked for an intuitive model for the (t, theta) space. Also Snell's law shouldn't be used as a given, but should be the consequence of the shortest path optimisation in that space.
I thought he was asking for another solution to the problem as in another shortest-time curve other than the cycloid.
This channel has a superb standard of videos and never fails to present new insights into old problems. Thanks for all the effort!
The animation at 4:39 ~ish using a space filling curve is magnificent, and for such a one off display is magnificent
I heard that you were doing a essence of calculus series. You should consider also doing a essence of multivariable calculus. Multivariable is a class that many people take in all of maths, engineering and science and is surprisingly poorly understood. Thanks! :)
Perhaps I will in the future. I did many videos on MVC for Khan Academy (khan style, not 3b1b style), which you may want to check out.
What aspect of MVC do you see as being poorly understood?
pretty much everything, basic facts about the gradients don't seem to be even remember by lots of people I know and things like Hessian, why its a matrix, jacobians etc seem to be words lots of people don't actually understand.
yea! I saw those, they are nice, but the most powerful style is 3b1b style ;)
Hell yeah!!
"Heeyy, Vsauce, Michael here" brought me here.
I wonder when he's gonna kill himself.
Marie Loiseau wat
Same
Same bro
@@___xyz___ whyy??
Who's here after the vsauce video?
Yep. But I had already seen it before so does it count?
I have already seen this channel but not this particular video...
I actually saw 3Blue1Brown's brachistochrone vid long before Vsauce's take on it. I'm just here to see if any Vsaucers ended up here.
same
yup haha
It is interesting to note that Huygens in 1672 showed by graphical means, that a cycloid was a tautochrone, that is that starting at any point on the curve an object sliding along it will reach the bottom in the same time, thus a cycloid should be the ideal path for a pendulum on a clock.
Hugens partly got to the solution as the result of a good guess, the mathematics of a cycloid is pleasingly simple and given that Mersenne had already showed that a circle didn't quite work, the cycloid was the next obvious and easy curve to try.
The two problems are clearly linked and I am sure that Newton knew this and proceeded along similar lines.
Interestingly to make a tatochrone pendulum all you need to do is have a flexible string half the arc length of the desired cycloid holding the bob and then cut a solid cycloid of the desired length in half, placing each half either side of the pendulum (curves down small ends meeting at the pivot flat halves away from the pivot). As the pendulum wraps around the cycloid halves the bob traces a tautochronous cycloid beneath. Or, rather than two halves of a cycloid you can use a cycloidic taurus with an internal diameter of 0 for a rather pleasing effect.
SPACKlick I get the first part of your comment, but what's a cycloidic Taurus? Google doesn't yield any useful results.
But Newton was pretty darn savage tho
Cycloidic taurus: I think, a donut shape, where instead of the cross-section having a circular shape, it has a cycloid. That way, the pendulum line can come through the hole in the middle, and the pendulum can swing in any direction.
spur3 oh I see, so the hole would be just a point. I think it's torus rather than Taurus, Google was confused 😝
Glad you're posting more videos. These are some of the most beautiful on TH-cam.
easily one of the coolest math ch. vocal , video quality, use of math language... everything real nice
Could you have an intuitive way to see that "rate of change" and "area under the curve" are opposites?
I'll do that and much more in the "Essence of calculus" series.
Can't wait for that. Your Essence of algebra is brilliant! Thanks for what you're doing.
_That_ will be amazing!
One way you could think of it is that the function is the rate of change of the area under itself. Imagine the graph of a function, and pick two points in the X axis. You can see the area under the curve between these two points right? Now what if you nudge one of them just a bit? The area changes just a little, but you can see that, the higher up the function is at the nudged point, the more the area in that interval will change.
In other words, how much the area under a curve changes is not only based on how much the length of the interval changes, but also on the value of the function at the end points.
Another explanation for those more familiar with discrete mathematics is that an integral is just an infinite sum of differences. If you have something like
f(n) = g(n) - g(n-1)
What you basically have is that f is the analogous to the "discrete derivative" of g (difference in height divided by difference in length). So what happens when you sum all the consecutive values of f(n) from 1 to, say, m?
f(1) + f(2) + f(3) + ... + f(m)
(g(1) - g(0)) + (g(2) - g(1)) + (g(3) - g(2)) + ... + (g(m-1) - g(m-2)) + (g(m) - g(m-1))
-g(0) + g(1) - g(1) + g(2) - g(2) + g(3) - g(3) + ... + g(m-1) - g(m-1) + g(m)
And you can see the all the terms, except for g(0) and g(m), cancel out, and you end up with
g(m) - g(0)
This is a little more difficult to imagine on a continuous function, but the principle is the same (taking it to the limit to infinity and yadda yadda yadda). And, of course, the first value of n doesn't have to be 1, that was chosen for simplicity.
