Illusory works of art have a curious fascination. They represent a triumph of art over reality. They are illogic masquerading as logic.

Why do illusions capture our interest? Why have so many artists gone to the trouble to produce them? Mountain climbers say they scale mountains "because they are there." Perhaps we seek illusions because they aren't there.

Waterfall by M. C. Escher.

We have all admired the lithograph Waterfall by Maurits C. Escher (1961). His waterfall recycles its water after driving the water wheel. If it could work, this would be the ultimate perpetual motion machine that also delivers power! If we look closely, we see that Mr. Escher has deceived us, and any attempt to build this structure using solid masonry bricks would fail.

All M. C. Escher works © - Cordon Art B. V. - P.O. Box 101-3740 Ac - Baarn - The Netherlands. All rights reserved. M.C. Escher (TM) is a Trademark of Cordon Art. Used by permission.

ISOMETRIC DRAWINGS

The conventions of classical perspective are very effective at simulating such reality, permitting 'photographic' representation of nature. This representation is incomplete in several ways. It does not allow us to see the scene from different vantage points, to walk into it, or to view objects from all sides. It does not even give us the stereoscopic depth sensation that a real object would have due to the lateral separation of our two eyes. A flat painting or drawing represents a scene from only one fixed viewpoint, as does an ordinary monocular photograph.

One class of illusions appears at first look to be ordinary 'perspective' renderings of solid, three dimensional objects or scenes. But on closer examination, they reveal internal contradictions such that the three dimensional scene they depict could not exist in reality. These pictures have a special fascination for those of us used to the convention of depicting nature on a flat surface of paper, canvas, or in a photograph.

Isometric illusory art was created as early as 1934 by Swedish Artist Oscar Reutersvärd with the impossible arrangement of blocks shown here. The colors in this version are not to be blamed on Oscar. This design has been widely used, and even appears on a Swedish postage stamp.

THE PENROSE ILLUSION

The picture appears to depict three bars of square cross section joined to form a triangle. If you cover up any one corner of this figure, the three bars appear to be fastened together properly at right angles to each other at the other two corners—a perfectly normal situation. But now if you slowly uncover a corner it becomes apparent that deception is involved. These two bars that connect at this corner wouldn't even be near each other if they were joined properly at the other two corners.

The Penrose illusion depends on 'false perspective', the same kind used in engineering 'isometric' drawings. This sort of illusion picture displays an inherent ambiguity of depth, which we will call 'isometric depth ambiguity'.

Isometric drawings represent all parallel lines as parallel on the flat page, even if they are tilted with respect to the observer in the actual scene. An object tilted away from the observer by some angle looks the same as if were tilted toward the observer by the same angle. A tilted rectangle has a two-fold ambiguity, as demonstrated by Mach's figure (right), which may be seen as an open book with pages facing you, or as the covers of a book, with the spine facing you. It may also be seen as two symmetric parallelograms side by side and lying in a plane, but few people describe it that way.

The Thiery figure (above) illustrates the same artistic deception.

Schroeder's reversible staircase illusion is a very `pure' example of isometric depth ambiguity. It may be perceived as a stairway that one could ascend from right to left, or as the underside of a stairway, seen from below. Any attempt to draw this with proper perspective vanishing points would destroy the illusion.

The illusion can be enhanced by adding recognizable figures, as in the version at the right is © 2001 by John C. Holden. It should carry an OSHA warning: Caution: Illusory stairways can be hazardous.

Mach's figure, or the open book illusion, can be the basis of even more deception. This construction shows the intimate symbiotic relation between mathematics and physics.

The simple design below looks like three faces of a string of cubes, seen either from the outside, or the inside. If you put your mind to it, you can see them as alternating: inside, outside, inside. But it's very hard, even if you try, to see at as simply a pattern of parallelograms in a plane. This is the same as the 'tumbling blocks' design sometimes used in quilts.

Blackening some facets enhances the illusion, as is shown below. The black parallelograms at the top are seen either as from below, or from above. Try as hard as you can to see them as alternating, one from below, one from above, and so on, left to right. Most people can't.

The design at the right uses the tribar illusion relentlessly in strict isometric drawing style. This is one of the 'hatching' patterns of the AutoCAD (TM) computer graphics program. They call it the 'Escher' pattern.

The isometric wire-frame drawing of a cube (below left) shows isometric ambiguity. This is sometimes called the Necker cube. If the black dot is on the center of a face of the cube, is that face the front, or the rear face? You can also imagine the dot is near the lower right corner of a face, but still you can't be sure if it is the front or rear face. You have no reason to assume that the dot is in or even on the cube, but might be behind or in front of the cube, since you have no clue to determine the relative size of the dot.

If the edges of the cube are given a suggestion of solidity, as if the cube were made of wooden 2x4s nailed together, a contradictory figure results. But here we have used ambiguous connectivity of the horizontal members, which will be discussed in the next section. This version is called the 'crazy crate'. Perhaps it would serve as as the frame to build a shipping crate for illusions. Nailing the plywood faces onto the frame to complete the crate would be a real challenge, but necessary to keep the illusions from falling out!

