Question

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!
          

Answer 1

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.

Answer 2

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)}

Answer 3

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...

Answer 4

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

Answer 5

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.

Answer 6

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.