Intersecting spheres approaching a spherical shaped shared space

[quantum_wave]

Ending up with a spherical cow instead of starting with a spherical cow:

When two spheres intersect, the intersection forms a pair of spherical caps in the shape of a 3-D lens.

Two spheres, two spherical caps, 3-D lens shape overlap, two surfaces to the lens

If a third sphere intersects both of those spherical caps, there are six spherical caps and the shape of the intersection where all three spheres share space becomes a truncated 3-D lens.

Three spheres, six spherical caps, truncated 3-D lens shape of the shared space, three surfaces to the truncated 3-D lens

If a fourth sphere intersects with all six of those spherical caps, there are twelve spherical caps and the shape of the shared overlap is further reduced by a second truncation of the 3-D lens.

Four spheres, twelve spherical caps, doubly truncated 3-D lens shape, four surfaces to the shared space ...

Five spheres, twenty spherical caps, five surfaces to the shared space ...

Six spheres, thirty spherical caps, six surfaces to the shared space ...

And son on …

If the height of each cap is identical, call it h, and if of the radius of each sphere is identical, call it r, then each additional intersection optimizes the trend toward a spherical shaped shared volume, and the shape of the shared intersection where all of the n spheres overlap approaches a spherical shape as each addition sphere is added.

Is there a simple formula for the volume of the shared intersection of n spheres, each with r radius, that intersect so that the height of each spherical cap formed is h, and where the overlap approaches a spherical shaped limit? Yes or no would be good enough to satisfy me.

If yes, then is there a formula that will yield the shared space of n spheres to equal 1/n of the volume of one of the identical spheres, i.e. a formula for the height of the caps relative to the radius of each of the intersecting spheres where h/r would decrease as n increases and so that the shared volume always equals 1/n of the volume of one of the identical spheres?

Again, yes or no is good, but if no, what additional information would be needed for such a formula?

 

[Vkothii]

Ahem;
if I might suggest an algorithmical approach, which is what you have in fact done yourself, so to check over the steps you've made - the formulation and the induction, the trivial first case, and so on...?

The formula is geometric, but there are two kinds of spaces - surfaces which are spherical and continuous, and intersections which are a volume; call these s-spaces and i-spaces, which map to each other via the usual spherical parameters - pi, a radius, a circumference, angles and so on.

For equal radii of 2 intersecting s-spaces (spherical surfaces), the i-spaces (lenses), have a radius which depends on the part of the radius of the intersecting spheres, that is within the i-space itself, and there is a ratio between the spheres' radius and the radius of the lens. This is the trivial case, btw.
(hint) : So you need a formula for the radius of a lens L; the intersection of two spheres S1, S2 with identical radius rS; the intersecting part of the radius of each sphere, a, is half the width of a lens with radius rL, and axial width 2a, right? I'm sure pi and an angle or two are in there somewhere.

P.S. You can prove that the i-space formed by multiple intersecting circles with identical radius approaches a circle instead of doing it in 3 dimensions, then induce a proof in 3 (or possibly more) dimensions, you realise.

 

[quantum_wave]

I am using this link about Sphere-Sphere Intersections at mathworld.wolfram which walks through the math of how to calculate the volume of the intersection of two such spheres, i.e. the volume of the lens.

As additional intersections are added via idential spheres, the number of surfaces of the intersection increases and an algoritm developes.

I guess what I need to know is if the relationship between the radius of the identical spheres and the height of their caps, both measures are the same for each sphere and cap, is enough information to ultimately calculate the volume of the intersection that is common to n spheres. If the formula can be derived from the algorithm without need for any additonal parameters then I am happy.

If so, then I think that if each sphere had a different radius and the caps had different heights, the volume of the intersecton could still be calculated, with some effort and once the method was defined, any combination of spheres and cap heights could be plugged in to yield the volume of the joint intersection.