Thank you so much for making these available on TH-cam, I'm sure many children back in the day fell asleep to this, but I'm glad I have the freedom to watch it on my own accord.
i am a kid (12) learning from this and i don't feel like sleeping in fact i am watching this because i can't sleep thinking it might make me productive soo maybe not all kids
No mention of the series invented in India (I think) a long time ago, using fractions of 4 divided by odd integers: 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13... = Pi
Pi is a circle whose diameter is one. To find the circumference of pi whose diameter is one attach the pi to the side of square one. Next draw a line from the far corner of square one through the center of pi to intersect the point on pi. You’ll notice a triangle is formed inside of square one. Use Pythagorean theorem to solve for its hypotenuse. To this value add the radius of pi which is one half. You’ll note that this line has a length of Phi. Making the far corner of square one the center of a new circle make its radius phi. Extend the diameter of pi which is the side of square one to intersect a point on the circle whose radius is phi. From this point draw a line to the far corner of square one and use Pythagorean theorem to find the length of the rise of this triangle to be the square root of phi. The perimeter of square one is four. Divide four by the square root of phi to see the value of pi. Also the area of pi is the reciprocal of the square root of phi. Check me to see if it isn’t true.
@@johns5779 In the first problem, whether two numbers _m_ and _n_ are co-prime or not depends on the slope of a line passing through (0,0) and (m,n). And the arc-tangent function establishes a one-to-one relationship between slope and direction. The probability of a needle crossing a line in the second problem also varies depending on the direction in which the needle is pointing. And through group theory, the locus of all possible directions forms a circle.
@@denelson83 You are right, but its a stretch to discuss co-prime numbers without explaining how these relate to the coordinates of a visible lattice point. I know how they relate, now that you mention it, but I always forget that this is the case .
The story of ip - 0:43 the girl are measuring circle things like backet etc . But I am think about that the area of circle is 2 π r and circle is made on coplaner , it is not a 3 degree figure
I'm missing something and it's bothering me. At 3:06 the volume of a cone is found by rotating a right triangle by 2pi. But if you take that same right triangle and duplicate it, (then flip it to make a rectangle of double area) and then rotate THAT by the same 2pi, you produce a cylinder with the same base and height. Since the rectangle we just rotated is double the area of the origional triangle, why is the volume of the resulting cylinder not double the volume of the cone? Will someone please explain the fault in my reasoning? I know I'm messing up I just don't know how.
Rotating congruent figures around an axis may produce different solids, with different volumes depending on how the figures are oriented with respect to the axis of rotation. As a simple example, picture a 2x1 rectangle. Revolve it around the short edge and you get a tuna can shaped cylinder with radius of 2 and height of 1...volume of pi*(2^2)*1 = 4pi. Revolve around the longer edge and we see a soda can shape with a different volume...radius 1, height 2, volume of pi*(1^2)*2= 2pi. In your case, the two right triangles, oriented as they are with respect to the axis, sweep out different solids with different volumes when revolved. The easier one to envision is the cone. The other triangle does not form a cone when revolved about the same axis. For lack of a better term, it forms "the rest of a cylinder"...a cylinder minus a cone, a different solid with a different volume. One more common sense thought...in the second case, more of the area of the triangle is a greater distance from the axis of rotation, so one might expect a different volume upon rotation.
For the combined duplicate triangles, you're rotating them about an axis that passes midway through the base of each triangle, which generates a different solid than would be obtained if you'd rotated them (together) about an axis along the side ('height') of just one triangle.
Hmmm...I have a circle with the following dimensions: *C = 1 inch* *D = 1 inch* Following the formula... *Circumference ÷ Diameter = π* *1 inch ÷ 1 inch = 1* Therefore π = 1 Looks like this simplistic theory has been refuted and shows the value of π to be dynamic rather than static.
The praying mantis 3d model on the last video was probably the most difficult. These are simple 2d graphics and it's the late 80's nothing really groundbreaking at that point.
Think about this more simple case: If you rotate a square of side length 1, then you get a cylinder with volume pi. If you rotate a rectangle with height 1 and side length 2 (basically two of our original squares sitting side by side), then we get a cylinder with volume 4pi. Clearly the area of the rectangle is twice that of the original square, yet the respective cylinders have a ratio of 4. This concept is talked about in the first episode of the series. You should not expect that doubling an area of some 2d shape should correspond to a doubling of volume of some generated 3d shape. In a lot of cases, this is not true!
![The biggest hand calculation in a century! [Pi Day 2024]](/img/n.gif)
Thank you so much for making these available on TH-cam,
I'm sure many children back in the day fell asleep to this, but I'm glad I have the freedom to watch it on my own accord.
