A sudden smiley? :-)

Glad that MO is up and running again.

Following the suggestion by Michael Murray one can also produce more than just one sudden smiley:

I guess, that a higher number of images is also possible. But probably some structure may be visible from other directions as well in this case. By the way: The problem seems to be a bit related to tomography...

Surely you can draw the 2-D image in the XY plane so it consists of points of the form (x, y, 0) and then give each point in it a random non-zero Z co-ordinate. So it should look like a mess except viewed looking in along the Z-axis.

Perhaps the digital sundial of Falconer is what you need: http://www.researchgate.net/publication/225574085_Digital_sundials_paradoxical_sets_and_vitushkins_conjecture

Here is a photograph of a working model: http://apod.nasa.gov/apod/ap120626.html
     
     _{(Image added by O'Rourke)}