Limited Area, Unlimited PerimeterDate: 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/ |
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