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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:
     Smileys: 3 views
(Somehow my analytical smiley has a Halloween evil glint!)

PS(31 Oct 2012). Happy Halloween!
           PumpkinSmiley

show/hide this revision's text 4 Better word choice!

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:
     Smileys: 3 views
(Somehow my analytical smiley has a Halloween evil taint!glint!)
show/hide this revision's text 3 added 328 characters in body

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:
     Smileys: 3 views
(Somehow my analytical smiley has a Halloween evil taint!)
show/hide this revision's text 2 added 12 characters in body
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