Generate a mathematical visualization video showing the fractal evolution of a cube with the following recursive rules: 1. **Initial State**: Start with a single large cube. 2. **Iteration 1**: - Cut the cube in half along its central x-axis (parallel to the yz-plane). - **Remove the right half** completely. - **Replace the removed space** by filling it with 16 identical smaller cubes, each with side length 1/4 of the original cube. 3. **Recursion**: - Apply the same process to every new smaller cube: - Cut each along its central x-axis, - Remove its right half, - Fill the space with 16 even smaller cubes (side length = 1/16 of the original). 4. **Depth**: Visualize 3 iterations (original → 16 cubes → 256 cubes). **Key Requirements**: - Use **3D animation** to show cutting, removal, and filling steps clearly. - Color-code each iteration (e.g., original=blue, 1st-gen=green, 2nd-gen=red). - Add dynamic labels: "Iteration 0", "Iteration 1", etc. - End with a rotating view of the final fractal structure.

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Welcome to fractal cube evolution. Fractals are fascinating geometric shapes that exhibit self-similarity at different scales. We start with a single blue cube and apply three recursive rules. First, we cut the cube along its central x-axis, parallel to the y-z plane. Second, we completely remove the right half. Third, we replace the removed space with 16 smaller cubes, each with side length one-fourth of the original. This process creates beautiful fractal patterns through simple recursive operations. Now let's execute the first iteration step by step. We begin with our original blue cube labeled as Iteration 0. First, we visualize the cutting plane moving through the cube's center, parallel to the y-z plane. Next, we highlight the right half in red to show the portion that will be removed. Watch as the right half disappears completely. Finally, in the vacated space, we place 16 smaller green cubes arranged in a 4 by 2 by 2 grid pattern. Each small cube has a side length of one-fourth the original, so their total volume is significantly less than the removed portion. This creates the characteristic fractal property where the structure becomes increasingly sparse with each iteration. Now we enter the second iteration, demonstrating the exponential complexity growth. We start with 17 total pieces from iteration 1: one blue left half and 16 green cubes. The recursive rule applies simultaneously to all existing pieces. Watch as cutting planes appear through every cube, showing where each will be divided. We remove the right half of each piece, leaving 17 left halves. Then comes the dramatic multiplication: each removed space is filled with 16 tiny red cubes at one-sixteenth the original scale. This gives us 17 times 16 equals 272 new red cubes, plus the 17 remaining left halves, totaling 289 pieces. The fractal structure is becoming increasingly complex and sparse. We now execute the third and final iteration, reaching the target depth of our fractal evolution. Starting with 289 pieces from iteration 2, we apply our recursive rule one last time. Each of the 289 pieces is cut along its central x-axis, the right halves are removed, and the vacant spaces are filled with 16 yellow cubes each. These new cubes have a side length of one sixty-fourth the original scale. The mathematics is staggering: 289 times 16 equals 4624 new yellow cubes, plus the 289 remaining left halves, giving us a total of 4913 pieces. Our fractal has evolved from a single cube to nearly 5000 pieces, demonstrating the explosive growth characteristic of recursive geometric processes. Here we see our completed fractal structure in all its mathematical beauty. The final result contains 4913 individual pieces, each precisely positioned according to our recursive rules. Let's analyze the key mathematical properties. The surface area grows exponentially with each iteration, while the total volume decreases dramatically from 1 to approximately 0.004. The fractal dimension is approximately 1.89, indicating a structure more complex than a line but less than a solid. The self-similarity factor of one-fourth means each smaller structure mirrors the larger pattern. Watch as we rotate the fractal to appreciate its intricate geometry from all angles. This demonstrates how simple recursive rules can generate structures of extraordinary complexity and beauty, embodying the fundamental principles of fractal geometry.