I've been working on a proof to determine the surface area of a cube based on how many times you cut it vertically in half (cut the width in half) and then cut the resulting prisms in half a given number of times. I think that the equation is exponential, based on the graph that I generated in a java program that I wrote, but when I try to find the exponent based on the "y = Pe^(rt)" equation, the r value is consistent between x values, but not between y values.
For example, if I have a cube with side length 3, and cut it and its prisms in half 0 times, the resulting surface area of all remaining objects will be 54. If I cut it one time the total surface area becomes 72. The resulting r value for this when using regression is around 0.287682. However if I use regression for the next set of points (1, 72) and (2, 108), the value jumps to 0.405465.
However, if I have a cube with side length 4, and cut it and its prisms in half 0 times, the resulting surface area of all remaining objects will be 96. If I cut it one time the total surface area becomes 128. The r value is STILL 0.287682. And when I use the next set of points (1, 128) and (2, 192), the value is STILL 0.405465.
Why does this value change? Are .287682 and .405465 special in some way, or is that just the way that exponential functions work? My real question is, if this function isn't exponential, then how can it be written?
As your graph demonstrates, the total area after $n$ cuts is $6A+\sum_{k=1}^n 2^k A = (4 + 2^{n+1})A$, where $A$ is the area of one face. Here I have used the formula for the finite geometric sum
$$\sum_{k=1}^n r^k = \frac{r-r^{n+1}}{1-r}.$$
This is not exponential until you subtract off the $4A$, which is why the successive ratios increase with $n$. On the other hand, the successive ratios also do not depend explicitly on $A$, which is why you got the same ratios when you took two different choices of $A$. It turns out that the exponential factor $r$ in the usual exponential regression (which is to say linear regression applied to $\ln(y) = rt+\ln(P)$) depends only on these successive ratios and not on the absolute numbers. The absolute size of the numbers is entirely absorbed into the prefactor $P$.