How Archimedes Trapped Pi
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- เผยแพร่เมื่อ 19 มิ.ย. 2024
- Archimedes was a brilliant Greek Mathematician living around 250 BC. He got a value for Pi whose accuracy would not be passed for another 400 years. The way he did it was to calculate the perimeter of a hexagon inside a circle, and one outside a circle. He then developed a formula to calculate the perimeter of a shape with twice as many sides, so 6 became 12 sides, became 24, 48, and finally 96 sides.
This video was made with Python & Manim
It was inspired by this amazing post by Better Explained
betterexplained.com/articles/...
Some of the heavier math on what Archimedes did can be found here
itech.fgcu.edu/faculty/clinds...
History of the computation of Pi
en.wikipedia.org/wiki/Chronol...
Images used in this video
Pi - pixabay.com/illustrations/pi-...
Archimedes By Domenico Fetti - archimedes2.mpiwg-berlin.mpg.d..., Public Domain, commons.wikimedia.org/w/index...
Ptolemy ptolemy image en.wikipedia.org/wiki/Ptolemy...
Parthenon
en.wikipedia.org/wiki/Ancient... - วิทยาศาสตร์และเทคโนโลยี
Clever fellow. I've also heard that he had deduced that the area surface of an inscribed sphere of a unit cube is about 52% of the surface area of that cube. The same can be said for the volume of a unit cube and the volume of an inscribed sphere of a unit cube. He compared measured volumes for the unit cube and sphere. Don't recall how he did the surface area.
Congratulations. I think your first video trial with Manim is very successful. Do you use Manim under Linux, Mac or Windows? I am having difficulty installing on Linux Mint, with anaconda.
I used it on Windows. I didn't have any trouble installing it except for Latex. My first MikTek installation didn't seem to work, and I had to uninstall it and follow a tutorial by TheoryOfBeethoven doing the full installation. But Manim itself went fine for me
@@dubiousinsights4008 Question, hkw do we know for sure Archimedes estimate was 99.9 percent accurate when we stil today don't know the full value of pi since it's an irrational number..no one knows all of its digits and I don't think any computer can ne programmed to know it exactly yet right since we don't know it??
@@leif1075 there are many ways to define pi. One way is to define an infinitely long summation in which you add smaller and smaller numbers following a set pattern. Say I have an approximation of 3.1415. Since you know the values to add are getting smaller and smaller, it's possible to tell that the amount of digits that you already have will not be affected by the newer terms. For example, the next term to add might be 0.00009. Thus we can confidentally say that the digits we already have are correct.
In the case of the 99% thing, you're right to think that it is not completely accurate. It's calculated by subtracting Archimedes' value from the digits of pi we know for certain and finding out how much it differs. But we obviously have to chop off that value somewhere in order to compare the two numbers. So as you add more digits to the true value of pi, the accuracy of the percentage difference also changes. Say 99.4555%. there's just not a lot of reason to care about 0.00000000001% difference so we say the approximation was 99% accurate.
Hope that's clear.
Are you still having trouble? I faced a lot of difficulty as well, but if your trouble is the same as mine, I may be able to help :)
Nice history lesson and excellent explanation of Archimedes' method for estimating the value of pi.
This is a really good video for a first hit at using manim - I am well impressed, it is much better than what I have achieved with it when I have played with the library. How long many hours do you think you put into it?
It has inspired me to find some time to play with it again.
I definitely started it after April 30 which is when I published my previous video, so less than 5 days of calendar time. I was up late a few of the nights, so my best guess is 10-12 hours. Thanks for the kind words!
@@dubiousinsights4008 At 1:50, why do you say puter hex is equilateral when those sides atent ewual?, and where iin earth do you get 1/radical 3...why woukd you just post that without deriving since no one would see where that's from..not even archimedes
Excellent video, just used it in my class :))
Wow I had no idea! This is so epic!!! 🔥
How many sides does a circle have? 2:40
Google: 0 (no sides)
This video: 96
About 3.14 if you're willing to approximate.
Infinitely many
The information was useful and I heard you made this video with python.... awesome
Yeah, all python. The library is called "Manim" for Mathematical Animation. It was originally developed by the 3 Blue 1 Brown guy th-cam.com/channels/YO_jab_esuFRV4b17AJtAw.html and has spun off into a community edition
docs.manim.community/en/stable/
I'm still learning it
Archimedes used calculus, although he did not realize it
Basically. He used BASIC math.
He did not realize it, seriously? He basically came up with the basics of analysis on his own, and used them for proper proofs, none of the wishy washy stuff!
Fantastic explaination
I have read in few books about history of math about how Archimedes was basically so close to inventing calculus but now I know precisely what authors were alluding to ;)
Disappointed the video wasn't 3 minutes 14 seconds long
c/2 = pi*r, r=1 = pi = 2*sqrt(2) * infsum(0->1) sqrt[1-sqrt(1-x_a)*sqrt(1-x_b)-x_a*x_b], there x are the piecewise linear segments of the circle line circumference
01:50 The Outer Hexagram, as I understand it, was solved by the 30-6-90 rule which involves Trigonometry, in this case, namely the value of TAN 60 degrees,
Amazing
The man had his moments 🥰
Nice video
woah that's really cool
Phew, glad they caught Pi in time before he went on another rampage. Is his prison sentence still going on?