Out of these two, I prefer the first reasoning just because it's more intuitive, but the second one shows why antiderivatives have indefinite constants: if it's a constant term, it's going to be the same in g(m) and g(0), so they'll cancel out. Therefore, it doesnt matter which constant term you choose, there is no "right" one.
Whew, that was a wall of text. Sorry about that, but I wanted to make it as clear as I could in a single comment without edits.
dy / dx = f(x)
=> dy = f(x) dx
The RHS is the area of a tiny sliver of the curve
=> sum up all the slivers and get
integral of dy= integral of f(x) dx
=> y = F(x)
The first step isn't technically mathematically correct but I think it's intuitive enough.
You're insanely intelligent, and an inspiration to me. I'm going to try this.
If there's one thing I think I've learned from your videos, it's that nothing consistent is off-limits and there is always a different reference frame from which to view your problem.
This is amazing, honestly keep it up I need more insight into certain math problems and the history that surrounds them
Thanks! Usually, I don't focus on the history of problems, so I feel Steve was a real addition to the channel for making that part of the focus. I'll probably do more things like this in future videos as well now.
Yeah my mind basically exploded when snells law could be applied, such an amazing use of a particular physical law to reduce a seemingly challenging unrelated problem down to its bare mechanics
+Kenneth Goodall Right!?
Your channel is pure culture to enthusiasts of math like myself. Pure gold. Please do not stop.
My preferred and the most elegant solution I've seen to the brachistochrone is by that of the calculus of variations in which is kind of pops out.
Makes me feel special that I was subscribed to you before Vsauce. And I love that more people are tuning in, you need more exposure, your videos are simply the best math videos on the net.
Was subscribed here before VSauce too .. I'm a patron !
strontiumXnitrate lol, the worst kind
I now need to figure out how to work "brachistochrone" into a conversation.
This explantion is so clear and nice, also good and easy to see.
Really love it.
This was very fun to watch! Actually a nice recap with some additional information about a calculus of variations lecture some time ago. Thanks for that!
The animation is so good! his must take you such a long time to do, this was a very fun video to watch
I love the math and graphics and the cute pi sliding around
I know it may be late but I was looking for a satisfactory explanation of the warm up challenge in the comments and I could not find any. As so I have tried solving it myself and I finally came to a possible solution.
First we must label θ as the angle between the vertical at the center of the circle and the point on the circle's circumference which is tracing the cycloid. We can find the tangential velocity at that infinitesimal point in time if we consider the point of contact C of the circle and the horizontal surface as the instantaneous center of rotation and by using *v = wr* (Eq. 1). If we label the point where the particle is as L, then we can see this becomes *v = wl* (Eq. 2) where l is the length CL. Using the cosine rule we can see that the length l is simply *l = 2Rsin(θ/2)* (Eq. 3).
We can also equate the velocity if we consider conservation of mechanical energy since there is no friction. Thus K.E. = P.E. of the particle which is *1/2 mv^2 = mgy* (Eq. 4) where y is the vertical distance from the horizontal surface to the point L. This reduces to *v = sqrt(2gy)* (Eq. 5).
Substituting Eq. 3 and Eq. 5 into Eq. 2 yields *sqrt(2gy) = 2wRsin(θ/2)* which can be squared to become *2gy =4 w^2 R^2 sin^2 (θ/2)* (Eq. 6). We can substitute y here with the parametric equations of the cycloid. If we take the y axis downward as positive then it varies slightly that the one in the video as it will be *y = R(1 - cosθ)* (Eq. 7). Substituting into Eq. 6 gives us *2gR(1-cosθ) =4 w^2 R^2 sin^2 (θ/2)* (Eq. 8).
Lastly, we can use the trig identity *sin(θ/2) = sqrt((1-cosθ)/2)* (Eq. 9) and transform Eq. 8 into *2gR(1-cosθ) =4 w^2 R^2 ((1-cosθ)/2)* from which we can cancel out the various term to reduce into *g = R w^2* which finally yields *w = sqrt(g/R)* (Eq. 10). This shows that the angular velocity is constant and that the particle's trajectory due to gravity does follow that of the trajectory of a rotating wheel with constant rotation. I hope this helps anyone in the future who might revisit this wonderful video.
