Visual curiosities and mathematical paradoxes
When your eyes see a picture they send an image to your brain, which your brain then has to make sense of. But sometimes your brain gets it wrong. The result is an optical illusion. Similarly in logic, statements or figures can lead to contradictory conclusions; appear to be true but in actual fact are self-contradictory; or appear contradictory, even absurd, but in fact may be true. Here again it is up to your brain to make sense of these situations. Again, your brain may get it wrong. These situations are referred to as paradoxes. In this article we'll look at examples of geometric optical illusions and paradoxes and give explanations of what's really going on.
Optical illusions are pictures that play tricks on our eyes and baffle our perception. They are not the result of faulty vision. Depending on light, viewing angle, or the way the picture is drawn, we may see things that aren't there – and often don't see what's right under our nose. These tricks of the eye and mind have been part of human experience since the beginning of history. The ancient Greeks made use of optical illusions to perfect the appearance of their great temples. In the Middle Ages, misplaced perspective was occasionally incorporated into paintings for practical reasons. In more recent times, many more illusions have been created and implemented in the graphic arts.
Architectural optical illusions
Figure 1: the Parthenon in Athens. Image: WPopp.
Illusions have been intentionally incorporated as architectural elements since antiquity, usually to counter the effects of visual distortion. The most famous example is the Greek Parthenon (figure 1).
The temple is based on horizontal and vertical lines which meet at right angles. However, it turns out that the human eye distorts these lines when looking at large constructs. Long horizontal lines, for example, appear to sag in the middle, while two parallel vertical lines seem to spread away from each other as they go up. To counter the effect, the Greeks replaced the most prominent horizontal line by a line that bows upwards in the centre. Every other horizontal line then has to be made parallel to this newly introduced curve. The columns of the Parthenon were made to lean together at the top, just a few degrees, to make them seem parallel. (See  in the reading list below for more information.)
Figure 2: horizontal lines curve upwards and vertical lines lean towards each other. (Image: Allgemeine Bauzeitung, III, 1838, plate CCXXXVIII.)
Ambiguous optical illusions
Figure 3: a Rubin vase (developed by the Danish psycologist Edgar Rubin).
Figure 3 is an example of an ambiguous optical illusion. It is very important that your visual system can interpret patterns on your retina in terms of external objects. To do this, it needs to be able to distinguish objects from their background, which most of the time is quite easy. Ambiguous optical illusions arise when an object is concealed through natural or artificial camouflage. In these cases, both the figure and the background will have meaningful interpretations, which cause a perceptual "flip-flop". (This is explored in detail in .)
In figure 3 you can see either the vase in the foreground or the two faces in the background. At any time, however, you can only see either the faces or the vase. If you continue looking, the figure may reverse itself several times so that you alternate between seeing the faces and the vase. The Gestalt psychologist Edgar Rubin made this classic figure/ground illusion famous.
Figure 4: the Penrose triangle.
In more recent times many more optical illusions have been created and implemented in the graphic arts. Among these are so-called "impossible objects" which make up a unique and fairly new strain in the world of illusions.
The first formal examples of impossible objects were published by Lionel and Roger Penrose in 1958 in their seminal article Impossible objects. They introduced the tribar, later known as the Penrose Triangle, and the endless staircase, later known as the Penrose staircase. It was their work that brought impossible objects into public awareness.
To understand what is going on in figure 4, the Penrose triangle, refer to figure 5. This physical model of the Penrose triangle works from only one special angle. Its true construction is revealed when you move around it, as shown in figure 5. Even when presented with the correct construction of the triangle, your brain will not reject its seemingly impossible construction (shown in the last picture in figure 5). This illustrates that there is a split between our conception of something and our perception of something. Our conception is ok, but our perception is fooled. (You can read more about these impossibles objects in .)
Figure 5: a physical model of the Penrose triangle in Perth, Australia. The construction appears as a triangle only from one angle. Image: Bjørn Christian Tørrissen.
The Dutch artist Maurits C. Escher used the Penrose triangle in his constructions of impossible worlds, including the famous Waterfall (click on the link to see the image). In this drawing, Escher essentially created a visually convincing perpetual-motion machine. It's perpetual in that it provides an endless water course along a circuit formed by the three linked triangles.
Figure 6: the Penrose staircase.
The Penrose staircase (figure 6) is not a real staircase – it's an impossible figure. The drawing works because your brain recognises it as three-dimensional and a good deal of it is realistic. At first glance, the steps look quite logical. It is only when you study the drawing closely that you see the entire structure is impossible. Escher incorporated the Penrose staircase in his lithograph Ascending and Descending. (You can see the lithograph by clicking on the link and you can read more about this in .)
