Imagine a 4D ball passing through our 3D world. We can only see one "slice" at a time. At the very middle, the slice is a full sphere — toward the edges, it shrinks to a point.
Slicing a 4-ball at coordinate w gives a 3-ball whose radius depends on how far the slice is from the equator:
Pythagoras in ℝ⁴: a hyperplane w = w₀ intersects B⁴(r) in a 3-ball of radius √(r² − w₀²). Setting w₀ = r·cos θ parameterises the slice as a function of polar angle.
r'(w) = \sqrt{r^2 - w^2},\quad |w| \le r
As w sweeps from −r to +r, you watch the 3-ball appear, swell to full size at the equator, then vanish.
Try moving the W slider — watch the sphere grow then shrink.