Older computer screens don't have have precisely linear geometry. The stereo illustrations look best when viewed on an LCD screen, or from a printed copy. I have used 16 color GIFs to minimize loading time. These were derived from source files created with DesignCad, and interested persons can request source files in that format, or in DXF format. They have been converted to bitmaps for presentation here.

Disclaimer: Some of the isometric illusions below were creations of Swedish Artist Oscar Reutersvärd. My specific stereo interpretations of them are not to be blamed on him. I am not the first to do stereo versions of these. I would welcome any documented references on the history of these illusions. Those that are my original creations are so indicated.

### ILLUSIONS WITH PERSPECTIVE

Since many of the common illusions seem to depend on "false" perspective, or on the lack of true stereoscopic depth, we might inquire how illusions could be constructed with true perspective, and possibly even in full stereo depth. Conventional wisdom holds that such illusions are destroyed when depicted with accurate stereoscopic depth.

The ultimate challenge would be to create illusory sculpture. Some illusions can be realized as sculptures, but must be viewed from one particular point with just one eye. Such is the case with the Necker cube, above, sometimes drawn as shown above right, and called the "crazy crate."

We will begin by considering whether some of the common illusions could be altered to include perspective.

### ISOMETRIC ILLUSIONS

Many classic "tribar" illusions are conventionally drawn in isometric fashion, in which parallel lines are rendered parallel on the page, there being no convergence toward a vanishing point. Objects of the same size are also the same size on the drawing surface no matter how far away they are. Engineering drawings are often isometric to make it easy to preserve relative lengths no matter what the inclination of a line to the observer's line of sight. Isometric drawings do not preserve angles. Any angle of 90° is rendered as 120°. Perspective drawing doesn't preserve angles either. Cartesian coordinate systems are often depicted isometrically, with one axis vertical, and the other two axes making angles of 120° with it.

Isometric drawing is essential for some of these illusions. The ambiguous staircase illusion would lose its illusory character if drawn in true perspective. But a few of the isometric illusions can be successfully rendered in perspective, as I will demonstrate.

In fact the prototype of them all, the "Penrose" illusion, was first presented with a distinct suggestion of perspective.[1] We show it that way here. Each bar has convergence toward a vanishing point. The original drawing, in the Penrose paper, was shown with some shading as well.

Isometric illusions depend upon two deceptions.

1. The false perspective in which even supposedly receding parallel lines remain parallel. There's no size change with distance.
2. Ambiguous or impossible connection between picture elements. Picture elements are connected, even though perspective cues tell us they should not be near each other.
The example below left is shown in isometric style. The conventional wisdom has been that perspective rendering of such pictures is not possible, or would destroy the illusion.

The same illusion is shown at the right in perspective, using two vanishing points. The illusory nature of the object is certainly not diminished by this presentation, though I don't claim that it is greatly enhanced.

Some persons experience an interesting effect while comparing these two pictures. Look at the perspective view for a while, then shift your attention to the isometric view at the right. The isometric view may now appear "wrong" or "warped", and you may feel that the top and bottom of the frame are not parallel, and diverge at the rear of the frame!

### STEREO DRAWING

Stereo drawings require two pictures, one for each eye. To view such drawings requires some practice. Generally two methods are used: (1) parallel viewing and (2) cross-eyed viewing. Both methods require one to learn a new visual skill.

Normally when we look at the "real" world, our eyes converge on an object and they also focus on the same object. We habitually do this, and our brains have become accustomed to a one-to-one correspondence between focus and convergence.

We can learn to "unlock" focus and convergence, enabling us to view stereo pairs without optical aid. In this document we use a display method which can be adapted to either parallel-eye viewing or crossed-eye viewing. Three drawings will be shown side by side. The middle drawing is to be viewed with the right eye. The other two are identical and one is to be viewed with the left eye. Here's how to view them.

(1) Parallel viewing. This is sometimes called wall-eyed viewing. Use the left and middle pictures only. Look at a distant object then bring your eyes down to the paper trying not to converge or focus on the paper. You'll see a blurred double image. Consciously try to bring the double image into one. Now try to focus your eyes on it without allowing the two images to drift apart.

Parallel viewing is limited to pictures about 2.5 inches wide, the spacing of the typical human eye.

(2) Cross eyed viewing. Use the right and middle pictures only. Hold your finger exactly halfway between your eyes and the page. Focus on your finger. Your eyes will be converged on the finger also, and you should be conscious of the two pictures out of focus behind your finger, but probably nearly coincident. Move your finger a bit until the two pictures are fully coincident. Now hold the convergence while refocusing from your finger to the page. Finally, you can remove your finger from the field of view.

Crossed eyed viewing can be used with large pictures. Paintings have been presented this way. Salvadore Dali painted some of this kind. For those who can learn the skill of viewing, this is one of the most effective ways for viewing stereo without special glasses.

