This feels misleading to me.
Directly comparing volumes in different dimensions doesn't make any sense because the units are different. It doesn't make sense to say that a quantity in m^3 is larger or smaller than a quantity in m^4. Because it doesn't make any sense to compare the area of a circle with the volume of a sphere.
> More accurate pictorial representations of high dimensional cubes (left) and spheres (right).
The cube one is arguably accurate -- e.g. in 100 dimensions, if the distance from the center of a cube to the center of a face is 1, then the distance from the center of the cube to a corner is 10.
But the sphere one, I don't know. Every point on a 100-dimensional sphere is still the same distance away from its center. The sphere is staying spherical in an intuitive way, it's just that the corners of the enclosing cube have gotten so much further away.
So what is accurate to say is that the proportion of volume of a sphere relative to that of its bounding cube keeps decreasing. Which, rather than being supposedly "counterintuitive", makes perfect intuitive sense -- because every time you add a dimension, you can think of it as "extruding" the previous sphere into the new dimension and then shaving it round, the way a 2D circle can be extruded into a cylinder in 3D and then shaved down to make it into a sphere. Every time you add a dimension, you shave off more.
The article suggests that a 3D sphere has greater volume than a 2D circle -- with a unit radius, the sphere is 4/3π while the circle is just π. But again, they're in different units, so it's a meaningless statement. It makes much more sense to say that a 2D circle takes up (1/4)π≈0.79 of its bounding square, a 3D sphere takes of (1/6)π≈0.52 of its bounding cube, a 4D sphere takes up (π/32)π≈=0.31, and so forth. So no, the volume doesn't go up and then down -- it just goes down every time when taken as a unitless proportion (and proportions are comparable).