Ah gotcha. I'm not even sure it's meaningful to compare the "relative pointiness" of angles between different dimensions. It's like trying to compare whether the area of a circle is bigger or smaller than the volume of a sphere. And then what if I only have 70% of a circle, but 80% of the sphere?
You could make up some metric like "the unit sphere occupies a greater volume fraction of the unit cube compared to the area fraction of a unit circle in a unit square", and compare like that if you only have some fraction of each circle/sphere instead of the whole thing, but is there any value in making that comparison?
> I'm not even sure it's meaningful to compare the "relative pointiness" of angles between different dimensions.
I think this is actually something we can't avoid. The problem with dodging the question this way is that it's really, really easy to translate a low-dimensional angle up into a higher-dimensional space.
Imagine I have two 2D vectors, U and V, with an angle between them. I can think of them as representing an infinite 1D surface consisting of the points (m·u_1 + n·v_1, m·u_2 + n·v_2) for all nonnegative real m and n.
It's trivial to define a 2D surface in 3D space representing exactly the same angle: it is the points (m·u_1 + n·v_1, m·u_2 + n·v_2, a) for all nonnegative real m and n and all real a. When the angle between U and V is θ radians, this surface will always express an angle of 2θ steradians. I don't think it's a stretch to say that the two angles, θ radians and 2θ steradians, must be equally sharp. (And indeed, the flat-line / flat-plane example is a special case of this one.)
I need to correct myself: I've defined my surfaces as linear combinations of the vectors forming the angle, which is wrong. Rather, the 1D surface expressing the 2D angle consists of the points in the set (mU ∪ nV) for all nonnegative m and n, and the 2D surface extending the angle into 3D space consists of the set [(m·u_1, m·u_2, a) ∪ (n·v_1, n·v_2, a)] for all nonnegative m, n and all a.
You could make up some metric like "the unit sphere occupies a greater volume fraction of the unit cube compared to the area fraction of a unit circle in a unit square", and compare like that if you only have some fraction of each circle/sphere instead of the whole thing, but is there any value in making that comparison?