PHOTOGRAPHING ILLUSIONS

The crazy crate cannot be made of lumber. Yet the photograph shown here is of something made of lumber, something that certainly looks like the crazy crate. It is a cheat. One piece, that seem to pass behind another, is actually two pieces with a break, one nearer, one farther than the crossing piece. This only seems to be a crate from one particular viewing point. If you looked at the real thing from near this point, your stereoscopic vision would give the trick away. If you moved your head away from the viewing point for which it was designed, you'd see the trick. In museum displays of this you are forced to look through a small hole in a wall, using only one eye.

To make such a photograph, one has to engage in deception. If an ordinary camera is used, the more distant lumber pieces subtend a smaller angle than the nearer ones. So the more distant ones must be made physically larger, and those that have one end nearer than the other end must be tapered in size from one end to the other.

There's another way to accomplish this for smaller objects. The small model below left is made of plastic Quobo ® bricks, 1 cm high. The entire model is over 7 cm high. Notice that there's a size disparity where the nearer yellow horizontal tier touches the more distant red brick. But in the picture to the right, there is no size difference there. Note also in the picture to the right, that all bricks subtend the same angle, opposite edges of the green base are parallel and all other parallel lines of the model are parallel on the picture. This is an isometric photograph.

The normal photo on the left shows the chair and lamp behind, as well as other clutter of a small workroom. It was taken with a digital camera with the subject only about 30 cm from the lens.

The photo on the right was taken with the same camera, and approximately the same subject distance. But a telecentric optical system was used, consisting of a large 13 cm diameter lens placed with its focal point very near the camera's own lens. This particular large lens didn't have high quality (it was molded, not polished), so the resolution of the picture is poorer. Such systems suffer from the problem that any dust or scratches or other defects on the lens can show in the final picture. Use of a single lens also produces "pincushion" distortion that renders straight lines as slightly curved.

Telecentric lens systems of high quality are used in industry for product inspection, and in microscopy, for increased depth of focus (DOF). They are limited to photographing small objects smaller than the diameter of the front surface of the lens. See: Telecentric systems.

For some subjects one can "get away" with this kind of deception by using a telephoto lens of high magnification and placing the subject very far away from the camera.

AMBIGUOUS CONNECTIVITY

Some use the general term "undecidable figure" to describe these pictures. That term is so broad that it could be applied to nearly all illusions.

Here's an illusory musical tuning-fork, with only two tines. The figure on the right shows its perspective, with vanishing points.

ILLUSIONS OF SHAPE

These illusions take advantage of the way our brains process information containing repeating patterns. One regular pattern can dominate so strongly that other patterns appear distorted.

A classic example is the pattern of concentric circles with a superimposed square. Though the sides of the square are absolutely straight, they appear curved. The straightness of the square's sides may be checked by laying a ruler along them.

Perfectly straight and parallel lines superimposed on a pattern of radial lines seem to be curved, and the strength of the curvature depends on how near the straight line is to the center of the radiating lines. This is one of the Herring illusions.

ILLUSIONS OF SIZE.

Some have 'explained' this illusion as a result of our experience with perspective in photographs and works of art. We interpret the two lines as 'parallel' lines receding to a vanishing point, and therefore the circle not touching the lines must be nearer, and hence larger.

The same picture is shown (above right) with darker circles, and the parallel lines have become part of dark triangles. If the 'receding parallel line' theory were correct, this illusion should be weaker. You be the judge.

The width of the brim of this hat is the same as the hat's height, though it doesn't seem so at first. This is the classic "plug hat" illusion. Try turning the picture on its side. Is the illusion the same? This is an illusion of relative dimensions within a picture, which is a distortion of shape.

A related illusion is observed in 3d. Find a drinking glass, pill bottle or other cylindrical container. The "zombie" glasses are perfect for this demonstration as a "bar bet". Try to judge by eye whether the cylinder's circumference is larger or smaller than its length. With a tape measure or just a piece of ribbon, string or tape, measure the circumference. Then lay off the ribbon along the length. You may be surprised, for we usually judge the length to be greater then the circumference—until we measure them.

The picture shows several cylindrical containers that can be used to demonstrate this illusion. The 26 oz peanut can on the left shows the illusion very well. The length of the vertical red twist-tie (from supermarket lettuce) is exactly equal to the can's circumference. This can be verified by wrapping it around the can. The other containers include common prescription pill bottles.

All of these illustrate the fact that our visual judgment of length is flawed. We just can't visually compare lengths of lines of different shape reliably. The clearest demonstration of this is shown here.

ILLUSIONS OF ALIGNMENT

AMBIGUOUS ELLIPSES

Improper connectivity is also an essential element of this ambiguous ring illusion:

And here's a more elaborate version of it.

Cover about one third of the picture at either end, and the rest of the picture looks like part of a normal ring or washer. This may remind you of a Möbius strip model made by giving a paper strip a half twist and joining the ends. But this is different. Two colors have been used, for this figure, unlike a Möbius strip, has two faces. You may wish to think of this as a Möbius strip model made from thick, flexible material, its face one color and its edge another color.

