answer:The Hausdorff dimension of the Cantor dust can be calculated using the formula: Hausdorff dimension (D) = log(N) / log(1/r) where N is the number of self-similar pieces and r is the scaling factor. For the Cantor dust, the initial segment consists of one interval of length 1. After the first step, we remove the middle third, leaving two intervals of length 1/3 each. So, N = 2 and r = 3. Now, we can calculate the Hausdorff dimension: D = log(2) / log(3) D ≈ 0.6309 So, the Hausdorff dimension of the Cantor dust is approximately 0.6309.

answer:The Cantor dust set is a fractal, and its Hausdorff dimension can be calculated using the formula: Hausdorff dimension (D) = log(N) / log(1 / r) where N is the number of self-similar pieces in each iteration, and r is the scaling factor (the ratio of the size of each piece to the size of the original set). In the case of the Cantor dust set in 2-dimensional space, each iteration removes 1/3 of the previous segment length, so the scaling factor r is 1/3. In each iteration, the set is divided into 4 self-similar pieces (2 in each dimension). So, we can plug these values into the formula: D = log(4) / log(1 / (1/3)) D = log(4) / log(3) Using a calculator, we find that the Hausdorff dimension of the Cantor dust set in 2-dimensional space is approximately 1.26186.

answer:The Hausdorff dimension of the Cantor dust can be calculated using the formula: Hausdorff dimension = log(N) / log(1/s) where N is the number of self-similar pieces and s is the scaling factor. For the Cantor dust, at each iteration, the line segment is divided into 3 equal parts and the middle one-third is removed, leaving 2 self-similar pieces (N = 2). The scaling factor is the length of each remaining piece, which is 1/3 (s = 1/3). So, the Hausdorff dimension of the Cantor dust is: Hausdorff dimension = log(2) / log(1/3) ≈ 0.6309 The Hausdorff dimension of the Cantor dust is approximately 0.6309.

answer:The Cantor dust, also known as the Cantor set, is created by iteratively removing the middle third of a line segment of length 1. To find the Hausdorff dimension of the Cantor dust, we can use the formula: Hausdorff dimension = log(N) / log(1/r) where N is the number of self-similar pieces and r is the scaling factor. In the case of the Cantor dust, at each iteration, the line segment is divided into 3 equal parts, and the middle third is removed, leaving 2 self-similar pieces (N = 2). The scaling factor is 1/3, as each remaining piece is 1/3 the length of the original segment (r = 1/3). So, the Hausdorff dimension of the Cantor dust is: Hausdorff dimension = log(2) / log(1/3) ≈ 0.6309 Thus, the Hausdorff dimension of the Cantor dust is approximately 0.6309.