Spiral Illusion

tmex wrote:Yes, Hokie. It is humbling, but we need that from time to time. It helps to keep a perspective on what we do.  You should really check out the Bertrand Paradox. It is another two decade thought process for me.

tmex wrote:The "moon illusion" is another unexplained human perception failing. Why does the moon look bigger on the horizon that it does above the horizon even though it is closer to us when directly above us? It has never been explained by physicists, astronomers, medical doctors,...nada. Nobody knows the answer.

tmex wrote:Yes, I have seen that explanation and many others. Do you believe it? That is the key question. I do not. I have looked at the horizon moon many times, and cannot reconcile it with an optical illusion. Could be. What else could explain it? Something is miswired in our brains to be sure.

The Doppler effect is observed whenever the source of waves is moving with respect to an observer. The Doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for observers towards whom the source is approaching and an apparent downward shift in frequency for observers from whom the source is receding. It is important to note that the effect does not result because of an actual change in the frequency of the source. Using the example above, the bug is still producing disturbances at a rate of 2 disturbances per second; it just appears to the observer whom the bug is approaching that the disturbances are being produced at a frequency greater than 2 disturbances/second. The effect is only observed because the distance between observer B and the bug is decreasing and the distance between observer A and the bug is increasing.

The Doppler effect can be observed for any type of wave - water wave, sound wave, light wave, etc. We are most familiar with the Doppler effect because

tmex wrote:It could also be the "Dopeler Effect" - the faster new ideas come at you, the better they sound.  

Hokie wrote:

tmex wrote:After reading dozens of pages from mathematicians to metaphysicians to linguists, I have concluded that the correct answer lies in Edwin Jaynes' principle of "maximum ignorance" which he developed in response to Bertrand Paradox (wiki that one if you want something that will ruin your day). In a nutshell, the principle of maximum ignorance advises not to "overthink" a problem. Keep it simple.

Even though even expected value theory is correct and used all the time without much thought, there is no reason to apply it to the two envelope problem. Just because a theory is correct does not mean it does not run the risk of providing a nonsensical result from time to time. I know that sounds weird, but it is the mathematical equivalent of quantum mechanics. I have no other way to explain it. It all boils down to the intricacies of how a problem is posed, and it is a wonderful example to show how wrong you can be while standing in the path of righteousness.

So which is the correct choice to maximize earnings?
keep the first envelope or switch?

or is there not an answer?

tmex wrote:@Hokie

I think the Bertrand Paradox is truly deep. References go to 2010 and beyond. This is not a game, and challenges our perceptions of math. The reality is, no one has a clue. That is deeply disturbing to me given that all three approaches seem perfectly valid. Again, the principle of  "maximum ignorance" seems to prevail. I am going with it.

tmex wrote:@Hokie

I think the Bertrand Paradox is truly deep. References go to 2010 and beyond. This is not a game, and challenges our perceptions of math. The reality is, no one has a clue. That is deeply disturbing to me given that all three approaches seem perfectly valid. Again, the principle of  "maximum ignorance" seems to prevail. I am going with it.

tmex wrote:The "moon illusion" is another unexplained human perception failing. Why does the moon look bigger on the horizon that it does above the horizon even though it is closer to us when directly above us? It has never been explained by physicists, astronomers, medical doctors,...nada. Nobody knows the answer.

   The scientific challenge has been to explain why those equal angular sizes look unequal.

This article is long for three reasons.
1. It advances the relatively new idea (1965, 1970, 1985, 1986) that, for most people, the moon illusion begins as an angular size illusion which has several possible outcomes.

2. It reviews in detail the very few explanations of the illusion that vision scientists are paying the most attention to (and still researching). These theories are not simple.

3. It reviews in detail the latest theory (1985, 1986, 1989) that the moon illusion is an example of the less familiar, but ubiquitous, "size" illusion known as oculomotor micropsia/macropsia. Explanations for oculomotor micropsia then are reviewed.
For the moment, it seems to be the most satisfactory explanation.

   Oculomotor Micropsia

The angular size illusion has defied explanation for decades.  McCready suggests that the moon illusion is a special case of a lesser-known phenomenon known as oculomotor micropsia, which literally means “looking small”.  It is intimately related to oculomotor macropsia, which translates as “looking large”.

It is believed that the way we focus our eyes is affected by changes in the visible patterns near an object.  These patterns or secondary objects are known as distance cues.

Oculomotor macropsia is an angular size illusion caused by changes in the activity of eye muscles.  When one views the horizon moon, the details in the landscape form distance cues that sends a “very far” signal to the observer.  This has the effect of making the eyes focus at a distance much greater than where the object is, called overfocusing.  Hence, a smaller angular size is perceived.

Oculomotor micropsia has the opposite effect.  The zenith moon offers few distance cues, and the eyes tend to adjust to a relatively near position, known as the resting focus position.  This is typically about one or two metres from the face.  The perceived angular size becomes smaller.  A closely related phenomenon is night myopia, where the eyes tend to underfocus in dark surroundings.  This causes temporary nearsightedness, also known as “night blindness”.

It must be strongly emphasized that oculomotor micropsia and oculomotor macropsia are physiological and not geometrical effects.  Furthermore, they occur when viewing all objects, not just the moon.

tmex wrote:The metaphor is not compelling KFF. On the horizon the moon is next to a much larger sphere, the surface of the earth (which you can barely even see). If the metaphor was valid, the moon should look smaller on the horizon as the circle next to the bigger circles in your example looks smaller the the same sized circle next to the smaller circle.

I do agree that it is some sort of illusion. I just have yet to see any plausible explanation or even a decent parallel.

    The Distance-Cue Connection.
The visual angle contrast theory can use the fact that for many of these classic illusions (including the moon illusion) the changes in the visual angle subtenses of the context images happen to also be distance-cue patterns.

For example, a possible pictorial illusion with the Ebbinghaus pattern is that the ten context circles portray discs of the same linear size (linear size constancy), so the five portrayed discs at the upper right look farther away than the five at the lower left, because they look angularly smaller (thus illustrating the perceived angular size cue to distance).
Therefore, it can be argued that these context patterns act as distance-cue patterns that educe the relative visual angle illusion, providing a small imitation of the moon illusion (McCready, 1985.

Moreover, this distance-cue connection allows the oculomotor micropsia/macropsia theory to be proposed as an explanation for that angular size contrast effect (McCready, 1985), and as discussed in detail later, in Section III.

shiftpro wrote:What a wonderful thread, you guys rock.