Dimensions used in cellular automata

Lecture

To estimate the area of an arbitrary area, you can use a simple

device - a palette - a transparent plate with a square grid applied on it.

If you count the number of grid squares that fall entirely inside the area, then

we obtain an estimate of the area below, if we calculate the number of squares, completely

covered the area - the estimate from above.

Obviously, the smaller the size of the square, the more accurate the estimate. If you evaluate

the area of a certain region of space with decreasing linear size

square ε, the number of squares will increase as N (ε) ≅ 1 / (ε) ^ 2, but if

the region, and the curve, then N (ε) 1 / (ε) (Fig. 11).

Fig. 11. Reducing cell size increases accuracy.

Summarizing these relations, we get:

where D is the dimension

the studied set. The dimension acts as a number characterizing

the growth rate of the number of cells covering this set with decreasing size

cells Next, we logarithm and direct ε to zero:

(3)

It is clear that the dimension of the line calculated by this formula will be equal to 1,

the dimension of the plane is 2, the dimension of the volume is 3. In general, the dimension

“Familiar” to us objects is expressed by an integer. But it turns out you can

construct such objects, the dimension of which will be fractional.

For example, examine the Cantor set, which is constructed as follows.

way: a single segment is divided into three equal parts and the middle part

ejected, then everything repeats for the remaining two parts, etc. Eventually

we have such a picture (Figure 12):

Figure 12. Cantor set

Such a set is called fractal. To estimate the dimension of this

sets as cells ε take intervals of length ε

. As can be seen from the construction, the coverage by intervals of length ε = 1/3 contains two elements, with

ε = 1/9 - four elements, and so on with ε = (1/3) ^ k (the number of elements will be 2 ^ k. Then by

dimension formula we have:

This is a fractional number! It should be noted that the dimension of fractal sets

generally fractional. For more information about the dimension, see [11].

11. Akhromeeva TS et al. Non-stationary structures and diffusion chaos, M. 1991

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