The Penrose stairway leads upward or downward without getting any higher or lower – like an endless treadmill. Escher drew his staircase in perspective, which would indicate another size illusion. The monks that are descending should get smaller and the ones that are ascending should get larger. They don't. In this case Escher was prepared to cheat a little bit. At first glance, the steps appear quite logical. It is only when one studies it more closely that one sees the entire structure is impossible. It is arguably the most reproduced impossible object of all time.
Figure 7: the space fork, also known as a blivet.
Another impossible object is the space fork (figure 7). One notices in the figure that three prongs miraculously turn into two prongs. The problem arises from an ambiguity in depth perception. Your eye is not given the essential information necessary to locate the parts, and the brain cannot make up its mind about what it is looking at. The problem is to determine the status of the middle prong. If you look at the left half of the figure, the three prongs all appear to be on the same plane; in other words, they seem to share the same spatial-depth relationship. However, when you look at the right half of the figure the middle prong appears to drop to a plane lower than that of the two outer prongs. So precisely where is the middle prong located? It obviously cannot exist in both places at once. The confusion is a direct result of our attempt to interpret the drawing as a three-dimensional object. Locally this figure is fine, but globally it presents a paradox. Sometimes this figure is referred to in the literature as a cosmic tuning fork or a blivet.
Paradoxes, sliding puzzles and vanishing pictures
A paradox often refers to an appearance requiring an explanation. Things appear paradoxical, perhaps because we don't understand them, perhaps for other reasons. As the mathematician Leonard Wapner (see ) notes, paradoxical statements or arguments can be categorised into one of three types.
Type 1 paradox: A statement which appears contradictory, even absurd, but may in fact be true.
Figure 8: the Banach-Tarski theorem - turning a marble into the sun.
The Banach-Tarski theorem involves a type 1 paradox, since there is a conclusion of the theorem that appears to contradict common sense; yet, the conclusion is true. The result is that, theoretically, a small solid ball can be decomposed into a finite number of pieces and then be reconstructed as a huge solid ball, by invoking something called the axiom of choice. (The axiom of choice states that for any collection of non-empty sets, it is possible to choose an element from each set.) This may sound like a perfect solution to your financial troubles, simply turn a small lump of gold into a huge one, but unfortunately the construction works only in theory. It involves constructing objects that, although we can describe them mathematically, are so complicated that they are impossible to make physically. You can read more about the Banach-Tarski theorem in the Plus article Measure for measure.
Type 2 paradox: A statement which appears true, but may be self-contradictory in fact, and hence false. Type 2 paradoxes follow from a fallacious argument.
Sliding puzzles and vanishing pictures are paradoxes of this type, as we shall point out later in this section.
Type 3 paradox: A statement which may lead to contradictory conclusions. This is also known as an antinomy and is considered an extreme form of paradox, perhaps having no universally accepted resolution.
Russell's paradox and one of its alternative versions known as the barber of Seville paradox is one such example. In this paradox, there is a village in which the barber (a man) shaves every man who does not shave himself, but no one else. You are then asked to consider the question of who shaves the barber. A contradiction results no matter the answer, since if he does, then he shouldn't, and if he doesn't, then he should. You can find out more about this paradox in the Plus article Mathematical mysteries: the barber's paradox.
Sliding puzzles are examples of type 2 paradoxes. These are fallacies which are often difficult to resolve. Let's consider a few of the more famous (or infamous) types of sliding puzzles.
The first type of sliding puzzle we consider is the Nine bills become ten bills puzzle shown in figures 9 and 10. It's an incremental addition/ subtraction sliding puzzle.
Figure 9: Nine bills.
Figure 10: Ten bills.
In figure 9 nine twenty-pound notes are cut along the solid lines. The first note is cut into lengths of one-tenth and nine-tenths of the original note. The second note is cut into lengths of two-tenths and eight-tenths of the original note. The third note is cut into lengths of three-tenths and seven-tenths of the original note. Continue in this fashion until the ninth note is cut into lengths of nine-tenths and one-tenth of the length of the original note.
In figure 10 the upper section of each note is slid over onto the top of the next note to the right. The result is ten twenty-pound notes, when originally there were only nine notes. Casual viewers may be tricked into thinking an additional note has been magically produced unless they measure the lengths of the ten new notes. The deception is explained by the fact that each new note has length nine-tenths of the length of the original twenty-pound note. The more cuts used in such an incremental sliding puzzle, the more difficult it is to detect the deception. Apparently, someone actually attempted this trick in pre-war Austria. (See .)
Another type of sliding puzzle appears to create a hole after sliding is completed. The paradoxical hole puzzle in figure 11 is an example.
