Discover the Pi Paradox: The Infinite Perimeter of a Circle with a Diameter of 1

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In summary: But here is the gist:The problem is that the integral you are approximating does not converge to a constant. Instead, it converges to a function which is a power of pi. So, even though the top of the rectangles get closer and closer to the curve, the total length of the top of the rectangles does not always equal 1/2.
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jamester234
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Suppose you have a circle with a diameter of 1. If you draw a square with all four sides touching the circle, the perimeter of the square is 4. Now suppose you indent each corner of the square so that they all touch the circle-this will make a cross shape, and the perimeter of it is still 4. Now suppose you indent each corner of the cross so that they again all touch the circle. You can see now that the square is becoming more like a circle, and yet the pereimeter is still 4. This can be done to infinity, therefore, pi equals 4!
 
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jamester234 said:
Suppose you have a circle with a diameter of 1. If you draw a square with all four sides touching the circle, the perimeter of the square is 4. Now suppose you indent each corner of the square so that they all touch the circle-this will make a cross shape, and the perimeter of it is still 4. Now suppose you indent each corner of the cross so that they again all touch the circle. You can see now that the square is becoming more like a circle, and yet the pereimeter is still 4. This can be done to infinity, therefore, pi equals 4!

the perimeter of your square would become roughly equal to the circumference if your circle. you still got some math to do before you get pi.
 
  • #3
Nope. You have not proved the length of the circle (your limiting curve) is 4. You have proved, instead, that the length of the limit is not equal to the limit of the lengths. No need to use a circle to prove this: For example, you can use stairsteps converging to a sloping line segment.
 
  • #4
That demonstration is obviously wrong because it is contradicted by the proof that [itex] \pi = 2 [/itex]. Consider a circle of diameter 1 with its center at the origin and look at the part of it that lies in the 1st quadrant. Divide up the area under the curve of the circle into small rectangles in the manner that people do when they approximate integrals. The tops of these rectangles become closer and closer to the curve as their widths approach zero. The total of the lengths of the top of the rectangles is always 1/2 no matter how small they are. So the total perimeter of the circle is 2 when we sum over all 4 quadrants.
 
  • #5
This exact problem was just thrashed to death in another math forum on here.
 

1. What is the Pi Paradox?

The Pi Paradox is a mathematical concept that describes the infinite perimeter of a circle with a diameter of 1. This means that no matter how small the circle, its perimeter will always be infinite, unlike any other shape.

2. How is this possible?

This paradox arises from the fact that the number pi (π) is an irrational number, meaning it cannot be expressed as a finite decimal or fraction. Therefore, the circumference of a circle cannot be measured precisely, and it will always be infinitely precise.

3. Why is this important?

The Pi Paradox is important because it challenges our understanding of infinity and the concept of measurement. It also has practical applications in various fields, including mathematics, physics, and engineering.

4. Can the Pi Paradox be proven?

While the concept of the Pi Paradox has been widely accepted, it cannot be proven mathematically. This is because pi is an irrational number, and its exact value cannot be determined.

5. Are there any real-life examples of the Pi Paradox?

Yes, the Pi Paradox can be seen in real-life examples, such as the coastline paradox, where the measured length of a coastline becomes longer as the unit of measurement decreases. It also has applications in fractal geometry, where the same pattern is repeated infinitely on different scales.

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