"Optical Illusion" in 4D
(Apologies in advance for the wordiness - not very mathematical, I know!)
I open my book of Escher optical illusions and look at the 2D page. "Aha!" my brain says - "That image doesn't make sense when I try to interpret it as a representation of a 3D scene. Isn't Escher clever?"
Conceptually, I could throw a Flatlander into the Escher picture and he could explore the 2D page. The shapes on the page wouldn't make much sense to him but he could move around them (and possibly peek inside them to explore their interiors, by cutting them open) all the same. Patiently I could explain that there's this third dimension unknown to him and if only he look at the shapes from this third dimension, he might understand the cleverness of Escher and how the perfectly-valid 2D shapes combine to make an optical illusion for 3D beings. If he's bright enough, he might actually be able to conceptualise the mathematics of viewing things in three dimensions.
So, by analogy, a hypothetical 4D being could open her "book" of optical illusions and look at the 3D page. Her brain would be suitably fooled and amused as she examined the 3D objects and tried to interpret them as a 4D scene.
If this statement is correct (and perhaps it isn't - perhaps an "optical illusion" in 4D is provably impossible?), then it should be possible for us to construct some 3D solids (possibly with non-trivial interiors) and arrange them in a way which we could convince ourselves would imply an optical illusion if only we were to look at them from a 4D point-of-view.
Would it be possibly to construct and arrange such 3D shapes, do you think? How we would go about proving that they'd form an "optical illusion" in 4D?
PS I'm not interested using time as a fourth dimension - strictly spatial dimensions, please.
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$\begingroup$We run into definitional trouble quickly.
I don't think we can usefully separate the concept of an "optical illusion" from the fact that your brain contains a mechanism for reconstructing a 3D scene given a 2D stimulus received at your retina. An optical illusion is when this mechanism almost but not quite can construct a 3D hypothesis that matches the mechanism. (A completely nonsensical jumble of lines, lights and shadows is not an illusion).
As such, the very idea of "optical illusion" is tightly dependent on the limits of, specifically, the human brain as a means of reconstructing 3D scenes, and it's not clear that we would even agree with other 3D creatures what good optical illusions are.
Note especially that the brain is fallible and sometimes creates a wrong model, or claims that the stimumus is impossible when it is actually produced by a real 3D scene. For example many of the images you get from a Google Image search for "impossible box" (try it) are photographs of real, physical objects that have been carefully constructed to look like illusions (when seen from the right angle and in the right lighting). So we can't just define "illusion" as a 2D stimulus that looks like it was generated by a 3D scene locally but can't possibly be created by a real 3D object.
But if we can't even agree with other 3D beings what an optical illusion is, what hope have we of predicting which kind of 4D scenes the brain of a hypothetical 4D creature will consider "reasonable enough" explanations of a 3D picture that they don't count as illusions to him?
Bonus musings: There are even arguments that it is not even biologically intrinsic to being human to be able to view a flat perspective image (say, at an angle, or from the wrong distance to work as an actual trompe-l'oeil) and decode its content accurately -- rather it is a learned skill that depends on familiarity with such images.
Even stronger, the illusory content of the "impossible fork" construction depends on conventions about contour drawings that certainly have a cultural component. It is technologically possible to imagine a future where line drawings are a lost art (or perhaps only appreciated by a small highbrow elite) and every picture of 3D scenes the average citizen sees in daily life aims for photorealism, whether they are actual photographs or CGI. Would someone who grows up in such a culture even be able to recognize the impossible fork as making sense locally?
$\endgroup$ $\begingroup$Yes! Scott Kim did exactly that back in 1978 with an article titled "An Impossible Four-Dimensional Illusion" It takes a Penrose Triangle and extends it to 4 dimensions. This can be constructed in 3 dimensions and is available at
Though the PDF's seem to be corrupted.
$\endgroup$ 1 $\begingroup$There are different definitions (or rather, categories) of what is an "optical illusion", but the example you are talking about is a special case of ambiguous information. In general, there is an infinite number of 3D interpretations from a 2D image. The basic reason is that you are missing the information on the "depth" direction. Some illusions result in a "local" structure that doesn't make sense when you try to interpret it in a global way, but even with that knowledge it is almost impossible to change the perceived local structure (but that is a limitation of our visual system because it not always look for a globaly consistent solution) . Our brain, or any 3D recovering algorithm tries to give the 2D image the most likely interpretation, based on past experience or on some a priori probability distribution. So yes, any 3D object has an infinite number of interpretations as a 4D object or scene, and depending on the observer's visual algorithms, even some that are inconsistent at a global level.
$\endgroup$ $\begingroup$Scott Kim here, author of the 1978 paper referenced above. The impossible triangle illusion, it turns out, is straightforward (if mind boggling) to generalize up to higher dimensions. To the extent that we can assume that 4d vision works like our 3d vision, the illusion works out just fine. The key is to realize that the impossible triangle traces a path that goes a fixed distance along each of three perpendical axes, traveling from (0,0,0) to (1,1,1) in coordinate space. So if you look down the line x=y=z toward the origin, (1,1,1) will appear to coincide with (0,0,0). That's true by analogy in any number of dimensions, though the 2d case (0,0) to (1,1) collapses to a line segment that is too reduced to make a good illusion).
What my paper doesn't mention is that one can also generalize the infinitely ascending staircase illusion up a dimension, and it works BETTER as a 4d illusion than as a 3d illusion. Many other types of optical illusions (like convex/concave illusions) generalize nicely up a dimension, while others don't so easily.
Here's the real kicker. Both of these classic illusions were discovered (or rather, re-discovered) by famous physicist Roger Penrose, working with his psychologist L. S. Penrose. He sent a paper they wrote to M. C. Escher, who worked these illusions into his art. Penrose, being a physicist, is very comfortable thinking in 4d, so when he invented these illusions he also worked out their 4d counterparts. So after I worked these out for myself I wrote to him, and he confirmed that he had already made a model of them!
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