Being that Pi is an irrational number, does this mean NOBODY knows what it's ACTUAL value is - all we can do is APPROXIMATE it's value?
But then if we HAVEN'T calculated it to infinty, how do we know for sure it's an irrational number - it may only APPEAR so due to our lack of computing power...
What if it suddenly became a pediodical number after 10 to the 1000th power digit?
10 t0 the 1 000 000 power?
10 to the 1 000 000 000 power and so on???
Irrational niumbers may not exist for all we know!
Am I missing something???
Actually, come to think of it, it MUST have a deternined value since the circumference of a circle DOES have a deternined value - that's not a theoretical problem... it can clearly be seen with the naked eye that the circumference of a circle has a DETERMINED value...
Yes/no, anyone?
dude your math is blowing my mind
stop YELLING
It has been proven mathematically. pi is an irrational transcendental number and goes forever
this is epic
How did he calculate the square root of something. That's more interesting
An octagon can obtain tan(22.5°).
Newton trapped Archimedes, who trapped the pi
Please show a sample calculation using any other system other than Hindu place value decimal system ( which was unknown to Greeks and Romans),
Please show a sample calculation using any system other than Hindu place value decimal system, which was not known to Greeks and Romans.
He trapped it using limits. In 400 BC! But no one calls his method limits. He did it again in the quadrature of the parabola.
0:56 that’s not a valid argument.
You can easily draw a shape such that the square is outside, but has a smaller perimeter.
It works with a circle, but not because the circle is inside.
You're right. That's because the actual argument involves the area of a unit circle, which is pi, and not the circunference of a circle with radius ½, which is also pi and you can actually use such an argument with areas.
I don't think Archimedes had access to trigonometry. You don't really show how you got your formula for going from 6 to 12 to 24 ... sides. I think this is a key step for understanding. Could you show where this came from? I do it a different way in my Archimedes pi video, but I would like to understand your method. (See th-cam.com/video/_rJdkhlWZVQ/w-d-xo.html for comparison.) Nice graphics. From other comments it sounds like you used a piece of software called Manim. Is that correct? I'll have to look it up.
smart boi
If we're gonna use the radius (as we should), the constant should be TAU!!! A hexagon has SIX sides, meaning a full circle. Using Pi only works for half the circle.
Try to slow down a bit
OK so how did Archimedes know the circumference 2pi radius? Without that one wouldn't even start to work out what pi was
Thats the definition of pi. We defined it such that its the ratio between the circumference and the diameter, which is 2x the radius.
@@feepentertainment6752 Agreed but somebody had to come up with that definition. So my question is who if not Archimedes and how was it shown to be correct
@@campbellmorrison8540 You just observe it's constant.
The circumference of a circle is just a bent line. Straighten it out and make it a leg of a right triangle with the diameter being the other leg. We scale this triangle and see that the ratio of C:d is constant.
@@feepentertainment6752 Im thinking my question is being misunderstood. Yes we know the ratio between the circumference and the diameter is a constant, BUT where did that fundamental fact come from? I could say the ratio between the area and the radius is a constant but who can show or disprove this? Where did Archimedes get his basic premise from is what I am asking
This is ridiculous the people that buult the Great pyramid knew about pi 2thousand before achrimdies
and? its not saying that he discovered pi
Everybody knew about pi, the only difference was how good was their approximation.
Go back in time and tell him its 355/113 😅
I thought he used Pokeball
Ha ha trusta cube in the thumbnail lmao ez L skill GD extreme demon infinity
how can you have 99.99% accuracy of a number that goes on forever? you cant even reach 1% accuracy of such a number.
I solved this problem 2 years ago. I wrote a paper.
ok
Archimedes wrote that paper 2000 years back 🥲
@@ashwinkumar9990 not with a square but with a hexagon.
Archimedes and Pi .........
Has the West ever heard the name " Aryabhatta " ?????
Show me ONE SQUARE OR CIRCLE OR A TRIANGLE ON EARTH ANYWHERE…
There are NO SUCH shapes
So basically, A circle is a 96-Sided Square lol
basically infinite sided
@@AvgCooki a circle has 0 sides
"a 96-sided square" please tell me you understand how this phrase is so wrong
you speak too fast
You can adjust video speed if you can't follow him. Which tbh is quite a normal pace.
Dramatic presentations should end now. I've calculated exact value of pi = 25/8.
Next you can go to the nearest machine shop and check circumference of 8 cm dia cylinder. It should measure 25 cm. 😅
🤡
Archmedes trapped it , Newton killed and gutted it .
yes bamb, period ma'am🤪
Knife goes in, guts come out
Mentioning those two names together is blasphemy.