A Controversial Maths Video.
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- Опубліковано 16 кві 2024
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Today we will show that circles and infinite sided polygons do indeed have a thing or two in common :) Let's solve some spicy limits and calculus, shall we? Enjoy! =D
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Most polygons are just finitely sided circles
brave
Damn 😂
This also means that polygons have circumferences and that attempting to correct a past word choice was pointless.
Bruh
Your mama's ass is just two finitely faced hemispheres
Lines are just circles of infinite radius so polygons are just collections of arcs of circles but also circles are collections of arcs of circles qed
all of them are just boundaries of convex sets
@@yonaoismeWhat if they're concave?
This was one of the first things I independently discovered back in high school. It was one of the early things that helped establish my love of mathematics.
I realized that a regular n-gon can be cut into n isosceles triangles and so I derived the relationship between the apothem (radius of the incircle) and the area and found it to be A = a² n tan(π/n). And the limit as n→∞ of n tan(π/n) is π. I later discovered that the perimeter is also given by p = 2n tan(π/n). For an interactive demonstration, see Desmos graph 6ijb6uc1yu
Suuuurrrreee buddy. Sure
@@Kloppin4H0rses No high schooler has ever taken a calculus class / liked doing math in their free time
By the end of senior year in high school, I too discovered this problem. I am now a first-year math major. How cool! Being in calculus at the time, I also realized that the integral of our result yielded the formula for the area of a circle as well.
Sameee :)
Aren't infinte sided polygons are called aperigons, they have the exact same properties as a circle, with the same area, the only difference is that they have interior angles which eachone 180 degrees, while a circle has none
Apeirogons, from the greek word for infinite "Apeiro" (Άπειρο). And they're very cool
With world apeirogon you just reminded me of mind bending clip of jan Mislai about regular polihedra. Starts with platonic solids, somewhere in the middle we land in honeycomb apeirogon to end in some stellated net of pointy helixes.
@@Filipnalepa I watched that one too!
Not really, i agree the distance converges, but some points on the border of the circle will never be touched, since the border of the circle is uncountable yet the points of the infinigon are only countable infinite.
For example lets go with aquidistant n-gon, the point on the border at the angle sqrt(2) will never be exactly touched by any n-gon.
You could say you take a setunion integral, but that's cheating. And not n gon convergence
The difference is that, for a circle, every point on the circumference is a radial point, and there are uncountably many points on the circumference, whereas an apeirogon only has a countably infinite number of radial points.
Also, polygons are kinky ;), circles aren't.
What a coincidence! I was just using this idea to prove the curved surface area of a cone formula (pi r l) from the limit of an n-gon based pyramid. Very cool!
niceeee :)
Good to know
"We're going to start off with a square." *draws a kite*
Love the vids, keep up the good work!
I did this last March on PI day. I calculated a/2 by splitting the triangle and made two right triangles and solved for a/2 using law of cosines. Try it, I think it's easier.
Let's go another video
I love seeing these arguments. Great video Flammy ma boi
My heart is boiling blood already, I'm shocked by the thumbnail 😢
I love your videos. They make me enjoy math.
I love your videos papa! Looking forward for the next video!
If you have a polygon and increase the number of sides, the perimeter will approach the circumference of a circle of the same radius and you can use this to compute digits of pi. 2^n * sqrt(2 - sqrt( 2 + sqrt (2 + ...) with n many square roots, plug in huge numbers for n to get a better approximation for pi.
What about substituting the Taylor series of cos instead of the double angle formula
~12:22 ← one can also apply the Cesàro-Stolz theorem from that point on…
It's like saying that all reality happens in my head.
Well, sort of. And this polygon is sort of a circle. And it sort of is not.
Papaflammy papapaflammy6969 love your videos…we need a video/series on numerical methods ❤️❤️ atmospheric science rulez btw ✨
I came up and did this problem a while ago in high school. very cool!
I don’t think using l'hospital would’ve been illegal here since a limit in a continuous variable x converging implies that it converges in the discrete variable of naturals n. The other way doesn’t work so we can’t use limits in n for limits in x but we can use limits in x for limits in n so we can use l'hospital for the x limit and then conclude it for this n limit as well.
