Math Forum - Ask Dr. Math
classes of optical illusions-illusions of two-dimensional . of the figure, since the relation of the first segments to processing of geometrical relationships. .. of the parallel line is present,. a reduction by half is strong support for the theory. Unfortunatley, I can only think of one way that optical illusions have to do with you some ways that mathematical concepts can help explain optical illusions. circles and space. Geometry,. arithmetics and algebra are the main mathematical concepts that directly. relate to optical illusions.
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 . The construction appears as a triangle only from one angle. 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.
Optical illusion, mathematical illusion
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. 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. 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 Paradoxes 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. A statement which appears contradictory, even absurd, but may in fact be true. 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.
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. 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: Sliding Puzzles 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 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.
Yet, for the most part, we perceive an accurate world of depth, surfaces and objects.
Optical illusion, mathematical illusion
This seems to be the source of some optical illusions. We see a two dimensional image on a page and our brains constructs a three dimensional image that may be an impossible object. Let my try to illustrate with an example from a fasinating book I just read.
The book is Donald D. How we construct what we see, W. Norton and Company, New York, Look at the three illustrations Your brain probably constructs a three dimensional cube from the image in the center but the images on the left and right look flat.
Visual curiosities and mathematical paradoxes
Hoffman uses mathematical concepts to try to explain how our brains construct the images we see. He calls them rules. Always interpret a straight line in an image as a straight line in three dimensions.
If the tips of two lines coincide in an image, then always interpret them as coinciding in three dimensions.
So, for example in the image on the left, the diagonal lines intersect in the image and hence your brain uses rule 2 to construct a three dimensional image where these lines intersect, forcing it to be flat. The "optical illusion" where this can be seen is the devil's triangle or impossible triangle.