Limited Area, Unlimited Perimeter

Date: 11/27/97 at 05:31:19
From: Rosa
Subject: Unlimited perimeter

Dr. Math,

There is a figure that has unlimited perimeter but limited area. What 
is the figure? Can you draw it for me?

Thank you very much!

Regards,
Rosa

Date: 11/27/97 at 12:27:01
From: Doctor Anthony
Subject: Re: Unlimited perimeter

You are probably thinking of a figure like the Koch Snowflake.  

To draw this you start with an equilateral triangle of side a.  
Now divide each side into three equal parts and on the middle third 
of each side construct an equilateral triangle pointing outwards from 
the original triangle. The total perimeter is now (4/3)(3a) = 4a.  

We now further subdivide each straight edge into 3 parts and construct 
equilateral triangles on the middle third of each side - again 
pointing outwards from the original figure. This process will enlarge 
the perimeter by a further factor of 4/3. There is no overlapping of 
the extra sides with those already present. The above process is 
repeated indefinitely, at each stage the perimeter being increased by 
a factor of 4/3, so we have:

 perimeter = (3a)(4/3)(4/3)(4/3) ......... to infinity.

Clearly the perimeter will increase without bound and become infinite, 
but the area of the figure will be less than the area of the 
circumcircle of the original equilateral triangle. So this figure has 
an infinite perimeter but a finite area. This is the defining property 
of a fractal shape that has self-similarity to an infinite depth.  
That is, you can enlarge a portion of the boundary to ANY magnitude 
and find shapes similar to the original figure.

For a picture and directions, see the Fractals unit on the Web by 
Cynthia Lanius:

  http://math.rice.edu/~lanius/frac/koch.html   

-Doctor Anthony and Sarah,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/