Geometry for Elementary School/Fractals

All the previous constructions we considered had one thing in common. The constructions were ended after a final number of steps. When one recalls that mathematicians actually used a ruler and compass in order to execute the constructions, this requirement seems to be in place. However, when we remove this requirement we can construct new interesting geometric shapes. In this chapter we will introduce two of them. Note that these shapes are not part of Euclidian geometry and were considered only years after its development.

Cantor Set

For a full overview of Cantor set see the article at wikipedia on which this section is based. The Cantor set was introduced by German mathematician Georg Cantor.

The Cantor set is defined by repeatedly removing the middle thirds of line segments. One starts by removing the middle third from the unit interval [0, 1], leaving [0, 1/3] ∪ [2/3, 1]. Next, the "middle thirds" of all of the remaining intervals are removed. This process is continued for ever. The Cantor set consists of all points in the interval [0, 1] that are not removed at any step in this infinite process.

What's in the Cantor set?

Since the Cantor set is defined as the set of points not excluded, the proportion of the unit interval remaining can be found by total length removed. This total is the geometric series

${\frac {1}{3}}+{\frac {2}{9}}+{\frac {4}{27}}+{\frac {8}{81}}+\cdots =\sum _{n=0}^{\infty }{\frac {2^{n}}{3^{n+1}}}={\frac {1}{3}}\left({\frac {1}{1-{\frac {2}{3}}}}\right)=1.$

So that the proportion left is 1 – 1 = 0. Alternatively, it can be observed that each step leaves 2/3 of the length in the previous stage, so that the amount remaining is 2/3 × 2/3 × 2/3 × ..., an infinite product which equals 0 in the limit.

From the calculation, it may seem surprising that there would be anything left — after all, the sum of the lengths of the removed intervals is equal to the length of the original interval. However a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing open sets (sets that do not include their endpoints). So removing the line segment (1/3, 2/3) from the original interval [0, 1] leaves behind the points 1/3 and 2/3. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So we know for certain that the Cantor set is not empty.

The Cantor set is a fractal

The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 1/3 and translated.

Koch curve

For a full overview of Koch curve see the article at wikipedia on which this section is based.

The Koch curve is a one of the earliest fractal curves to have been described. It was published during 1904 by the Swedish mathematician Helge von Koch. The better known Koch snowflake (or Koch star) is the same as the curve, except it starts with an equilateral triangle instead of a line segment.

One can imagine that it was created by starting with a line segment, then recursively altering each line segment as follows:

1. divide the line segment into three segments of equal length.
2. draw an equilateral triangle that has the middle segment from step one as its base.
3. remove the line segment that is the base of the triangle from step 2.

After doing this once the result should be a shape similar to the Star of David.

The Koch curve is in the limit approached as the above steps are followed over and over again.

The Koch curve has infinite length because each time the steps above are performed on each line segment of the figure its length increases by one third. The length at step n will therefore be (4/3)ⁿ.

The area of the Koch snowflake is 8/5 that of the initial triangle, so an infinite perimeter encloses a finite area.