Geometric sequences , cones and cylinders

I'll be happy to oblige you with an explanation about why these questions (if I understand their scope correctly) will always result in a geometric series. The questions, as I see them, involve inscribing shape $B$ into shape $A$, then inscribing $A$ into the copy of $B$, then inscribing $B$ into that copy of $A$, etc, etc. Each copy of shape $A$ has to be similar, as indeed must be the copies of $B$.

By "similar", I mean the mathematical term in geometry, which means that the two shapes can be identified by a process of scaling (i.e. stretching or shrinking the shape uniformly in all directions), and then applying an isometry (i.e. a map that preserves distances and angles, such as rotation, translation, reflection, etc). For example, all square no matter what size are similar, but no square is similar to a rectangle that is not a square.

It will also help if we are looking for the largest copy of $B$ that we can put in $A$, and vice-versa. Under these circumstances, we will always get a geometric series.

Now, let's suppose we have two shapes $A$ and $A'$ in $n$ dimensions (could be $2$ or $3$, or more, or even just $1$). This means we must be able to scale $A$, then move/rotate/reflect it until it becomes $A'$. The moving/rotating/reflecting will not change the volume of $A$, but the scaling will. Let's say we scale by a factor of $lambda$. Then, in $n$ dimensions, the new volume of $A'$ will change by a factor of $lambda$. That is, $$operatorname{Vol} A' = lambda^n operatorname{Vol} A.$$ For example, if $A$ is a sphere of radius $1$ in $3$ dimensions, and $A'$ is a sphere of radius $2$ in $3$ dimensions, then $A'$ has $2^3 = 8$ times the volume of $A$.

I claim that, if we inscribe similar shapes $B$ and $B'$ into $A$ and $A'$ respectively, each with maximum volume, then $operatorname{Vol} B' = lambda^n operatorname{Vol} B$. If say $operatorname{Vol} B' < lambda^n operatorname{Vol} B$, then applying whatever scaling/rotation/reflection/translation operations were necessary to take $A$ to $A'$ to $B$ instead will yield a shape $B''$, similar to $B$, inscribed in $A'$. Because the volume is determined by the scaling factor, which was $lambda$, we have $$operatorname{Vol} B'' = lambda^n operatorname{Vol} B > operatorname{Vol} B'.$$ This contradicts $B'$ being the largest copy of $B$ inside $A'$.

A similar contradiction can be worked out when $operatorname{Vol} B' > lambda^n operatorname{Vol} B$. Since each scaling/rotation/reflection/translation operation can be reversed, this is really just a symmetric case, switching the roles of $A, B$ and $A', B'$.

Of course, now when we look at putting copies of $A$ into $B$, we get another scaling factor, let's call it $mu^n$. Now, once a copy $A'$ of $A$ is placed into a copy of $B$, which is in $A$, we see that the volume of $A'$ is scaled by a factor $mu^n lambda^n$. Since $A'$ is similar to $A$, the next copy down (call it $A''$?) will be multiplied by the same factor $mu^n lambda^n$, etc, etc. In this way, we get a geometric sequence, with common ratio $mu^n lambda^n$. The volumes of the $B$s will also form a geometric sequence similarly, with the same common ratio.

The tricky bit is finding $lambda$ and $mu$! This will depend on the specific shapes you're considering.