What is the Relationship Between Perimeter and Area in an Infinite Staircase?

[macbowes]

Alright, so I was just browsing 4Chan and I came across this post.

[PLAIN]http://img121.imageshack.us/img121/5374/1291537737867.jpg

I realize infinitely making corners out of corners may result in an approximation of a perfect curve, however, it will always be jagged and thus result in the difference between 4 and pi.

My question is, can you maintain the same perimeter while infinitely reducing the area? Cause that's what it appears to be doing in the picture.

 

[disregardthat]

https://www.physicsforums.com/showthread.php?t=450364

I realize infinitely making corners out of corners may result in an approximation of a perfect curve, however, it will always be jagged and thus result in the difference between 4 and pi.

Approximations by regular polygons will also always be jagged, but the limit is pi.

 

[macbowes]

The perimeter will polygon will always equal 4. The area, of the polygon, however, will come infinitely close to the area of the circle. At least that's my understanding.

 

[HallsofIvy]

That's a variation on the problem where you take more and more "stairsteps" from (0, 0) to (1, 1) getting a figure very close to the straight line from (0, 0) to (1, 1) but showing that the total length is always "2", not the length of the straight line, \(\displaystyle \sqrt{2}\). Essentially, the problem is that the stairsteps do not converge uniformly to the line.

 

[disregardthat]

Essentially, the problem is that the stairsteps do not converge uniformly to the line.

They do converge uniformly to the line.