Feels great to understand this stuff/see the solution just about before it happens. Thanks papa flammy for making us more smarter :D
Alternative approache for the "cheatung" part
sqrt(1-cos(2π/n)) =
sqrt(1-cos(2π/n))*sqrt(1+cos(2π/n)) /sqrt(1-cos(2π/n))
=
sqrt(1-cos²(2π/n))/sqrt(1+cos(2π/4n) =
sqrt(sin²(2π/n))/sqrt(1+cos(2π/n)) =
|sin(2π/n)|/sqrt(1+cos(2π/4))
for n >1 , 0 1,
cos(2π/n) > -1
So we can conclude
sqrt(1-cos(2π/n)) =
sin(2π/n)/sqrt(1+cos(2π/n))
In conclusion
L[ n*sqrt(2)*r*sqrt(1-cos(2π/n)) ] =
L[ n*sqrt(2)*r*sin(2π/n)/sqrt(1+cos(2π/n)) ] =
L[ r*sqrt(2)/sqrt(1+cos(2π/n)) ] * L [ n*sin(2π/n) ] =
r * L[ 2π * sin(2π/n)/(2π/n) ] =
2πr
I derived that the area of regular polygons approach that of circles myself
A circle is an infinitely cornered polygon without any sides
Aperigon: 2 plus 2 plus 2 plus 2 plus 2 plus 2 plus 2 plus 2
Numberblock 18: AHH!!!
It's somewhat easier with a = 2rsin(π/n),
so lim na= lim 2nr sin (π/n) = 2πr, using that lim sin x = x as x approaches 0.
Is a perimeter a number or a set of edges?
a number
Well if 0.999...=1 it's not all that strange. Like there's a reason mathematicians avoided infinity for a long time, - it doesn't follow normal rules which makes it weird.
Infinitely-Sided, already the interior .... :) :) :),
Did I hear you cannot use l'Hospital for a sequence? You can do it. There is nothing wrong about it, if you can extend the sequence to a function.
its easier with a non regular polygon. you can always draw your polygon in a way, such that the edges are perpendicular to the radius. And then its just pythagoras. And if ypu take the inscribed polygon and compare it to the outscribed one,you will see, that they dont differ for n->∞.
Feels like you might have saved yourself a whole lot of working by just bisecting your triangle to give a right triangle and using sin(pi/n) = a/2r . so a = 2rsin(pi/n) . Then you take n lots of that to get 2rn*Sin(pi/n) and then solve the limit as you did...
What I take from this video is that perimeter and circumference are the same thing!
KInda ye, that's probably why I messed up the terms before XD
I think polygons are a key to alot of things in math, I just used them to prove that sin^2(a+c) = sin(a)sin(b) + sin(c)sin(d) when a+b+c+d = pi
Each side length of a circle is epsilon. One over omega. Some infinitely small quantity.
But wait!
Unpopular opinion, there is only one size of infinity. This means the idea of infinitely large and infinitely small quantities no longer make sense.
Thus, a circle is not an infinitely sided polygon. However, it can be approximated as a polygon, getting an exact answer for the limit as the polygons side lengths go to zero. Precisely getting the circumference of the circle even though there are lengths involved, since they go to zero.
you said that using L'Hospital is not very clean, because we are dealing with discrete values. But if i'm not mistaken, to show that sin(x)/x approches 1 you also use L'Hospital and x is pi/n and therefore also discrete, so you run into the same problem
Noone ever uses L'Hospital to prove this limit lol
first papa flammy❤
"Infinitely sided polygon" mfs when i tell them about a oval
Brooo, you are good
Polygons are discreet circles
@12 min - LOL! Sorry but 1 divided by infinity is NOT zero. It may be infinitely close (as close as you like) to zero but it never quite gets there. You are perpetuating a fantasy similar to the one about 0.9999 ad infinitum being equal to 1 - which it isn't - they are ALWAYS different no matter what decimal place you wish to examine.
Things that are always different can never really be the same. But it can be, and is, useful to assume they are the same.
Limits are a 'convenient' mathematical concept that are useful in understanding. A circle with a 'rational' radius will have an irrational circumference because you multiple by Pi, an irrational number.
Likewise a circle with a 'rational' circumference will have an irrational radius.
As far as I know no one has ever proven/shown that a circle with an irrational radius can have a rational circumference. Put another way the result of multiplying two irrational numbers is NEVER, and can not be, rational.
Yet square roots (e.g. square root of 2) are useful conveniences.
I think a lot of the confusion arises because even mathematicians forget the difference between the contrived, and logical, world of mathematics and the physical world.
In the physical world there is no such thing as an 'exact' distance. How big is an atom? an electron, a proton? There is no such thing as a physical ruler that is exactly 12 inches long because physical things don't have exact dimensions.
damn
Did he say papa Pythagoras? :D
Geometry is beautiful :)
Tbh this is so cool I gonna come
nice
I fail to see the controversy
Flat-Earthers will like this video that argues that circles are fundamentally straight essentially everywhere.
Epic
As a physicist I just believe your mathematical ass if it says a circle is a polygon.
Disappointed you started with a 4 sided polygon. Everyone knows circles are just try hard triangles.
en.wikipedia.org/wiki/Pythagorean_cup