Block Fractals

Exercises - Five Block Configuration Volume

Here we compute the volume of the four cube configuration by finding a pattern in the volumes of the first few levels of the construction.

Level 1	Level 2	Level 3

Level Side Length Number of Cubes Cube Volume Total Volume 1 1 5 1 V1 = 5*1 2 1/2 52 (1/2)3 = 1/8 V2 = 52*(1/8) 3 1/4 53 (1/4)3 = (1/8)2 V3 = 53*(1/8)2 ... ... ... ... ... n 1/2n-1 5n (1/2n-1)3 = (1/8)n-1 Vn = 5n*(1/8)n-1 = 5*(5/8)n-1
		Total Vol vs level

Because V_n = 5*(5/8)^n-1 -> 0 as n -> infinity, the limiting shape has zero volume.

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