Visual illusions and mathematics
Math in Optical Illusions. Optical Illusions primarily use the Geometric mathematical level! How everything relates to math: Any drawings. Optical illusions have fascinated humans throughout history. Greek builders used an optical illusion to ensure that that their columns appeared. Latin root of illusion is illudere which means “to mock”; Optical illusions mock our trust in our senses; Suggest that the eye knowledge of physical relationships.
In many ways, his creations are what brought the concept of impossible shapes into the cultural mindset. Question 14 Which direction are the lines moving? Clockwise Counterclockwise Both We know it sounds insane, but this optical illusion was actually created using a robot and a Spirograph. While there are similar optical illusions to this one that can be produced and reproduced by human hands, this one is so precise that it must be created with exact measurements, along with a machine that can follow those exact measurements to the millimeter.
This Japanese mathematician's optical illusions will make you question reality
It just goes to show that even a child's toy can be used to create some absolutely amazing things. Question 15 Are the white lines the same size? Yes No While many used this image for years to easily present how perspective can change how our brains perceive the world around us, it was eventually proven to be a fake.
Somebody, using camera and photography trickery, was able to manipulate this image and move the train tracks closer together further down the line, thereby increasing the shock when the image was presented. While we all know parallel lines appear closer as they approach the horizon, this image is actually an exaggeration of that phenomenon. Question 16 How many colors do you see?
This both amazes and worries us, as it just once again highlights how unreliable our own bodies are. We really do live our lives under the misconception that this stuff will always have our back, but this sadly isn't the case. However, one of the best things about these illusions is that they can tell us so much about ourselves, which hopefully this will one day too.
Question 17 How many full circles do you see? If tie-dye and surrealist imagery are anything to go by, it's quite clear that LSD and similar drugs clearly leave people with the ability to create some mind-bending stuff, this illusion being one of them. As you look at it, the circles appear to be turning, despite the fact that they aren't moving. This was handed out to people to trip them out while they came up on drugs at the infamous festival.
Question 18 Both sides of the image are the same color and brightness. True False Yet another illusion that works with color rather than perspective, this highlights how two of the same color, when placed together can actually appear different due to the way our brain perceives them.
If you were to place a thin band of white between these two blocks of gray, it would become immediately apparent they are the same color, but for whatever reason, our brain is unable to see this once they're placed together with a mere dividing line between them. Question 19 This triangle is actually physically impossible in the world of geometry.
True False This image is used by mathematicians to teach students that when it comes to geometry, your eyes and knowledge can be easily fooled. While this image appears to take place over the same number of squares, meaning it shouldn't matter what position the shapes are placed in.
However, if you rearrange the shapes from their original placing, it is possible to end up with a missing square, which should make no sense from what we know of geometry. While there is a solution to this, it continues to stump students to this day.
Question 20 This is actually a woman's legs made to look like a lamp through photoshop and camera tricks. True False This one has done the rounds on the internet many times and for many years at this point, usually with a headline claiming that what you see the first time you look at this image will tell you whether or not you have a naturally dirty mind, but the truth of it is, an artist actually created this image to show how easily a natural form can be manipulated into something else in the modern world.
Far from dirty minded, you're actually seeing the truth. 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. Ambiguous optical illusions Figure 3: 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".
Visual curiosities and mathematical paradoxes | nickchinlund.info
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. Impossible figures Figure 4: 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 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 .
- Visual curiosities and mathematical paradoxes
- Mathematical Optical Illusions
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.
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.
This Japanese mathematician's optical illusions will make you question reality
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.