Yet another beautiful presentation of a very clever proof. It's been a very nice surprise to hear Steven Strogatz participating on one of your videos!
I was really in doubt about this interview concept when you first presented it, but after just 30 seconds with talking and animation in perfect unison, I was convinced. Great job!
Mads Nielsen Mind washing with pretty pictures...☺️-Don't worry, this kind of phenomenon occurred to me, too, (for another video). Since that, i am more focused on formal audible arguments, than on icons following icons.
Awesome. Your videos are fantastically well-made!
2:49 brachistochrone IS the arc of a circle
Strogatz, the chaos man. His Nonlinear Dynamics lectures made life much easier for me.
Are you gonna cover a solution to the challenge question at some point?
I must say, I love this channel, you are almost souly responsible for kindling my love of math, thank you
0:42 that joke right there earned an instant sub .... also the face that he timed it so it hit exactly at the 42 seconds mark .... this guy is awesome!
?
Brilliant! These videos of yours are just outstanding!
I watch your videos all day, I watch for knowledge and for fun! I can't get enough of this! Your expository is like Nothing I have ever seen before. I'm about to start my PhD in applied Mathematics.... I feel deeply blessed that I came across your channel. You sir, are a breath of fresh air.
As a Cornell alumnus, I know Prof. Strogatz is very famous among math and science students for being a great educator, if you could not tell from the video. It's always a pleasure to see/listen to his work.
If you can help me with a plan for this research as well
one of the best maths/physics videos i have ever seen
Your videos are unbelievably well animated and very interesting!!
I'd like to see Newtons solution
It's pretty standard honestly. If you tried to solve the problem elementally , you would most likely wind up with Newton's solution.
The answer can be found in the book "Newton's Principia for the Common Reader" by S. Chandrasekhar, pp. 571-578. He points out that Newton had already laid the groundwork for solving this problem in the first part of the Principia.
Here, hold my apple...
3b1b: challenge the audience
Audiences: meme about newton
the challenge at the end is just tooooo good, like its really interesting and exciting just think and ponder on it
I saw this problem in Physics olympiad national qualifier 25 years ago. Of course, I couldn't even tough the starting point. It has been forgotten for more than 2 decades, and suddenly pop up in front my eyes. One of the biggest mysteries of my life is solved. Thank you!
The Most Casual "Oh Interesting" I've ever heard 5:55
7:38 Case Sensitivity:
The g in the potential energy term, mgy, is not "the gravitational constant." The gravitational constant, G, is the proportionality constant in Newton's law of universal gravitation that relates the attractive force to the masses involved and their relative position. G = 6.67E-11 m^3/kg•s^2. You are referring to g, which represents the acceleration due to Earth's gravity measured at the surface, 9.8 m/s^2.
An innocent slip as I know you know this.
Another great video!
I know nothing about higher math yet I love watching 3Blue1Brown videos. Imaging how these concepts occur in the world is why I watch. This concept reminds me of alpine skiing and skateboarding.
I love your videos. The quality is impeccable and the content is inspiring but i just can't watch them yet.
Really awesome solution, I paused the vide and tried to solve it myself. My first instingt was to use calculus, but I didn´t do very good. However this is how far I came and if someone could help me to continue this calculation it would be really nice.
The variable that we want to minimize is the time, and the variable we are looking for is the function between the two points. So we need a relationship between the time and the function. We know that t=s/v, there is a general way to work out the distance between two points on a function. And it says that the distance from the point x=a to x=b on the curve f(x) is the integral from a to b of the function sqr(1+(f´(x))^2). The distance between the points doesn´t really matter so I´m going to say that it is one just to make things easier. This means that s=0integral1(sqr(1+(f´(x))^2))dx
We know that v=k*sqr(f(x)) in any given point, but since the speed changes over time we can not plug it in to the formula. Also I´m going to say k=1 just to make things easier. It won´t affect the final curve any way. Correct me if I am wrong but the average speed between x=0 and x=1 is going to be 0integral1(sqr(f(x)))dx.
So if we plug this into our formula t=s/v we get t=(0integral1(sqr(1+(f´(x))^2))dx)/(0integral1(sqr(f(x)))dx). So I have a formula for the relationship between the time and the function. Now I just have to find the derivative and set it equal to zero I guess, it will probably end up with i differential equation. There is just a little problem, I have no idea of how to take the derivative of that function, and I don´t even know if it is right for that matter. Is there even an algebraic way to do it? If not, then I have no idea about how to solve this problem algebraically. It would be easier to find the derivative if the variable was x, but it not is f, we are looking for an entire function here! Does anyone know how to continue this solution?
en.wikipedia.org/wiki/Calculus_of_variations
The knowledge you need to learn is "calculus of variations". You need to use the Euler-Lagrange equation.
