Intriguing ornament

Yes, Comrade, this actually does become quite intuitive if you meditate over it long enough (as most things do, in maths and otherwise!). Let me give you some hints on how I think about it, without giving it all away. It's some simple but beautiful math, well worth thinking about. As in many cases (think inverse square field!) it's a case of two countervailing tendencies cancelling each other out perfectly.

Let $S(z)$ be the surface area of a sphere of radius $R$ up to height $z$ , the lower cap if you will. The main point will be to explain why $\frac{dS}{dz}=2\pi R$ , a constant. The formulas for a cap or, more generally, a slice of a sphere are immediate corollaries. Let $r^2=x^2+y^2$ as usual, and $ds^2=dr^2+dz^2$ . Note that $dS=2\pi r ds$ (If you wish to draw a figure, work in the $rz$ -plane.) Now we have two similar triangles, one involving $R,z,r$ and the other $dr,dz,ds$ ; this is just an expression of the fact that a tangent to the circle is perpendicular to the radius at that point. It follows that $\frac{r}{R}=\frac{dz}{ds}$ so that $dS=2\pi rds=2\pi Rdz$ , as claimed.

As we go up, $r$ decreases and $\frac{ds}{dz}$ increases, and those tendencies are inversely proportional and cancel each other out. To put it in more geometrical terms: As you go up, the rings around the $z$ -axis shrink in diameter, but they become "flatter" and thus, wider.