Figure 11: The paradoxcial hole.
The square on the left of figure 11 is cut along the solid lines into three pieces, and then the pieces are rearranged as indicated, with the result that a hole appears in the square while the area apparently remains the same!
The deception is exposed in figure 12.
Figure 12: The paradoxcial hole explained.
When you rearrange the pieces of the original square as shown in figure 11 a small difference becomes evident. The resulting square B is in fact a rectangle, as shown in figure 12. Its vertical sides are a tiny bit longer than those of the original square A. The difference in area equals the area of the hole. Hence, no part of the original square A has disappeared, but the area of the hole is redistributed throughout the area of at the bottom of square B.
The vanishing egg puzzle (figure 13) is a hybrid of two types of sliding puzzles: it involves an incremental addition/subtraction and it has a hole appear after the sliding occurs. After cutting the picture and rearranging the pieces there is one less egg. Where did it go? Note that looking at the row of eggs from right to left, the eggs are clearly larger in the bottom picture, with the result that one egg has incrementally disappeared.
Figure 13: The vanishing egg puzzle. Image © G. Sarcone, www.archimedes-lab.org.
This articles touches on just a few of the many types of optical illusions and sliding puzzle paradoxes. If you're interested in more in-depth discussions and a vast array of other examples, have a look at the reading list below.
 Optical Illusions by Gyles Brandreth, Michael A. DiSpezio, Katherine Joyce, Keith Kay and Charles A. Paraquin, Sterling Publishing Co., 2003.
 Lateral Logic Puzzles by Erwin Brecher, Sterling Publishing Co., 1994.
 Architecture of Greece by J. K. Darling gives an interesting historical account of ancient Greek architecture and its utilization of optical illusions.
[xthree] The Magic Mirror of M. C. Escher by Bruno Ernst, Taschen America, 1994 (English translation), includes material on Escher's life, the development of his work, with chapters considering: Explorations into Perspective, Creating Impossible Worlds and Marvelous Designs of Nature and Mathematics, among others.
 M.C. Escher The Graphic Work Introduced and Explained by the Artist, by M. C. Escher, Barnes and Nobles Inc., 1994 (English translation), includes a number of his works, accompanied by his own explanations and comments.
 The World's Best Optical Illusions by Charles H. Paraquin, Sterling Publishing Co., 1987, contains a number of clever optical illusions.
 Impossible Objects, a Special Type of Visual Illusion, by Lionel and Roger Penrose, In the British Journal of Psychology, February, 1958 issue, introduced the concept of impossible figures. Their three examples have spawned an entire area of investigation in the graphic arts; among its most notable proponents being Maurits Escher.
 New Optical Illusions by Gianni Sarcone and Marie-Jo Waeber, Carlton Books Limited, 2005.
 Incredible Visual Illusions (2005) by Al Seckel is a virtual cornucopia of information and examples, with detailed explanations and what is going on, and a variety of types of optical illusions.
 The Art of Optical Illusions (2000) by Al Seckel discusses a large number of individual optical illusions. Each chapter includes a section on notes with interpretations and explanations, and in a number of cases, historical origins of the illusions.
 Joseph Hoffer and the Study of Ancient Architecture by J. Sisa in the Journal of the Society of Architectural Historians, December 1990 issue, includes detailed architectural drawings of the Parthenon.
 Impossible Objects, Amazing Optical Illusions to Confound and Astound by J. Timothy Unruh, Sterling Publishing Co., 2001, gives short but excellent explanations of a number of impossible objects including the three examples of the Penroses as well as material on Hybrid Impossible Objects and How You Can Make an Impossible Object.
 The Pea and the Sun: A Mathematical Paradox by Leonard Wapner has an interesting discussion of paradoxes. He categorises paradoxical statements into three types as discussed in this article. This book has been reviewed in Plus.
About the authors
Linda Becerra is a Professor of Mathematics at the University of Houston-Downtown. She and her co-author (Ron Barnes) wrote an article on the foundations of mathematics, The evolution of mathematical certainty, which was awarded the Mathematical Association of America's Trevor Evans award for Best Paper in the MAA journal Math Horizons in 2005.
Professor Ron Barnes is a statistician in the Computer and Mathematical Sciences department of the University of Houston-Downtown. His PhD is from Syracuse University. His current interests include participation in a 5-year joint biology and mathematics project, funded by the National Science Foundation, and he has been involved in several other multi-disciplinary grants. He has also served as a NASA Faculty Fellow in Reliability. He has given a number of invited talks and addresses on topics in the areas of mathematical puzzles, games, curiosities and swindles.