Say I have here a cube with side length $1$. If I wanted to double its volume, I would have to add a layer of material $0.16109$ units all around it. If I wanted to make that same cube have triple volume of the original, I would have to add a layer of $0.09469$ units thick of material on top of the first extra layer. I used these equations here:
$1=3(2x⋅1^2)+(2x)^3$ for the first layer,
$1=3(2x(1.32218^2)+(2x)^3$ for the second.
Now I want to graph volume to thickness of the added layer like $(1;0)$, $(2;0.16109)$, $(3;0.09469)$, etc.
I will take different variables:
let $s$ be the sidelength of the cube
and $2a$ the sidelength increase
(coefficient $2$ is explained by the fact we want to have a "$a$" increase on each side)
Let us express the condition under the form:
$$(s+2a)^3-s^3=1$$
which is equivalent to:
$$6s^2a+12sa^2+8a^3-1=0. \tag{1}$$
Let us consider (1) as a quadratic in variable $s$ and parameter $a$. Its positive root is:
$$s=\dfrac{1}{6a}(-6a^2+\sqrt{6a(1-2a^3)})$$
with the following representation:
Remark: coefficients $6, 12, 8$ in (1) are resp. the number of faces, edges, and vertices of the cube. The following figure shows the interesting interpretation between the terms of (1):