Question

8- Explain what happens to the number of elements in a line segment if we reduce the length of a ruler by a factor?

9- Explain what makes a fractal self-similar.

10- Explain what it means to have a fractal dimension between 1 and 2.

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Answer 1

9-Simply put, a fractal is a geometric object that is similar to itself on all scales. If you zoom in on a fractal object it will look similar or exactly like the original shape. This property is called self-similarity. An example of a self-similar object is the Sierpenski triangle show below.

As one looks closer we observe that the large triangle is composed of three smaller triangles half the size (side length) of the original, which in turn are composed of three smaller triangles, and so on, and so on. On all scales the Sierpenski triangle is an exactly self-similar object.

The property of self-similarity or scaling is closely related to the notion of dimension. In fact, the name "fractal" comes from property that fractal objects have fractional dimension.

A one dimensional line segment has a scaling property similar to that of fractals. If you divide a line segment into N identical parts, each part will be scaled down by the ratio r = 1/N (e.g. cut a line in two equal pieces and you have two lines each of half the original length). Similarly, a two dimensional object, such as a square, can be divided into N self-similar parts, each part being scaled down by the factor r = 1/N^(1/2) (i.e. if you cut a square into 4 equally-sized squares, then each new square is half the size (side length) of the original square).

The concept of self-similarity naturally leads to the generalization to fractional dimension. If one divides a self-similar D-dimensional object into N smaller copies of itself, each copy will be scaled down by a factor r, where

r = 1 / N^(1/D)

Now, given a self-similar object of N parts scaled down by the factor r, we can compute its fractal dimension (also called similarity dimension) from the above equation as

D = log (N) / log (1/r)

As an example, let us compute the dimension of the famous curve of Von Koch, which is sometimes referred to as the "Koch Snowflake." The Koch Snowflake is generated by a simple recursive geometric procedure:

• divide a line segment into three equal parts
• remove the middle segment (= 1/3 of the original line segment)
• replace the middle segment with two segments of the same length (= 1/3 the original line segment) such that they all connect (i.e. 3 connecting segments of length 1/3 become 4 connecting segments of length 1/3.)

To complete the shape, the above procedure is repeated indefinitely on each line segment on the side of a triangle. Images showing the procedure and complete Koch Snowflake are shown below.

Its fractal dimension is given from the definition of the curve: N = 4 and r = 1/3 (remember 4 segments each 1/3 size of the original line segment).

Dimension = log (4) / log (3) = 1.26

Another interesting property of the Koch Snowflake is that it encloses a finite area with an infinite perimeter.

10-In addition to visual self-similarity, fractals exhibit other interesting properties. For example, notice that each step of the Sierpinski gasket iteration removes one quarter of the remaining area. If this process is continued indefinitely, we would end up essentially removing all the area, meaning we started with a 2-dimensional area, and somehow end up with something less than that, but seemingly more than just a 1-dimensional line.

To explore this idea, we need to discuss dimension. Something like a line is 1-dimensional; it only has length. Any curve is 1-dimensional. Things like boxes and circles are 2-dimensional, since they have length and width, describing an area. Objects like boxes and cylinders have length, width, and height, describing a volume, and are 3-dimensional.

Certain rules apply for scaling objects, related to their dimension.

If I had a line with length 1, and wanted to scale its length by 2, I would need two copies of the original line. If I had a line of length 1, and wanted to scale its length by 3, I would need three copies of the original.

If I had a rectangle with length 2 and height 1, and wanted to scale its length and width by 2, I would need four copies of the original rectangle. If I wanted to scale the length and width by 3, I would need nine copies of the original rectangle.

If I had a cubical box with sides of length 1, and wanted to scale its length and width by 2, I would need eight copies of the original cube. If I wanted to scale the length and width by 3, I would need 27 copies of the original cube.

Notice that in the 1-dimensional case, copies needed = scale.

In the 2-dimensional case, copies needed = scale22.

In the 3-dimensional case, copies needed = scale33.