+Samuel Lo ok So if I understod it right then I now have to solve this equation:
d/df((0integral1(sqr(1+(f'(x))^2))dx)/(0integral1(sqr(f(x)))))-d/dx(d/df'(x)((0integral1(sqr(1+(f'(x))^2))dx)/(0integral1(sqr(f(x)))))=0
Because I am assuming that what they reference as L is the same as t in my equation. But still I can't solve that.
I had propably made a misstanke with the perenthases in the equation but what it says is dt/df-d/dx(dt/df')=0 I just replaced t with what I know that t is equal to in termes of f.
We need to use the instantaneous speed, not average speed.
v = sqr(f(x))
ds = sqr(1+(f'(x))^2) dx
dt = ds/v = sqr(1+(f'(x))^2) / sqr(f(x)) dx
Total time = integral [ sqr(1+(f'(x))^2) / sqr(f(x)) ] dx
L = sqr(1+(f'(x))^2) / sqr(f(x))
L is a function of f(x), f'(x)
Then we do the partial derivatives: ∂L/∂(f(x)) and ∂L/∂(f'(x))
Finally, solve the differential equation:
d/dx (∂L/∂(f'(x))) - ∂L/∂(f(x)) = 0
"By the claw the the lion is revealed"
10:24 this is also given in Thomas calculus . I have the Indian edition, page 654
Fantastic video! One of my favorites. More videos like this!!!!
6:48
"but for now, all you need to know"
come ON, i wanna know it all!!!
5:28 I loved your animations of light rays that included the oscillations of the wave itself. Very clever
your videos rule dude, that part on light as a solution was like, genuinely mind blowing
What a brilliant story. As you say, awe inspiring. When we learned Snell's law, it was taught without reference to that path being the fastest, hence the path light will naturally take. The facts were taught but not the fundamental principle or reason. Of course that leads to other questions.....
Is there a way to support you?
His patreon link where you can get an early look at videos yet to be released:
www.patreon.com/3blue1brown
Aida Bit I think the patreon account wasn't a thing back then.
Wow, i am amazed by every aspect of this video!!!; what software do you use to make these animations.
Great video - you keep interesting me in things that I had no idea about
Wow, very mind-bogglingly elegant solution! I'll have to work on the challenge now!
This is a fantastic explanation. I'm taking precalc this year, and I can intuitively understand this. That is so cool haha. Came from Vsauce. It was an alright video, but he never really explained the reasoning behind the property, which was what I wanted to know. Glad he introduced me to this channel though.
Physicist (upon seeing the solid normal line at 7:05): REEEEEEEEE
I qualify but don't understand.
@@FlyingSavannahs Normal lines are generally drawn as dotted lines, but the general may differ from place to place!
Ok. That's news to me.
This will be a hand wavy solution to the question but the way I see it, the straight line in the theta-time space indicates a constant rate of change in theta which is the smoothest possible path to the solution as a straight line is always the shortest path: i.e. add any more or less curvature than that exact curve and you'll distort the theta-time curve creating a lower than optimal area under the curve of that graph. Background is only a B.S. in math though so take my intuition with a few grains of salt. Cheers to the work you do, and the tools you've developed - it's all much appreciated!
super cool that you brought strogatz, im a fan of his "non linear dynamics and chaos"
This video is really great! Why cannot I watch it when I was an undergraduate! The first time I read the details for the solution of Johann Bernoulli was from Ernst Mach's Science of Mechanics.
I have research on this topic. Can you help with university references and notes? Do you have a research plan for this problem?
Wait, so what was Newton's solution? Was it the same as Johann's?
can't defeat funorange
Probably same conclusion but different method(?)
en.wikipedia.org/wiki/Brachistochrone_curve
I'm with Typo
What I absolutely loved about this video is, light is shown as a particle and as a wave! xD
Amazing content as always. Keep it
going.
I watched this way before vsauce, man i feel special!!
Woohoo, new video!!!!!!!
Never knew about the history of the problem .. great job digging into the origin of the problem
Wow - I love this (and your other content)!!!
Cool challenge. Now I'm going to be procrastinating my homework, and it's all your fault! D:
+1ucasvb
If it's maths homework, tell the teacher! It should be fine. :D
@@ZardoDhieldor I'm not sure about that tho.