I dont know why kids would fall asleep tbh. These videos are honestly great and really informative and the visuals/illustrations make it more engaging
i am a kid (12) learning from this and i don't feel like sleeping in fact i am watching this because i can't sleep thinking it might make me productive soo maybe not all kids
No mention of the series invented in India (I think) a long time ago, using fractions of 4 divided by odd integers:
4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13... = Pi
See video at 4:46 ... it's the same as the top infinite series after multiplying both sides by 4 ... except you've switched both sides.
17:44 I remember this problem from my Math525 class in college...
Great video! Peace out
Pi is a circle whose diameter is one. To find the circumference of pi whose diameter is one attach the pi to the side of square one. Next draw a line from the far corner of square one through the center of pi to intersect the point on pi. You’ll notice a triangle is formed inside of square one. Use Pythagorean theorem to solve for its hypotenuse. To this value add the radius of pi which is one half. You’ll note that this line has a length of Phi. Making the far corner of square one the center of a new circle make its radius phi. Extend the diameter of pi which is the side of square one to intersect a point on the circle whose radius is phi. From this point draw a line to the far corner of square one and use Pythagorean theorem to find the length of the rise of this triangle to be the square root of phi. The perimeter of square one is four. Divide four by the square root of phi to see the value of pi. Also the area of pi is the reciprocal of the square root of phi. Check me to see if it isn’t true.
15:49 - Believe it or not, those two problems have everything to do with circles.
How?
@@johns5779 In the first problem, whether two numbers _m_ and _n_ are co-prime or not depends on the slope of a line passing through (0,0) and (m,n). And the arc-tangent function establishes a one-to-one relationship between slope and direction. The probability of a needle crossing a line in the second problem also varies depending on the direction in which the needle is pointing. And through group theory, the locus of all possible directions forms a circle.
@@denelson83 You are right, but its a stretch to discuss co-prime numbers without explaining how these relate to the coordinates of a visible lattice point. I know how they relate, now that you mention it, but I always forget that this is the case .
The story of ip - 0:43 the girl are measuring circle things like backet etc . But I am think about that the area of circle is 2 π r and circle is made on coplaner , it is not a 3 degree figure
I'm missing something and it's bothering me. At 3:06 the volume of a cone is found by rotating a right triangle by 2pi. But if you take that same right triangle and duplicate it, (then flip it to make a rectangle of double area) and then rotate THAT by the same 2pi, you produce a cylinder with the same base and height. Since the rectangle we just rotated is double the area of the origional triangle, why is the volume of the resulting cylinder not double the volume of the cone? Will someone please explain the fault in my reasoning? I know I'm messing up I just don't know how.
Rotating congruent figures around an axis may produce different solids, with different volumes depending on how the figures are oriented with respect to the axis of rotation. As a simple example, picture a 2x1 rectangle. Revolve it around the short edge and you get a tuna can shaped cylinder with radius of 2 and height of 1...volume of pi*(2^2)*1 = 4pi. Revolve around the longer edge and we see a soda can shape with a different volume...radius 1, height 2, volume of pi*(1^2)*2= 2pi.
In your case, the two right triangles, oriented as they are with respect to the axis, sweep out different solids with different volumes when revolved. The easier one to envision is the cone. The other triangle does not form a cone when revolved about the same axis. For lack of a better term, it forms "the rest of a cylinder"...a cylinder minus a cone, a different solid with a different volume.
One more common sense thought...in the second case, more of the area of the triangle is a greater distance from the axis of rotation, so one might expect a different volume upon rotation.
For the combined duplicate triangles, you're rotating them about an axis that passes midway through the base of each triangle, which generates a different solid than would be obtained if you'd rotated them (together) about an axis along the side ('height') of just one triangle.
Can anyone else hear a very faint voice during the first segment of the video?
But why 22/7 was referred as pi
poonam tiwari because if you divide 22 by 7 you get 3.14
who's speaking in the background!
Hmmm...I have a circle with the following dimensions:
*C = 1 inch*
*D = 1 inch*
Following the formula...
*Circumference ÷ Diameter = π*
*1 inch ÷ 1 inch = 1*
Therefore π = 1
Looks like this simplistic theory has been refuted and shows the value of π to be dynamic rather than static.
Pie sure taste good❤
good
I pity the Caltech video editor who probably had to code all these graphics from scratch on some old MacII or something!. haha.
The praying mantis 3d model on the last video was probably the most difficult. These are simple 2d graphics and it's the late 80's nothing really groundbreaking at that point.
helloooooo?
Anybody? :(
Hello
yeet
hellooo
Think about this more simple case: If you rotate a square of side length 1, then you get a cylinder with volume pi. If you rotate a rectangle with height 1 and side length 2 (basically two of our original squares sitting side by side), then we get a cylinder with volume 4pi. Clearly the area of the rectangle is twice that of the original square, yet the respective cylinders have a ratio of 4.
This concept is talked about in the first episode of the series. You should not expect that doubling an area of some 2d shape should correspond to a doubling of volume of some generated 3d shape. In a lot of cases, this is not true!