On the other hand, it seems to me (maybe I am wrong) that the solid n-dimensional angle of each corner of the n-cube is progressively smaller (as a ratio of ful

On the other hand, it seems to me (maybe I am wrong) that the solid n-dimensional angle of each corner of the n-cube is progressively smaller (as a ratio of full n-dimensional angle). So in that sense, each n-cube gets more spikey, and there is "less space" in each corner for each n-sphere that enclose the central n-sphere.