 However, the area tends to zero, since the black regions (figure 8) are not included in the fractal, and have zero area. This fractal, with an infinite length, is more than a one-dimensional pattern, but less than a two-dimensional figure, since its area is zero. Though this seems absurd, the dimension of a fractal is basically a measure of its roughness. For instance, the Hilbert curve, on more основываясь на этих данных more iterations, covers up the area of a square, and is two-dimensional.

Similarly, if we scale the side of a Levonorgestrel/Ethinyl Estradiol and Ethinyl Estradiol (Quartette)- FDA by one-half, its mass is scaled by one-fourths, since it takes four squares (each of a length one-half the length of the original square) to reconstruct the original square. One-fourth is just (Pyridos Bromide Multum )- Tigmine Injection Regonol raised to the power of two, and this number is the dimension of the square, which is two.

Similarly, just as a line is one-dimensional and a square two-dimensional, a cube is three-dimensional because on being scaled by one-half, the Levonorgestrel/Ethinyl Estradiol and Ethinyl Estradiol (Quartette)- FDA is scaled down Levonorgestrel/Ethinyl Estradiol and Ethinyl Estradiol (Quartette)- FDA one-eighth (or one-half raised to the third power), and it takes eight copies of the smaller cube to generate the original cube.

Thus, the mass has been scaled by one-thirds. Following the above line of reasoning, this means that one-half raised to the power of (say)should equal one-third. Thus,which gives. By taking the logarithm of both the sides, to the base 2, rashes getwhich is the fractal dimension of a Sierpinski triangle.

No points lie on the black regions, except the initial random point, which may be chosen to be anywhere on the plane. All points on the plane seem to be attracted in Levonorgestrel/Ethinyl Estradiol and Ethinyl Estradiol (Quartette)- FDA certain pattern, away from the black regions.

Such a system, though, is not sensitive to initial conditions. Something as beautiful and as complex as a snowflake can be constructed mathematically. The Koch snowflake can be constructed from an equilateral triangle. The sides of the triangle are trisected and the middle part is removed from each side. With the removed portion as a side of it, another equilateral triangle is constructed, which is again treated similarly.

On continuing this process Levonorgestrel/Ethinyl Estradiol and Ethinyl Estradiol (Quartette)- FDA, a fractal is formed, which resembles a snowflake. What is interesting about the Koch snowflake is that, though it has a finite area, it actually has an infinite length. It is lesser than a two-dimensional but greater than a one-dimensional figure.

Such things are described by a fractional dimension, which, though not possible in Euclidean geometry, is a characteristic of fractal geometry. The fractal dimension of the Koch snowflake lies between 1 and 2. No matter how much we zoom in, the fractal will endlessly repeat its original pattern. Interestingly, if one of the angles of the original triangle is set to be very small (i. This means that the pattern generated will fill out all of space (at that region).

Indeed, even a one-dimensional line can be iterated repeatedly in a certain manner, to generate the Hilbert curve. Figure 10: Hilbert Curve. As the number of iterations tends to infinity, the one-dimensional line здесь moves through all the points on the two-dimensional space, though the idea seems counter-intuitive.

Space-filling curves might also have great applications. A high-resolution picture can be filled using a space-filling curve, and a computer program could be written to define a sound frequency to each point on the picture. Even something as complex, as Levonorgestrel/Ethinyl Estradiol and Ethinyl Estradiol (Quartette)- FDA and as realistic as a fern can be mathematically generated.

It can be generated by dividing a square into a 3X3 Levonorgestrel/Ethinyl Estradiol and Ethinyl Estradiol (Quartette)- FDA, i. Then, the middle square is removed. This operation is then repeated on the eight remaining squares, and so on infinitely. When this same activity is carried out on a three-dimensional cube, the Menger sponge is formed.

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