It's a 12 minute tour de force of nonsense and non sequiturs that rambles between ancient history, conspiracy theory, Biblical appeals, and Indiana Law from 1897 (HB 246).
But let's just take a quick look at what PI actually is and why it is certainly neither 4 nor 3.
PI is JUST the ratio between the circumference of a circle and the diameter of that circle. That's it, there is nothing magic about it. All circles, no matter their size, have the same ratio. Which is pretty cool because it means that if we know the radius of the circle we can figure out the circumference and area also, or vice versa. All it really means there is a relationship between these measurements.
Finding a close approximation of PI is very easy.
Just find something that is very round and then take a string that doesn't stretch (fine thread works) and wrap it around your round cylinder (a tube or pipe that is uniform works, the larger the better) and mark it exactly once around -- measure that length with a finely marked ruler. That is the Circumference. You can also try to "roll" it along a flat surface and carefully mark & measure "once around".
Now measure the Diameter of the outside of that cylinder very carefully - just make sure you go as straight across as possible.
Now divide the length of the string by the diameter of the cylinder: C / D or Circumference / Diameter.
That's it - you have an approximation of Pi and it will very clearly be neither 3 nor 4.
Depending on how accurately you measured you'll find it's pretty close to 3.14 etc etc. The more accurately you measure it, the closer you'll get to 3.14159265 etc etc
I used a soda can. I cut a strip out of it so I could flatten that strip and measure it, it was ~207.5mm. I then measured across the can and got 66mm; 207.5/66 = 3.1439...
Beverage Can wiki says it's actually 2.6" wide which is ~66.04mm, which gives us: 3.1420 - a very good approximation without any hard work. I did this in under 5 minutes. We're far enough from 3 that PI is clearly not 3 - and we're very far from 4 so that's just ridiculous.
I went back today and remeasured everything in inches and took pictures. This time I got approximately 8 5/32" by 2.6" which gives me a ratio of 3.137. Even though measuring things super accurately is hard, we are nowhere even near 3.
Little bit of error introduced here because the edge didn't quite line up:
For our circumference, I laid the strip out flat and measured it - was a little off here:
And on the other end we have 1/32nd inch marks (that's why I shifted it by 1/4"):
Can you get closer? Try to borrow some very accurate calipers and see.
And it's also very easy to verify that doesn't matter if you measure in millimeters, inches, or anything else. The ratio doesn't depend on the units used -- just make sure you use the same units when you measure (or convert them to the same units).
You can also get it using an approximation of PI using the function |sqrt(1-x^2)| which gives you an area of π/4 over the range x = 0 to x = 1 (because it makes a little 1/4 circle).
Take few values and just treat them like little rectangles and add up the area - like this image:
I did a little example in Excel, I use the average point between each sample for my height and each width is 0.1 units. Here are the Excel details so you can recreate it. I'm just doing one quarter of the circle - same function as above but x just goes from 0 to 1.
This gives us the following calculation:
All we've done here is add up all the little areas of each rectangle.
We're off, as expected, a little bit because our pieces are too big. Break those up into 100 pieces and you get much closer (I got 3.140417032). Break it up into 1000 pieces and you get even closer. So you are adding up smaller and more accurate pieces -- when you get to an infinite number of pieces you'll get the exact value of PI. But it doesn't even matter what the EXACT value of PI is here -- the point is it just absurd to say it's 3.
There are lots of other ways to approximate Pi
But let's just take a quick look at what PI actually is and why it is certainly neither 4 nor 3.
What is PI?
PI is JUST the ratio between the circumference of a circle and the diameter of that circle. That's it, there is nothing magic about it. All circles, no matter their size, have the same ratio. Which is pretty cool because it means that if we know the radius of the circle we can figure out the circumference and area also, or vice versa. All it really means there is a relationship between these measurements.
Approximating PI
Finding a close approximation of PI is very easy.
