Geometri Limit

A Geometri Limit

Star with a regular triangel of perimeter. The midlines of the triangel cut it into four equal triangles,each being half as big as the original one. There, the perimeter of each, the midle one in particular. Now apply the same procedure to the middle triangel. The small triangel at the center will have the parimeter. Continuition of the proses generates a squence are not of triageles inscribed ino each other with diminishing periods,for the ( n + 1 )st triangel.

The prosedur can be modified without inflicting any change in the sequence ofperimeter. One can readily observe that the triangeles in the sequence are not only inscribed into each other but olso incribed into inscribed circle of their immediate predecessors. Inside those  circles te triangel may be rotatedwithout changing perimeter.

So what we are doing is this. In given triangel we inscribe a circle. Into tht circle we inscribe a triangel, than again inscribe a circle, and again inscribe a triangel, and so n. Perimeter of the resulting riangel oform a sequences, obviously the jenerc term p/2N of this sequence tends to 0 as N grows. The mathematical notation for that fact is

 

       Lim         p  = 0

 N→∞       2 N

It reads, the limit of p/2N as N tend to infinity is 0.

When we keep inscribing circles into triangles and triangles into circles,circles shrink into a point.