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Considere el fractal conocido como Esponja de Menger en dos dimensiones. La siguiente figura los pasos 1 a 3, y debajo el 4, de su construcción: Halle la dimensión fractal D para la esponja de Menger. Use la función N(s)=s^-D para calcular el número de los cuadraditos blancos más pequeños que se ven en la figura del cuarto paso. 

The image you've uploaded shows the construction steps of a two-dimensional version of the Menger Sponge fractal. In this context, the problem asks for the fractal dimension \( D \) and the number of the smallest squares (white squares) visible in the fourth step.

### 1. Fractal Dimension Calculation
The fractal dimension \( D \) for a self-similar fractal can be calculated using the formula:

\[
D = \frac{\log(N)}{\log(s)}
\]

Where:
- \( N \) is the number of self-similar pieces,
- \( s \) is the scaling factor (how much each piece is scaled down compared to the original).

For the Menger Sponge:
- At each step, the original square is divided into 9 smaller squares, and 5 of them are removed, leaving 4 squares.
- So, \( N = 4 \) and \( s = 3 \) (because each side is scaled down by a factor of 3).

Thus,

\[
D = \frac{\log(4)}{\log(3)} \approx \frac{1.386}{0.477} \approx 1.2619
\]

### 2. Number of Smallest Squares in Step 4
Using the formula \( N(s) = s^{-D} \), we can calculate the number of the smallest squares in step 4.

However, to use this formula, we must first calculate the number of iterations.

- In step 1, there are \( 4^1 \) remaining squares.
- In step 2, there are \( 4^2 \) remaining squares.
- In step 3, there are \( 4^3 \) remaining squares.
- In step 4, there are \( 4^4 \) remaining squares.

Therefore, in the fourth step, the number of the smallest white squares:

\[
N = 4^4 = 256
\]

Thus, there are 256 of the smallest squares visible in the fourth step.

### Summary:
1. The fractal dimension \( D \) of the two-dimensional Menger Sponge is approximately \( 1.2619 \).
2. The number of the smallest squares visible in the fourth step is 256.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How is the fractal dimension interpreted in real-world scenarios?
2. What is the general process of constructing the Menger Sponge in three dimensions?
3. How does the fractal dimension differ between the 2D and 3D Menger Sponge?
4. What role does self-similarity play in determining the properties of fractals?
5. Can the fractal dimension be a non-integer value, and what does it signify?

**Tip:** Understanding the relationship between scaling factor and the number of self-similar pieces is crucial for calculating the fractal dimension in any fractal geometry.

Learn how to calculate the fractal dimension of the two-dimensional Menger Sponge using the formula D = log(N) / log(s) and find the number of smallest squares visible in the fourth step. Detailed step-by-step solution provided.

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