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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).
  Dimensions used in cellular automata
Fig. 11. Reducing cell size increases accuracy.
Summarizing these relations, we get:
  Dimensions used in cellular automata 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:
  Dimensions used in cellular automata (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):
  Dimensions used in cellular automata
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:
  Dimensions used in cellular automata
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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