Area identical portions of a sphere

Question

Let's say I have a cake in form of a sphere and I want to cut it in equal and identical $N$ parts.

I'll use spherical coordinates $(r,\theta,\phi)$.

One way to do it is to do that each part is defined by : $\phi_i\in[0,\pi],\theta_i\in[i(2\pi/N),(i+1)(2\pi/N)]$ with $i\in\mathbb{N},0<i<N+1$. In this case each part has two neighbours. Like this

By neighbours I mean that they share a side (and not only a node).

I was wondering is it was possible for some $N$ to cut the sphere in identical parts so that each part has 4 neighbours.

Is there a possibility to say for which $N$ this could be possible, and the details of how $\phi_i$ and $\theta_i$ would be defined ?

Thanks

Answer 1

For N=6, imagine putting a beach ball inside an elastic cubic frame; when you expand the ball, the faces of the cube will be outlined by the frame and should satisfy your condition.

Edit: You could do the same thing with other icosahedra, though I don't think this gives you every type of partition you're interested in.