If anything, it seems like higher-dimensional cubes are spiky, not spheres. At the vertex of a square, the figure takes up 1⁄4 of the local area around the vertex, for a cube vertex it’s only 1⁄8 of the space, for a tesseract only 1⁄16, in 10 dimensions only 1/1024 etc.
Quite true! Also, for the same reason you mentioned, higher dimensional cubes have a lot of right angled corners, since they always have to add up to 360 degrees, so they definitely qualify as “spiky”.
If anything, it seems like higher-dimensional cubes are spiky, not spheres. At the vertex of a square, the figure takes up 1⁄4 of the local area around the vertex, for a cube vertex it’s only 1⁄8 of the space, for a tesseract only 1⁄16, in 10 dimensions only 1/1024 etc.
Quite true! Also, for the same reason you mentioned, higher dimensional cubes have a lot of right angled corners, since they always have to add up to 360 degrees, so they definitely qualify as “spiky”.