Self-similarity

We now examine objects of several different sizes. First, figure 2 presents five triangles labeled 1 through 5, which are similar right-angled equilateral triangles. We find that two neighboring triangles become smaller as we look from left to right in the figure, except for from triangle 3 to triangle 4. As shown in table 1, triangle 1 is reduced to triangle 2 by scaling factor ¹/₂ and triangle 2 to triangle 3 by the same scaling factor. But, triangles 3 and 4 are the same, so the scaling factor is 1. Again, triangle 4 is reduced to triangle 5 by the scaling factor ¹/₂. Here, we clearly see that triangle 4 is an odd one; it breaks the uniform pattern of scaling factors listed in table 1. Nevertheless, all triangles are similar in figure 2.

Figure 2.  Similar triangles

Table 1.  Scaling factor of triangles in figure 2

Figure 3 has five similar triangles also labeled 1 through 5, which however decrease in size uniformly, so that the neighboring triangles are reduced by scaling factor ¹/₂, as we move from left to right in figure 3. When all similar objects vary by a fixed scaling factor as in table 2, they are said to be self-similar.Hence, by definition the self-similar objects have no beginning or ending, and they form an endless sequence of similar objects. In particular, to the right of triangle 5 there follow triangle 6 with side 1/32, triangle 7 with side 1/64, and so on, all of which decrease by scaling factor ¹/₂. Also, to the left of triangle 1 there are triangle 0 with side 2 and ever-increasingly larger triangles by scaling factor 2, as we move from right to left in figure 3. In this way, labeling triangles in figure 3 by 1 through 5 is totally arbitrary. For instance, figure 3 looks just the same as figures 4 and 5, which are obtained by 50% reduction (scaling factor = 0.5) and 200% enlargement (scaling factor = 2) of figure 3, respectively. For self-similar objects, we can actually move up or down along the ladder of size and see the same sequence of objects. This is somewhat akin to probing small objects by a microscope and distant objects by a telescope, and the nature never shows vacuum.

Figure 3.  Self-similar triangles

Table 2.  Scaling factor of triangles in figure 3

Figure 4.  50% reduction of figure 3

Figure 5.   200% enlargement of figure 3

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