You can draw a line of that length, but how would you differentiate between 3" and 3.14159", when you only have capability to measure to the closest .1".
If all you have is a ruler marked for Hundredths of an inch, how would you know if the line is 3.14159"? The best you could do is say the line is 3.14", and you don’t know if it’s 3.14159" or 3.14172".
See what I’m saying? You may be able to calculate to 27 decimal places, but with a ruler that only has Hundredths of an inch, the rest is nonsense to you. You can measure to 2 decimal places, and the remaining 25 decimal places may as well be truncated, you have no purpose for them!
The only issue is drawing it and knowing that it is the right length. And you can do that in a variety of ways.
The first proof that an irrational number can be drawn was by some bloke back in the Pythagorean days when the establishment said that it couldn’t be done - but he simply drew a 1 unit square with a diagonal and proved that the diagonal had to be an irrational number - (\sqrt2) (because it had to be both an even and also an odd number at the same time).
To get (\pi) - just roll a 1 unit disc across a paper exactly once and you have exactly (\pi) length.
Pi is not a length, it is a ratio of the length of the diameter to the circumference.
So if you say the radius is 1, then the diameter is 2. Now, in order to figure out the length of the line, you have to multiply Pi x 2. What number are you going to use for Pi to multiply by 2?
A diameter 1 circle has a .5 radius. A diameter 1 circle has a circumference of Pi x 1. In order to find the length of the line of 1 revolution of that circle, you have to multiply Pi x 1. So what number are you using for Pi, 3.14? 3.14159? How many units is the line in length? 3.14 units? 3.14159 units?
How long is the line if you roll it 5 revolutions?
In order to have a unit length of the line, you have to have a number for Pi. Use 3.14, or 3.14159, it’s up to you how accurately you are going to measure the unit length.
If you use 3.14 for Pi, 5 revolutions and the line would be 3.14 x 5 units in length.
If you use 3.14159 for Pi, 5 revolutions and the line would be 3.14159 x 5 units in length.
Pi is just a symbol, a placeholder for the ratio of the diameter to the circumference. The ratio has to have a number, like the gear ratio in a gear box has a 3.14:1 ratio. You have to specify the ratio, you can’t just say it has a ratio, and leave it at that.
If you want to try to measure (\pi) to put it into decimal form - that is your problem - and good luck with that. The puzzle is merely to draw one - not to try to measure it into decimal form.
You know that a 1 centimeter disc has exactly (\pi) length circumference. You know that because that is what defines what (\pi) means. If you want to get a micrometer to measure it with - I don’t know why - but that is a problem for you to figure out. So far in doing that they have found something like 10 million digits or more so far.
Pi is a RATIO, it is not a length. If you know the length of the diameter of a circle is 1 meter, and you know that the ratio of the circumference of a circle to the diameter of that circle is 3.14159:1, then you know that the circumference of that circle must be 3.14159 x 1 meter. So a 1 meter diameter circle has a circumference of 3.14159 meters. If you use 3.14:1 for Pi, then the circumference is 3.14 of that circle. You used 2 decimal places for pi, so that is what you get for the circumference too. You want to use 29 decimal places for Pi, then the circumference has 29 decimal places too. It’s how accurate you want to get, and how accurate you can measure.
If you say the circumference is 3.14 meters, because you measured the line of 1 revolution of that circle, is the diameter of the circle exactly 1 meter? If the line is 3.14159 meters long, is the diameter 1 meter of that circle. It goes both ways!
1/3 is not a ratio, it is a fraction. If 1/3 was a ratio then it would be 1/3:1, like the gear ratio is 3.73:1. 3.73 revolutions of the input shaft means 1 revolution of the output shaft. It is the ratio 3.73:1. See how that is expressed, as 3.73 to 1?
1/3 is not a ratio. It is not 1/3:1, it is 1 part of 3 parts.
Pi is a RATIO, it is 3.14159…:1. For every 1 meter of diameter, the circumference is 3.14159 times greater. If the diameter is 3, then the circumference is 3.14159 x 3. Pi x D. Pi needs a number, and D needs a number, then you do the math to get the circumference.
The circumference string is 3.14159 times longer than the diameter string. The input shaft spins 3.73 times the revolutions as the output shaft. Ratio! 3.14159:1 and 3.73:1. Two ratios!
Circumference / D = Pi.
Measure the circumference. Measure the diameter. Divide the circumference by the diameter an you will find the ratio Pi, which is 3.14159:1 and on and on. Of course, you won’t be able to measure past a dozen or so decimal places when measuring the diameter or the circumference.
Why go to all that trouble, though? Why reinvent the wheel? Just use the known ratio 3.14159:1 for Pi, measure the diameter and do the math to find the circumference.
Or roll the disc 1 revolution, measure the line, and divide by 3.14159 to find the diameter.
The square doesn’t have the same area as the circle, so it’s not a “squared circle.”
Notice in mine, the purple area is shared by both. The remaining red parts and blue parts need to have the same area. So it boils down to if 1 red area is the same as 1 blue area, since the 4 red parts are the same area, and the 4 blue parts are the same area (4 red square corners, and 4 blue circle protrusions).
So do you think 1 red part has the same area as 1 blue part? If you could reshape a blue area into the shape of a red shape, while maintaining the same amount of blue area, just reshaped differently to the shape of a red shape, would it fit perfectly?
If we knew the area of a blue part and the area of a red part we could see if they are equal area. But I don’t know how to calculate that. Maybe somebody can help??
That would be pretty easy except because of the way you drew it. Because you just guessed until it looked right - you can’t know exactly where those vertical center lines are.
If I just assume that those center lines are separated by one radius - I can calculate any of those areas.
The angle of the arc indicated by the red would be (30^o). That is 1/12 of the whole circle.
So the area of the sector would be (\frac{\pi}{12} )
The side length of the square would then be (2 * sin(60^o) )
And the straight edge of the red would be (sin(60^o) -0.5)
The area of the purple portion of the square would be (4*(0.5*sin(60^o) + \frac{\pi}{12}))
The area of the square would be ( (2 * sin(60^o))^2 )
And that means that one of your red areas is (\frac{ (2 * sin(60^o))^2 - (4*(0.5*sin(60^o) + \frac{\pi}{12}))}{4} )
I didn’t guess at anything. EVERY line is formed from an intersection. The corners of the square are the intersections of those crossing lines at the corners. Every line is dead nuts accurate with intersections.
Every line is point to point somewhere along those lines. They are all DEAD NUTS ACCURATE, no guessing involved.
