And in that case n is about 1,964,262,400. According to Wikipedia the overall height of the Empire State Building is 443.2m. That's 44320cm. So we need a cube t

And in that case n is about 1,964,262,400. According to Wikipedia the overall height of the Empire State Building is 443.2m. That's 44320cm. So we need a cube that -- corner-to-opposite-corner -- is that many cm long. If the cube is 1 cm on a side, its corner-to-opposite-corner distance is the square root of n, so the number of dimensions of the cube needs to be about 1.96 billion for [a line the same length as] the Empire State Building to fit inside.