I have an idea to design a type of _Galton's Board_ to "draw" a relief map of a given two-dimensional function $z=f(x,y)$. A typical <a href="http://en.wikipedia.org/wiki/Bean_machine">_Galton's Board_</a> drops, say, ping-pong balls through a series of evenly spaced pins into vertical bins to demonstrate that the balls distribute according to the binomial distribution, approximating the normal distribution: <br /> <img src="https://i.sstatic.net/rODuS.jpg" alt="Galton's Board" /> <br /> (See this <a href="http://animation.yihui.name/prob:bean_machine">this link</a> for an animation.) First I would like to generalize this design to approximate an arbitrary function $y=f(x)$, which leads to my first question: <b>Q1.</b> Which class of functions can be represented as a convex combination of normal distributions? I know these functions are called <a href="http://en.wikipedia.org/wiki/Mixture_distribution">mixture distributions</a>, but I have not found a description of the total class representable. I am hoping that (say) any smooth function can be approximated. <b>Q2.</b> Given a function $f(x)$ to approximate, how could one work backward to a pin distribution that would realize the approximation? The result would be a type of user-designed <a href="http://en.wikipedia.org/wiki/Pachinko">Pachinko machine</a>. <b>Q3.</b> Can the above be generalized to two-dimensional functions $f(x,y)$? Presumably the answer is _Yes_. If so, one could imagine a potentially mesmerizing <a href="https://mathoverflow.net/questions/50343/">Museum of Math</a> display in which some famous visage emerges slowly as a ping-pong relief map. <br /> <img src="https://i.sstatic.net/ri3A4.jpg" alt="Imprint Toy" /> <br /> <b>Q4.</b> This final thought raises the question of which mathematician's face would be simultaneously most appropriate and most recognizable. :-) <a href="http://en.wikipedia.org/wiki/Galton">Sir Francis Galton</a> is certainly appropriate...