(PDF) Geometric–optical illusions and Riemannian geometry
course follows the general goals and objectives of the regular Geometry course. Right triangle relationships. Explore shapes and optical illusions. Hence the geometrical–optical illusions show promise as analytical tools in .. of scholarship in geometry for help in delineating the relationship between .. of simple structures should begin to approach the goal of the Gestalt movement. Visual illusions and mathematics New variations of spiral illusions. .. The relationship between the "Rotating snakes" illusion and age has.
Some components of geometrical-optical illusions can be ascribed to aberrations at that level. Even if this does not fully account for an illusion, the step is helpful because it puts elaborate mental theories in a more secure place. The moon illusion is a good example.
Geometric–optical illusions and Riemannian geometry
Before invoking concepts of apparent distance and size constancyit helps to be sure that the retinal image hasn't changed much when the moon looks larger as it descends to the horizon. Once the signals from the retina enter the visual cortex, a host of local interactions are known to take place.
In particular, neurons are tuned to target orientation and their response are known to depend on context. The widely accepted interpretation of, e. The most famous example is the Greek Parthenon figure 1.8 MIND-BLOWING Optical Illusions - THESE ARE ALL STRAIGHT LINES
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.
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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.
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.
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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. 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. The basic idea permitting an extension to curved targets consists in equipping the target itself with a geometry Ehm and Wackermann, In the present paper we elaborate on this approach, using the fact that half circles represent geodesics in a suitable model of hyperbolic geometry.
The elements of this framework are presented in Section 2. The details are deferred to the mathemati- cal appendices.
An experiment intended to verify the predictions and to measure the magnitude of the illusory distortion is reported in Section 3. It concludes with the above indicated proposal for a general approach covering gois of the Hering and the Ehrenstein—Orbison type as special cases. The connection to minimum principles comes via the concept of a geodesic, which is a path of minimal length, measured in the respective metric, that connects two given points.
According to our basic surmise the percept of the circular target can be modeled as a geodesic in a suitable geometry that is perturbed by the context if such is present. It is known that a 2D Riemannian geometry admitting only circles and straight lines as geodesics must have constant curvature Khovanskii,