(2) Mirror method. For completeness, we illustrate a method rarely used in printed books. Popular Photography magazine experimented with it for a while. One picture of the pair is printed normally, ad the other is printed beside it, but reversed left/right. That picture is viewed through a mirror which re-inverts it. We will not use that method here.

For practice, try this illustration from Sir Charles Wheatstone's book The Stereoscope.[2]

Don't expect to succeed the first time. This skill takes conscious effort and concentration. When you do succeed, you'll see the pictures snap into full three-dimensional depth. The picture will look like a wire-frame box. You'll actually see two 3-D images, one with normal depth, one with inverted (pseudoscopic depth). On either side of these you'll see fainter, phantom images with no depth. Ignore them.

Here's some more practice examples:

If you use cross eyed viewing on the pair intended for parallel, or vice versa, you will see a "pseudoscopic" depth, in which near and far are reversed. In the first picture, the pseudoscopic view appears as if you are looking down onto a truncated pyramid. In the second picture, the cone seems upright in the normal view, but tilted back and viewed from its base in the pseudoscopic view. Wire-frame stereo drawings often look interesting either way.

Here's another example for practice.

This coiled spring is more difficult to view:

Now, can illusion pictures be drawn this way? Some can. The three-tined fork illusion, sometimes called "Schuster's conundrum," succeeds remarkably well. This is strictly an illusion of ambiguous connectivity; there's no depth ambiguity at all.

Here's my color 3d rendition of the classic "Crazy Crate".

Let's try the Penrose Illusion (impossible triangle). Here we use the fact that a horizontal line has ambiguous depth even in stereo. So we've oriented the triangle with one side horizontal. The other sides have been given true stereoscopic depth, but no perspective depth cues are used.

But now try viewing this version. Here we haven't used the cheap trick of horizontal lines. We've used a different cheap trick. We've simply expanded the horizontal dimension of one picture by about 5%.

Why should this work at all? It seems to defy logic. Let's try the same trick with some other isometric pictures.

And another:

Feel free to view any of these pseudoscopically. It doesn't seem to matter a lot. You get a vague sensation of stereoscopic depth either way!

Some people have a weak perception of depth in such drawings even when both pictures are identical! This may be due to the artificial method for viewing them, particularly the slight keystone distortion of each picture when cross-eyed viewing is used. The absence of focus cues may play a role also.

I haven't prejudiced you by suggesting what you should see in these examples. Generally one experiences the same ambiguity of depth, as in the "flat" isometric version, but there's an added cue of stereoscopic depth as well. The stereoscopic depth seems to fluctuate depending on where one fixes one's attention within the picture. Clearly we are getting a conflict of depth and solidity cues. The stereoscopic cues and the isometric perspective cues do not agree.

### ILLUSIONS OF SHAPE

There's a large class of illusions called pattern-dominance or pattern-conflict illusions. They fall within a larger class of illusions of shape.

Pattern-dominance illusions, as usually presented, seem to be strictly due to conflict of overlapping patterns in a single plane. Our perception of the geometry of one pattern is altered by the presence of the other pattern. The illusion seems not to rely upon any suggestion of perspective in the drawing.

Most people judge that the circles in the left drawing a bit off-round, being gently flattened at four places, near the corners of the squares. Few would say that the circles distort the squares in this case.

We can test this by making another drawing (on the right) in which the squares dominate the background field of view, while a lone circle competes with that. Will the circle show distortion, but the squares remain square? Yes, that's what most people see.

Both versions of the illusion persist in stereo rendering even though the circles and squares now lie in different planes when seen in depth.

Some explain the following illusion by claiming the radial lines are interpreted by the brain as parallel lines receding to a vanishing point. This supposedly makes one circle (usually the right one) seem smaller, though they are drawn the same size and therefore subtend the same angle to the eye. Again, I find this explanation unpersuasive.

The Ehrenfels illusion presents a perfect square upon a background of radial lines. The square seems tilted forward. (Or, it appears to be a rhombus, with the top edge longer than the lower edge.) It still seems tilted or distorted when the square is drawn on a transparent sheet held some distance in front of the plane of the radial line pattern. In stereo rendering there's a strong illusion that the square is tilted, the top edge nearer than the lower edge, even though there are no stereoscopic cues to support this interpretation.

A related illusion, the Herring illusion, presents parallel lines against a background of radial lines. The parallel lines appear bowed or bent. They still appear bent if they are on a transparent sheet some distance in front of the plane of the radial line pattern. This fact comes through in stereo rendering also.

This one is repeated below in larger scale, for crossed-eye viewing only.

Some argue that the pattern of radial lines is suggestive of perspective–of parallel lines receding to a vanishing point. I don't find that explanation persuasive. But it's hard to devise pattern conflict illusions in which one or the other of the patterns can't be interpreted as having some characteristic of perspective.

