Yes. As the number of dimensions n increases above 3, the interior angles at each vertex of an n-cube get smaller and smaller[a], while the n-cube's hypervolume

Yes. As the number of dimensions n increases above 3, the interior angles at each vertex of an n-cube get smaller and smaller[a], while the n-cube's hypervolume gets more and more concentrated near its hypersurface, which is itself a hypervolume of n-1 dimensions.[b] Lines and areas are different animals; we cannot reason about them apples-to-apples; we know this intuitively. Areas and volumes are different animals too; we cannot reason about them apples-to-apples; we know this intuitively. Similarly, n-dimensional objects and (n+1)-dimensional objects are different animals; we cannot reason about them apples-to-apples. As human beings, we find it so difficult to reason "visually" about higher dimensional spaces, in part, I believe, because our puny little brains have spent a lifetime learning to model three dimensions (with a fourth dimension, time, flowing only in one direction). -- [a] See this comment by scatters: https://news.ycombinator.com/item?id=29969181 https://news.ycombinator.com/item?id=29969181 [b] See this old thread for intuitive explanations about how and why this happens with n-spheres as we increase the number of dimensions n: https://news.ycombinator.com/item?id=15676220 https://news.ycombinator.com/item?id=15676220