### Instruction: What is the perimeter of the 10th iteration of the Gosper curve, given that the side length of the initial equilateral triangle is equal to 1?### Response: The Gosper curve, also known as the Gosper Island or the flowsnake, is a fractal curve that can be constructed by recursively replacing each segment of an equilateral triangle with a specific pattern of smaller segments. 

For each iteration of the Gosper curve, the number of segments increases by a factor of 7, and the length of each segment is scaled down by a factor of (2 + √2). Therefore, after n iterations, the total number of segments is 7^n, and the length of each segment is (1 / (2 + √2))^n.

To find the perimeter of the 10th iteration of the Gosper curve, we can use the following formula:

Perimeter = (Number of segments) * (Length of each segment)

For the 10th iteration:

Number of segments = 7^10
Length of each segment = (1 / (2 + √2))^10

Now, let's calculate the values:

Number of segments = 7^10 = 282475249
Length of each segment ≈ (1 / (2 + √2))^10 ≈ 0.000015259

Perimeter ≈ 282475249 * 0.000015259 ≈ 4313.73

So, the perimeter of the 10th iteration of the Gosper curve is approximately 4313.73, given that the side length of the initial equilateral triangle is equal to 1.