I would think the easiest thing to do would be to just increase all of the measurements in the instructions by whatever factor that you want, so that you keep the ratios the same.
So for example, if you wanted the whole harmonograph to be twice as big, just multiply all of the measurements by 2. So like the 3x3 table top would then be 6x6, and you would do that for all of the pieces. Is that what you're asking?
Now, for describing the actual pictures, volumes and shapes, that looks like straight up calculus. A good math program would be useful for sure.