All Science Fair Projects

Title: Endless Snowflake: Constructing Shapes with Infinite Perimeters and
Finite Areas

Objectives/Goals

To determine whether it is possible to construct shapes with infinite perimeters
but finite areas.

Methods/Materials

Using the Geometer's Sketchpad program and a Koch curve, I began with an
equilateral triangle and grew
a snowflake through various generations which, if taken to infinity, would have
an infinite perimeter but a
finite area. Using power series, I calculated what that area would be. I then
decided to develop a new
fractal curve based on a square shape rather than the traditional triangular
Koch curve.

Results

I calculated that this fractal pattern for the triangle would produce a figure
of infinite perimeter, but whose
area is only 1.6 times the area of the original triangle. Meanwhile, for the
square, the area for an infinite
perimeter shape would be 2.0 times the area of the original square.

Conclusions/Discussion

Ordinarily, when shapes are magnified, area grows faster than perimeter.
However, using the idea of
convergent series, it is possible to add ever-smaller increments of area such
that while the perimeter grows
to infinity, the sum of the areas remains finite. It may be possible to
generalize this approach to three
dimensions, producing a shape of infinite surface area and finite volume.

Summary Statement

My project uses the ideas of fractals and power series to construct a Koch
snowflake, and to explore new
families of curves with infinite perimeters and finite areas.