In looking at Waterfall, Climbing and Descending, and Belvedere, I am
convinced that there is a unifying perspective principle at work in
these drawings. I haven't quite figured out what it is yet. Escher
uses many tricks, yes, but it is important to analyze the drawings and
figure out what those tricks are. Only when we know exactly what
causes these tricks, can we generalize them to producing 2d illusions
in continuous 3d space.
BTW, these three illustrations can all be found at
http://www.oir.ucf.edu/louvre/paint/auth/escher/stairs/
For example, the waterfall effect
is based on the facts that a) we know water always flows down hill, b)
things further back in a scene will tend to move "up" the page.
Actually, I find it very difficult to read the water as flowing
downhill and back into the page, because of the way the aqueduct
climbs the two vertical towers. I have to deliberately ignore the
towers to see the water flowing downhill.
If one were to translate this into 3d, it would lose one of these
effects.
This I don't automatically buy. In general I think these illusions
are sustained by the artificial introduction of a number of cyclical
height cues, and then hiding the point of discontinuity in the cycle.
This would imply that the illusion depends on the cyclical relations
of all parts of the structure, and is not necessarily limited to 2d.
Even if one point of the cycle becomes discontinuous after a 90 degree
rotation, perhaps the discontinuity can be hidden at that point, and
another one cyclically emerge to take its place. This would allow
full 360 degree traversal of an illusory object.
For example, I've been trying to analyze "Climbing and Descending" in
terms of successive wedged ramps. The small end of the wedge
indicates the low end, and the big end is the top. Now if you try to
draw four successive wedges in ordinary perspective, it doesn't work,
because the continuity of the image forces your wedges to always keep
rising. At some point, one of your wedges has to descend rather than
ascend, in order to look physically correct. But note that just below
the leftmost corner of the staircase, there's a roof that obscures the
foundation plane of the entire staircase structure. This allows the
top-leftmost staircase to begin its ascent from farther up, vertically
speaking, than would be possible if you were paying attention to the
foundation plane.
This brings out a further point: it is not necessary to construct a 3d
model, but merely to hide the discontinuities of the model by
distracting the viewer's attention. Someone mentioned that DOOM is
only 2d. True enough, but it does a good job of distracting you from
noticing this in many circumstances.
I am pretty sure there is a more precise mathematical perspective
analysis underlying all of this. Whether something looks like a
wedge, definitely depends on whether the perspective lines converge or
remain parallel, and whether the 2d projection looks like an ascending
or descending line. But I haven't figure it all out yet.
Can an engine be constructed such that if one were to walk to
the back of the waterfall structure, the illusion wouldn't be
shattered? This would rely upon being able to create the tricks of
perspective and assumptions about the real world for any possible
position of the viewer.
Probably not, but you might be able to walk 3/4 around it. That would
make the experience pretty much 3d.
Cheers,
Brandon