A sudden smiley? :-) This is a vague question, and I will no doubt be (properly!) chastised for posing it.
I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which
has the property that, viewing $S$ under orthogonal projection along a random direction $\vec{u}$ results
in a more-or-less generic, undistinguished cloud of points. But, there is some specific projection
direction $\vec{u^*}$, where suddenly (if one were 3D-rotating the points under mouse control) the cloud resolves itself, through unlikely point alignments, to paint a recognizable image, e.g.,

           

Is this an impossible :-) hope?
Update.
Following Michael Murray's recipe, with $10,000$ points within a cube in $\mathbb{R}^3$, three different viewpoints:

     
(Somehow my analytical smiley has a Halloween evil glint!)
PS(31 Oct 2012).  Happy Halloween!

          


Another addition (23Jun2018):

          


          

(Image from John Urschel / MoMath video.)


 A: If you have no desire for compactness, you can choose the set of points to be all of the points in $\mathbb R^3$ which lie in the inverse image of :-) under the orthogonal projection. Then the image under a different orthogonal projection will always be a union of intervals times $\mathbb R$, giving no indication of the shape of the smiley. 
On the other hand, if you demand that the points lie in a ball of radius $R$, then if you shift the angle of the projection by $\epsilon$, then you shift the image of each point by $O(\epsilon R)$, so the new image will be very similar: each point of the new image is within $O(\epsilon R)$ of a point of :-), and each point of :-) has a point of the new image within $O(\epsilon R)$ of it. In particular for $\epsilon R$ small enough this will be visually indistinguishable.
A: I would like to see such a subset of $\mathbb{R}^3$ having the property that  several of its shadows are recognizable.  For example, I would like to see sets of points whose shadows realize the actions of two or more generators in a Cayley graph.
See this question for more details.
A: Or the other way, a Sudden 2013 :)
A: 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...
A: 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. 
A: 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)
A: Google "shadow sculptures".
Here are some links:
http://www.mymodernmet.com/profiles/blogs/incredible-shadow-scuptures-16
http://fractalenlightenment.com/722/artwork/shadow-optical-illusions-ix
http://www.subtielman.com/shadow-sculptures/
http://www.feeldesain.com/shadow-sculptures-tim-noble-sue-webster.html
http://www.technovelgy.com/ct/Science-Fiction-News.asp?NewsNum=297


(source: ning.com)

A: A Painting Made From Pieces of Glass

