Math and optical illusions relationship quotes

Optical illusions

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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 [12] 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. Apparently, someone actually attempted this trick in pre-war Austria.

Another type of sliding puzzle appears to create a hole after sliding is completed. The paradoxical hole puzzle in figure 11 is an example. 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 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 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: 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. The vanishing egg puzzle.

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.

Further reading You can read more about the mathematics of perspective and about M. Escher's work in Plus. Paraquin, Sterling Publishing Co. Darling gives an interesting historical account of ancient Greek architecture and its utilization of optical illusions.

Escher by Bruno Ernst, Taschen America, English translationincludes material on Escher's life, the development of his work, with chapters considering: Escher, Barnes and Nobles Inc. Their three examples have spawned an entire area of investigation in the graphic arts; among its most notable proponents being Maurits Escher. Each chapter includes a section on notes with interpretations and explanations, and in a number of cases, historical origins of the illusions.

Sisa in the Journal of the Society of Architectural Historians, December issue, includes detailed architectural drawings of the Parthenon. Timothy Unruh, Sterling Publishing Co. A Mathematical Paradox by Leonard Wapner has an interesting discussion of paradoxes.

Mathematical Optical Illusions

He categorises paradoxical statements into three types as discussed in this article. This book has been reviewed in Plus. 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 given a number of invited talks and addresses on topics in the areas of mathematical puzzles, games, curiosities and swindles.

Yet, for the most part, we perceive an accurate world of depth, surfaces and objects. 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.

Mathematical Optical Illusions

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.

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.