Just find something that is very round and then take a string that doesn't stretch (fine thread works) and wrap it around your round cylinder (a tube or pipe that is uniform works, the larger the better) and mark it exactly once around -- measure that length with a finely marked ruler. That is the Circumference. You can also try to "roll" it along a flat surface and carefully mark & measure "once around".
Now measure the Diameter of the outside of that cylinder very carefully - just make sure you go as straight across as possible.
Now divide the length of the string by the diameter of the cylinder: C / D or Circumference / Diameter.
That's it - you have an approximation of Pi and it will very clearly be neither 3 nor 4.
Depending on how accurately you measured you'll find it's pretty close to 3.14 etc etc. The more accurately you measure it, the closer you'll get to 3.14159265 etc etc
Coke Can Method
I used a soda can. I cut a strip out of it so I could flatten that strip and measure it, it was ~207.5mm. I then measured across the can and got 66mm; 207.5/66 = 3.1439...
Beverage Can wiki says it's actually 2.6" wide which is ~66.04mm, which gives us: 3.1420 - a very good approximation without any hard work. I did this in under 5 minutes. We're far enough from 3 that PI is clearly not 3 - and we're very far from 4 so that's just ridiculous.
I went back today and remeasured everything in inches and took pictures. This time I got approximately 8 5/32" by 2.6" which gives me a ratio of 3.137. Even though measuring things super accurately is hard, we are nowhere even near 3.
Little bit of error introduced here because the edge didn't quite line up:
Another small amount of error introduced because my cut across the can wasn't perfect so it's slightly tilted:
For our circumference, I laid the strip out flat and measured it - was a little off here:
And on the other end we have 1/32nd inch marks (that's why I shifted it by 1/4"):
Can you get closer? Try to borrow some very accurate calipers and see.
Units Don't Matter to PI
And it's also very easy to verify that doesn't matter if you measure in millimeters, inches, or anything else. The ratio doesn't depend on the units used -- just make sure you use the same units when you measure (or convert them to the same units).
Approximating PI in Excel By Adding Up Little Rectangles
You can also get it using an approximation of PI using the function |sqrt(1-x^2)| which gives you an area of π/4 over the range x = 0 to x = 1 (because it makes a little 1/4 circle).
Take few values and just treat them like little rectangles and add up the area - like this image:
I did a little example in Excel, I use the average point between each sample for my height and each width is 0.1 units. Here are the Excel details so you can recreate it. I'm just doing one quarter of the circle - same function as above but x just goes from 0 to 1.
| Field | Excel Formula |
|---|---|
| |sqrt(1-x^2)| | =SQRT(1-(A2^2)) |
| Area: w x h | =(ABS(A3-A2)*ABS((B2+B3)/2)) |
| area*4 | =SUM(C3:C12)*4 |
This gives us the following calculation:
| x | |sqrt(1-x^2)| | Area: w x h |
|---|---|---|
| 0.0 | 1.00000000 | |
| 0.1 | 0.99498744 | 0.099749372 |
| 0.2 | 0.97979590 | 0.098739167 |
| 0.3 | 0.95393920 | 0.096686755 |
| 0.4 | 0.91651514 | 0.093522717 |
| 0.5 | 0.86602540 | 0.089127027 |
| 0.6 | 0.80000000 | 0.083301270 |
| 0.7 | 0.71414284 | 0.075707142 |
| 0.8 | 0.60000000 | 0.065707142 |
| 0.9 | 0.43588989 | 0.051794495 |
| 1.0 | 0.00000000 | 0.021794495 |
| Area*4 | 3.104518326 |
All we've done here is add up all the little areas of each rectangle.
We're off, as expected, a little bit because our pieces are too big. Break those up into 100 pieces and you get much closer (I got 3.140417032). Break it up into 1000 pieces and you get even closer. So you are adding up smaller and more accurate pieces -- when you get to an infinite number of pieces you'll get the exact value of PI. But it doesn't even matter what the EXACT value of PI is here -- the point is it just absurd to say it's 3.
There are lots of other ways to approximate Pi
No comments:
Post a Comment