acousticsdesignvibrations
A few questions about the design of a common table fan, such as below.
Suppose the leg spacing for a square and triangular chair is the same then the positions of the legs look like:
If we call the leg spacing $2d$ then for the square chair the distance from the centre to the edge is $d$ while for the triangular chair it's $d\tan 30^\circ$ or about $0.58d$. That means on the triangular chair you can only lean half as far before you fall over, so it is much less stable. To get the same stability as the square chair you'd need to increase the leg spacing to $2/\tan 30^\circ d$ or about $3.5d$ which would make the chair too big.
A pentagonal chair would be even more stable, and a hexagonal chair more stable still, and so on. However increasing the number of legs gives diminishing increases in stability and costs more. Four-legged chairs have emerged (from several millennia of people falling off chairs) as a good compromise.
There are not really any structural reasons for several layers of different softness/hardness. I believe that a hard surface as the outer layer is only added for a better user-feel.
If you only seek good protection, then you have two parameters to aim for:
The softer and thicker the better. To prevent collision damage, you are looking at Newton's 2nd law:
$$\sum F= \frac{\Delta p}{\Delta t}$$
Your phone hits the floor and there will be a momentum-change $\Delta p$ from it's impact speed to zero. You can't reduce that. But you can change the duration $\Delta t$ of this impact. Larger collision duration means smaller forces $\sum F$ on the phone.
Increasing $\Delta t$ means that you must cover it in a material that absorbs the kinetic energy; materials that deform at collision. Therefore soft. Not too soft, because it should be able to absorb the entire amount of kinetic energy. By making a thicker layer of this soft material, you again acheive more energy absorbed. A pillow is perfect.
(Just remember that you don't want to add too much mass by adding this cover, since mass raises $\Delta p$. But luckily, lightweigth and softness usually follow along.)
So, there's not really any argument for more layers. The softest material able to absorb the whole amount of kinetic energy with the given thickness that you will allow for, is the prefered solution. Any other materials are merely for design or user-feel purposes.
As the PO points out in the comments, there may very well be an advantage in having a hard outer shell, because the energy then can be dispersed by a large area. Maybe we shouldn't rule out such outer shell completely then. We are entering the engineering field then, since the design of the shell and evenly distribution of soft material under evenly spaced outer material is important here.
Best Answer
I won't address the details of the fluid dynamics that cause the cage to produce noise because it is well out of my area of expertise. But I have performed an experiment in my office. The small desk fan I bought was too noisy. Since it lives on the top of a tall bookcase and well out of reach, I removed the cage completely. It is a lot quieter now.
My guess is that turbulence around the bars of the cage causes the noise. With the cage removed, the air stream throws further. There is still some noise, so I'm guessing that the fan's blades are not optimally shaped.