If you are viewing this illusion in stereo from the monitor screen you may see that one of the parallel lines seems nearer than the other. If you view it from the printed page, they seem to be at the same distance. This is due to horizontal non-linearity of the monitor's display. Also, if you have uncorrected astigmatism in one or both eyes you may notice that the radial lines do not appear equally distinct. This is similar to the standard astigmatism test chart, which also has a radial pattern of lines.

### MORE 3D ILLUSIONS

[April, 2002] I finally got around to rendering my gear illusions in stereo.

Phantom Gears 1. © 2002, DES

It's more dramatic when larger. Here's the version for cross-eyed viewing only.

Phantom Gears 1. © 2002, DES

Yet another gear illusion:

Phantom Gears 2. © 2002, DES

This gear illusion in 2D has been "ripped off" by several publishers without credit to me. As I was the originator of it, I'm the best person to explain it's logic.

Phantom Gears 3D. © 2002, DES

As with most illusions, it uses several forms of deception. The following picture shows in the ovals, the central illusion, which is nothing more than Mach's "open book" illusion, shown to the right. It can be seen as facing pages of an open book, or as the front and back cover of a an open book seen from its back. This simple isometric illusion is the basis of the ambiguous staircase illusion as well as many others.

Phantom Gears 3D. © 2002, DES

The 2D version of course came first. This raised the question "Could this be rendered in 3D". The drawing was isometric, which has no classical perspective (no vanishing points, which give apparent depth to flat pictures). But you can still employ stereo parallax in isomemtric drawings, for the stereo disparity overrides weaker depth clues. That's easily accomplished with ellipses (and gears) by altering their width to height ratio. The two gears of the left eye picture are fattened horizontally, and the larger gear of the lelft eye picture is made narower. This is done only on either side of the vertical line passing trhough the illusory meshed gear teeth. In the CAD drawing, this breaks some of the points where lines join, and these must be repaired by hand. All this surgery was done only on the left eye picture. Then when all was fixed in the CAD drawing, it was converted to a GIF, and colorized with a paint program. The colors help to emphasize the illusion, but must be done consisently. Notice that the left gears have teeth with orange tops, but the right gear has orange valleys between the teeth. This is necessary because in the ambigous regions where they teeth mesh, the orange top and valley are the same parallelogram. Similar constraints apply to yellow and blue faces. Coloring these can warp one's mind, leading to mistakes. If any mistakes remain, please let me know.

One might even say that the "colors transfer from one gear to the other" where the teeth mesh. Now if someone wants a real challenge, how about making an animated GIF of these gears rotating in 3d. (I can supply the original CAD structure in DXF format if anyone wants to try.)

### A CHALLENGE

My ambiguous ring illusion (below) makes use of the inherent ambiguity of circles and ellipses seen in perspective. Cover the left or right third of the picture and everything seems conflict-free.

Can anyone do a stereo version of this which still preserves the illusion?

[April, 2002] Abdollah Sadjadian accepted the challenge, exercising his AutoCad wizardry to produce this mind-bending result (for cross-eyed viewing).

Here's another example of ellipse ambiguity. View this either cross-eyed or parallel. One set of rings will seem perfectly normal, with the rings lying in planes nearly perpendicular. But the other one, when examined carefully, will soon begin to seem "wrong", and finally will seem to have an unnatural twist where the rings link. Here's a conflict between stereo depth cues and the drawing cues which tell us which part of the ring is "in front". This is what happens when a magician messes up the Linking Rings trick.

Stereo rendering of illusions can be useful for testing hypotheses intended to explain these illusions. This is a tool which has more possibilities than have been previously exploited.

— Donald E. Simanek, April, 2002.

## References

[1] L. S. Penrose and R. Penrose, "Impossible Objects: A Special Type of Visual Illusion," British Journal of Psychology, 1958. Vol 49, pp. 31-33.

[2] Wheatstone, Sir Charles. "Contributions to the Physiology of Vision. Part the First; On Some Remarkable, and Hitherto Unobserved, Phenomena of Binocular Vision," Philosophical Transactions of the Royal Society, 1838, Part 1, pp 371-94. Reprinted in The Scientific Papers of Sir Charles Wheatstone, London, 1879, pp. 225-259. Online copy, complete.

[3] Seckel, Al. The Art of Optical Illusions. Carlton Books, 2000.

[4] Seckel, Al. More Optical Illusions. Carlton Books, 2002.

Al Seckel's books are, in my biased opinion, the best general illusion collections published, and are very reasonably priced. See these descriptions, and order them from your favorite book source.

This document is an ongoing project, for which feedback is welcomed by the author, who hopes that these drawings can stimulate an exchange of ideas. Use the address shown here. Expect to see additions and changes in this section of my web pages in the future.