Me too xD
where do you get ideas for your videos? I imagine it takes awhile to organize your thoughts and results to make something you're content with
Well, I keep a growing list of different interesting topics I think where I think there's room to provide a new, or at least under-discussed, perspective. Unfortunately I can't always spare the time to knock items *off* the list as quickly as they get added.
visualisation of calculus of variations? please?
Your videos are so rare, that I auto like them as soon, as they posted
Interesting and lovely told. Thanks
4:38 what is with the hilbert curve at the top left?
The circle used to graph the brachistochrone is rotating at a constant rate, so it makes sense that it would give a straight line in the angle v time graph.
that's a rationalization, not a proof :P
Snell's Law is an example that Bartosz Milewski gives of the kind of global thinking that is taken to the extreme in category theory, with its universal constructions. A math instructor of mine also mentions that these kinds of global optimizations (optimizing with the telescope, rather than the microscope, you might say) are done in calculus of variations. I find this duality of local vs global approaches to optimization to be quite interesting.
thanks for what you're doing, your videos are awesome
This is not rigorous and I'll probably revisit this later, but I do have a general idea:
The lateral force imparted on the object due to gravity is gsin(theta). As the slope is constantly changing, mapping the axis to sin(theta) renormalizes the curve in a way that reduces it to a straight line, as it undoes the encoding of the function. In other words, the function does not look linear on a normal cartesian set of coordinates is because the coordinate system does not factor in the constantly changing lateral force of gravity. By mapping the function into a set of coordinates that cleverly cancels out the constantly changing force of gravity, the optimization problem becomes linear.
I also want to thank you for making these videos. I rarely find videos or textbooks that explains things intuitively, and the animations help so much in this regard.
I am so happy for you that vsauce discovered your video xD
Wow. Just wow. This video was the highlight of my day.
I have only the warmest of feelings for Strogatz after listening to his Great Courses course on chaos.
You say "challenge", I say "homework"
My educated guess for your challenge is that it minimizes the change in energy. I remember in engineering school we had some examples that were similar where the integral of the work done was minimized when it was a straight line. This also seems to make physical sense in that light will want to minimize its energy state.
Really nice animation and illustrations
As the described path is a cycloid, it must fill the conditions for every point in the circle, not just the first point you are watching. That's why if the rotation speed isn't constant, then other points in the circle wouldn't fill the condition for creating the cycloid. It's because the huge symmetry of a circle that it must rotate at a constant velocity.
Your videos are inspirational. They have allowed me to see mathematics, or as Keith Devlin would say, "mathematics has made the invisible visible." And your videos do that for me in a unique way that has not been replicated by any other learning source. I am studying math education, and I would actually like to learn how to make videos like this so that I can make learning more enjoyable for my students. I'm not much of a programmer, but I'd be willing to learn. Is it a trade secret or can you point me in the right direction? Thanks, Grant.
-Ben
.
He said he uses Python, just watch some online course about python and google everything you don't know
or you can just google his github ;)
First time I'm seeing a video with no dislike.
+Morel EBELLE Some idiots ruined it
I ruined it :(
you can always "undislike"
I’m not sure if I’m looking at the problem you gave us correctly but I love puzzles. It looks like a straight line because t-0 graph is measuring how long it takes to get into a certain position each starting at zero. If you look at the graph side ways where t=y and 0=x. The end points makes a curve.
I am by no means a mathematician and my understanding of physics is rudimentary, but when you started talking about expressing the problem in term of time over theta, space time diagrams kept popping in my head. every curve or motion is actually a straight line in space time. I don't know maybe i am, reaching but i feel like there is a connection here.
It's because time and space are an illusion...
Great visuals/graphs in the video... You are really great at teaching!
Exactly what is "it" in your comment? How can an illusion predict the mathematics of a curve? Calling time and space an illusion is only true in the philosophy of nondualism, in which pure consciousness is all that truly exists. But such an insight cannot possibly predict any of the laws of nature, including the equations of curves having certain properties. In other words, you have taken a true statement in one model of reality and erroneously applied it to another.
isn't it in some way a shortest path problem in 4 dimensional spacetime?
also, what software are you using to make those great animations?
Discord Python
what no
length of a cycloid is much more than a straight line
minimizing both spatial and temporal coordinates would be a different problem
Are you talking about incorporating relativity to this problem? Its effect would of course depend on the acceleration field present and the size of the cycloid. For most practical purposes, this effect would be negligible. It's hard for me to imagine a situation where you would have a high enough acceleration field/large enough cycloid to really have any measurable effects due to relativity.
Great explanation, thank you!
wow that melts my brain totally...what a piece of great math