One reader says this isn't an illusion, for you can make one of flexible material, and he mailed me a model made from foam strip. However, while you can do this with a strip of square cross section, my illusion above seems to have a rectangular cross section. The wide face magically morphs into a narrower face. That is the subtle aspect of this illusion that most people don't notice right away. He makes a valid point, however, he'll have to go to a lot more trouble to make each face change width as you go around 180°.

When I devised this picture I thought that it might be an entirely original illusion. But then I noticed an advertisement with the corporate logo of the Canstar corporation [above left], a manufacturer of fiber optics. This is another case of two great minds independently inventing the non-existent wheel! If we dig deeply enough, we'd probably find even earlier examples.

Now [October 2003] I find this ring illusion [above right] on the web, without any credit to me, even though the proportions and layout match mine perfectly. At the site where I found this version, there was no clue who drew it. Such is the Internet. If the person who borrowed this idea will come forward, I'll acknowledge that person here. At least it indicates that someone was taken by the idea. I have changed the color of the version I found, because I considered it ugly.

Impossibly linked ambiguous rings. © 2004 by Donald Simanek.

Finally [Dec, 2004], this illusion evolves into something more interesting. Here two ambiguous rings are ambiguously linked. All of these illustrations are available in higher-resolution versions on request. Readers have suggested several names or captions for it: "the interconnectedness of everything," "a new atomic theory," "tying mental knots," "super-colliding synchronous orbitals," "illusory quantum entanglement," (I like that one.) and "virtual unreality."

THE ENDLESS STAIRCASE

The other classic Penrose illusion is the "impossible staircase". This illusion is often rendered as an isometric drawing, even in the Penrose paper. Our version is identical to that of the Penrose paper, except for its lack of shading. The color version to the right allows you to follow a particular color on a step through the layers below. You discover that there aren't enough layers for all the steps.

Ascending and Descending by M. C. Escher.

This could be drawn with vanishing points in full perspective. M. C. Escher, in his 1960 lithograph Ascending and Descending, (above) chose to construct the deception in a different manner. He placed the staircase on the roof of a building and structured the building below to convey an impression of conformity to strong (but inconsistent!) vanishing points. He has the right vanishing point higher than the left one.

One task artists have not yet successfully addressed is to draw an illusion picture with its shadow. Just as shading could kill an illusion, its shadow could also give away the illusion. Possibly an artist could be clever enough to place the light source in such a location that the shadow would be consistent with the rest of the picture. Maybe the shadow could become an illusion itself! The possibilities boggle the mind.

SEEING ILLUSIONS

Others can see that 'something is wrong' with the picture, but are not fascinated enough to inquire how the deception was accomplished. These are people who go through life never quite understanding, or caring, how the world works, because they can't be bothered with the details, and lack the appropriate intellectual curiosity.

It may be that the appreciation of such visual paradoxes is one sign of that kind of creativity possessed by the best mathematicians, scientists and artists. M. C. Escher's artistic output included many illusion pictures and highly geometric pictures, which some might dismiss as `intellectual mathematical games' rather than art. But they hold a special fascination for mathematicians and scientists.

It is said that people in isolated parts of the world, who have never seen photographs, cannot at first understand what a photograph depicts when it is shown to them. The interpretation of this particular kind of visual representation is a learned skill. Some learn it more fully than others.

Historically, artists learned geometric perspective and used it long before the photographic process was invented. But they did not learn it without help from science. Lenses became generally available in the 16th century, and one early use of lenses was in the camera obscura. A large lens was put in a hole in the wall of a darkened room so that an upside down image was cast on the opposite wall. The addition of a mirror allowed the image to be cast onto a flat floor or table top, and the image could even be traced. This was used by artists who experimented with the new `European' perspective style in art. It was aided by the fact that mathematics had developed enough sophistication to put the principles of perspective on a sound theoretical basis, and these principles found their way into books for artists.

It is only by actually trying to make illusion pictures that one begins to appreciate the subtlety required for such deceptions. Very often the nature of the illusion seems to constrain the whole picture, forcing its `logic' on the artist. It becomes a battle of wits, the wit of the artist against the strange illogic of the illusion.

Display Case For Illusions, © Donald E. Simanek, 1996.	Belvedere by M. C. Escher.

Now that we've discussed some of the deceptions that may be used in artistic illusions, you may use them to create your own illusions, and to classify any illusions you run across. Soon you'll have quite a collection, and will need some way to display them. I've designed an appropriate glass display case, shown on the left.

The reader may wish to check the consistency of the vanishing points, and other aspects of the geometry of this drawing. By analyzing such pictures, and trying to draw them, one can gain a real understanding of the deceptions used in the picture. M. C. Escher used similar tricks in his architecturally impossible Belvedere (above right).

ADDITIONAL READING

• Oscar Reutersvärd, founding father of impossible objects.
• Oscar Reutersvärd.

-- Donald E. Simanek

References:

Page created 1996, slight revision, 2014.

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.

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