Solved: Exercises 54 and 55 refer to the Menger sponge, a threedimensional cousin of the
Chapter 12, Problem 54(choose chapter or problem)
Exercises 54 and 55 refer to the Menger sponge, a threedimensional cousin of the Sierpinski carpet. (See Exercises 2326.) The Menger sponge is defined by the following recursive construction rule.Assume that the seed cube of the Menger sponge hasvolume 1. (a) Let C denote the total number of cubes removed ata particular step of the construction, U the volume ofeach removed cube, and V the volume of the sponge atthat particular step of the construction. Complete theentries in the following table. C U VStart 0 0 1Step 1 7 1272027Step 2Step 3Step 4Step N (b) Explain why the Menger sponge has infinitesimallysmall volume
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