Q:

What equation describes the face of a cube?

Picture a huge cube shaped planet of edges of many thousand miles. Assume that the distribution of the material (say oak wood) within the cube is homogenous.
Now, imagine the oaklings doing geodesy. They do not have the slightest idea about the shape of their planet. They just walk around and try to mathematically describe the surface. This is how they picture their world before they would discover that they actually live on the surface of a huge cube:

The red sketch is an artistic reproduction of the planet, as one of the oaklings pictured before the great discoveries. 
(I have been trying to write up the equation with no success. As far as I can imagine the oaklings measured the angle between the axis of their body and the floor. From this angle (changing everywhere) I couldn'n make up the equation.)
EDIT
What do I mean by an equation? Let's say, the length of an edge is $2d$. Let the origin of our coordinate system be located at the center of a face of the cube. An equation would look like this one below:
$$\operatorname{height}(x,y)=\begin{cases}
x^2+y^2&\text{ if } (x,y)\in [-d,d]\times[-d,d]\\
\text{undefined}&\text{otherwise}.
\end{cases}
$$
Unfortunately this is not the right equation.

A:

I’m not sure that you’ve provided enough information to come up with a definitive equation, but here’s one possibility. It looks like you’ve got square tiles with “altitude” given by the distance from the cube’s center.  
For a single tile, consider a cube with side length $2s$ centered at the origin and aligned with the coordinate planes. Using spherical coordinates, the “altitude” that the oaklings would measure along the top face of this cube is simply $s\sec\theta$. We also have $\tan\theta=R/s$, where $R$ is the distance from the center of the face, so relative to the center of a tile the altitude is given by $$s\sqrt{1+(R/s)^2}=\sqrt{s^2+R^2}=\sqrt{s^2+(x-x_c)^2+(y-y_c)^2}$$ where $(x_c,y_c)$ are the coordinates of the tile center in the “flattened” version of the world.

