Physics Friday 141

To answer the questions left lingering last week, we begin with the Sierpinski carpet:

We see that it is composed of eight smaller carpets, each with side length one-third that of the overall carpet; thus, if I is the moment of inertia of the total carpet of mass M and side length a, then each of the smaller carpets has moment about its center of
Now, we use the parallel axis theorem again. We see that four of the sub-carpets, the “edge” ones, have centers with distance from our axis of . The other four, the “corners”, have centers with distance .

Thus, we have

Now, for the Menger Sponge. We have 20 sub-units, each with edge-length one-third that of the overall sponge, so
Finding the distances, we see that there are 12 which lie along edges parallel to our axis (comparable to the “corners” in the carpet case), and thus have distance . The other 8 are analogous to the carpet “edge” units, with distance . Thus,


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