Fractal Geometry

Fractal Dimension

Ordinary dimension is always given as an integer. For instance, a line is one-dimensional, a square is two-dimensional, and a cube is three-dimensional. However, fractal dimension can be a non-integer, i.e. fractal dimension can be a fraction!

9 self-similar squaresTo understand why fractal dimension can be a non-integer, let's first determine why the dimension of a square is two. If we join the midpoints of opposite sides of a square, we will decompose the square into 4 self-similar copies of itself. Each self-similar copy has a magnification factor of 2. That is to say that if we magnify any of the copies by a factor of 2, we get the original square.

If we trisect the sides of the square, we decompose the square into 9 self-similar copies of itself, each with a magnification factor of 3. By continuing the process, a pattern emerges. See the table below.

Decomposition of a Square
Self-Similar Copies (S) Magnification Factor (N)
4 2
9 3
16 4
25 5

Now suppose that on each face of a cube, we join the midpoints of opposite edges. The cube is decomposed into 8 self-similar copies of the itself, each with a magnification factor of 2. If we trisect opposite edges of each face, we decompose the cube into 27 self-similar copies of itself, each with a magnification factor of 3.

Continue the process and the pattern becomes clear. The number of self-similar copies equals the cube of the magnification factor, i.e. S= N 3 .

For a square, a 2-dimensional figure, the exponent of the magnification factor is 2. For a cube, a 3-dimensional figure, the exponent of the magnification factor is 3. It appears that the exponent of the magnification factor is the dimension. In general, if a figure is decomposed into S self-similar parts, each with magnification factor N, then the exponent of the magnification factor is the dimension. Let D = dimension and we have

N D =S

Solving for D we find

log N D = log S D log N = log S D = log S log N

Thus, we can define fractal dimension as follows.

dimension= log( number of self-similar copies ) log( magnification factor )

Dimension of the Sierpinski Triangle

One of the simplest geometric fractals is the Sierpinski triangle. It is constructed by joining the midpoints of the sides of an equilateral triangle and removing the triangle formed. Repeat the process to get successive stages of the Sierpinski triangle.

Stages of the Sierpinski triangle
Stages of the Sierpinski triangle.

The second figure consists of 3 self-similar copies of the original triangle, each with a magnification factor of 2. The third stage of the Sierpinski triangle consists of 9 self-similar copies, each with magnification factor 4. This kind of self-similarity is a defining characteristic of fractals. So what is the dimension of the Sierpinski triangle?

Since the Sierpinski triangle consists of 3 self-similar copies, each with a magnification factor of 2, we compute the dimension as

D= log3 log2 1.5850

However, the third stage of the Sierpinski triangle consists of 9 self-similar copies, each with magnification factor 4. So we can compute the dimension as

D= log9 log4 = log 3 2 log 2 2 = 2log3 2log2 = log3 log2 1.5850

As before, the dimension is roughly 1.59.

Dimension of the Koch Snowflake

The Koch snowflake is constructed by first trisecting each side of an equilateral triangle, forming 3 self-similar parts. The self-similar parts each have a magnification factor of 3. Each side of the triangle is then reconstructed using 4 self-similar parts. Thus, the dimension of the Koch snowflake is

D= log4 log3 1.2619

Conclusion

This brief introduction to fractal geometry barely scratches the surface of this intriguing branch of modern mathematics. One often gets the impression that all mathematics originated in the distant past. Names such as Isaac Newton (1642 - 1727) and Euclid (circa 300 BC) come to mind as the Who's Who of mathematicians. However, fractal geometry would not have been possible without the advent of modern computer technology. Fractal geometry is a new and still emerging branch of mathematics. Diverse, groundbreaking, and completely unexpected practical applications of fractal geometry are constantly surfacing.

Hopefully you are now inspired to dig deeper and learn more about this relatively new and fascinating area of mathematics. If not, then watch this PBS video about fractals. That should do the trick.

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