How can an object grow so that the ratio of surface area to volume remains constant?

Question

Obviously there are many applications for this and many solutions. I am also interested in closed curves that have the same ratio of area to arc length as we "grow" them, growth is done by augmentation in both cases.

THIS IS NOT ABOUT SCALING.

Answer 1

An example: Start out with a square of size 1 and now grow one side as $x$ and one side as $y(x)$, then you want $(x+y)=2x y$ or $y(x)=\frac{x}{2x-1}$.

So after some time, one side stays constant $1/2$ and the object grows "only in one direction".

I doubt that you can get much more precise answers without having a more precise question, but it is possible that someone else knows the right question.

Answer 2

Take a rectangle (with sides $a$ and $b$). The "volume" is given by $V=a b$; the "surface" is given by $S=2(a+b)$. Fix the ratio to $$r = \frac{S}{V} = \frac{2(a+b)}{ab}.$$ You can solve this equation for $$b = \frac{2a}{a r -2},$$ valid for $ar>2$. Plugging it into the expression for the volume, we obtain $$V= \frac{2 a^2}{a r -2}.$$ You can see that the volume grows arbitrarily large as $a\to2/r$.