Darrell Hougen

I would have liked to see Janice Rogers Brown appointed to the Supreme Court, though she is getting a little old and is now retired. https://www.nytimes.com/2005/06/09/politics/new-judge-sees-slavery-in-liberalism.html https://reason.com/blog/2017/07/12/janice-rogers-brown-americas-most-libert Darrell

When I saw the original post, I thought you were referring to a different Roger Simon. https://pjmedia.com/columnist/roger-l-simon/ Darrell

Who you gunna believe, me or your lyin' eyes?

Hi Jonathan, To be honest, I'm not sure what you're doing above myself. I know the equations of perspective projection for a pin-hole camera: X = x/z Y = y/z where (X, Y) are image coordinates and (x, y, z) are world coordinates. But, that doesn't help much if I don't know where the camera is positioned or what the viewpoint is. Alternately, I know that in the absence of distortion, straight lines in the world produce straight lines in the image. I also know that a rectangular solid in general position generates three vanishing points --- 3 point perspective. Using the two posts at the ends, it would be pretty easy to find one vanishing point. However, I'm not sure if there is enough depth information to accurately calculate the positions of the other two. So, perhaps you have some method based on triangles or something for drawing the relevant lines. What lines are required for determining the foreshortened shape of the wheel when it gets to the right-hand side? I know it should be an ellipse, but I don't know how to determine the eccentricity with the information given. Anyway, I could look it up, but I'm just curious what you're doing. Darrell

Okay guys, I'm here now. Now that this problem has been beaten to death, I'm ready to jump right in.

In case I don't get on here again until sometime next year ... Darrell

Let me take this discussion in a slightly different direction. It turns out that people have widely varying levels of ability to recognize faces. There was a 60 Minutes episode on this topic. At one end of the spectrum are the so called "super recognizers" who are able to walk down the busy streets of New York City bumping into people and remember virtually all of the people they meet. They might run into someone in the afternoon and say, "Oh, I saw that guy over on 23rd street this morning." One woman who was being interviewed was shown a high school year book picture of someone and she figured out right away that it was a picture of Mike Wallace. At that time, I believe, Mike Wallace was dead and gone so she couldn't have seen him recently. On the other end of the spectrum are people who have a very difficult time recognizing faces. Some people have a difficult time recognizing friends. Some have trouble recognizing their own family members. Some men had difficulty recognizing their own wives. And some people even had difficulty recognizing themselves in a mirror. But, even the people who had difficulty recognizing themselves were able to recognize ordinary objects --- a cup, a table, a chair, a car, etc. So, it seems like facial recognition is a very specific mental function. It is a function that is handled by a very specific part of the brain. That makes sense because facial recognition is very important to humans so having a particular part of the brain dedicated to facial recognition makes it possible to recognize subtle differences between faces that might not be immediately obvious with regard to other kinds of objects. Although we might learn to recognize particular apples, for example, differences between apples aren't as immediate and obvious as differences between faces. One man on the 60 Minutes program was discussing how he had learned to recognize himself by concentrating on individual parts of his face. He would look at his lips, his mustache, his nose, his eyes, etc., and could convince himself that he was looking at himself by studying his face carefully. Presumably, he could apply the same method to recognize other people as well --- I have a big nose; my wife has a small nose, etc. From the foregoing conversation, it would seem that visuospatial/mechanical reasoning is another specialized mental function. So, one has to wonder whether a person that lacks the ability to easily and naturally perform such reasoning can learn to answer questions about mechanics by concentrating on simple aspects of the problem and reasoning at a higher, conceptual level about their interrelationships. I should say that know that I have limitations of my own. I'm lousy with people's names. When I was young, I realized I didn't know the names of the some of other students in one of my elementary school classes and made the unfortunate decision at that time that it wasn't important and that I didn't care. As I grew to adulthood, I realized that my inability to remember people's names was a definite handicap, so I reversed my earlier attitude and attempted to get better at remembering. When I watch a movie, I attempt to name the actors and actresses in it. At the end of the movie, I watch the credits to try to learn new names. When I meet people, I focus on getting to know their names. Sometimes, I still forget to pay attention, but I try. Not everyone has such difficulty with names. My own daughter has a natural ability to learn people's names. She's in her twenties now, but when she first started kindergarten, she learned the names of all of her classmates before the first week was out. She must have gotten that gene from her mother. At any rate, I don't know the extent to which cognitive deficits can be compensated for, but I find the question interesting. I also wonder if there are other kinds of common cognitive deficits. Darrell

Hi Tony, There are basically three scenarios being discussed: 1. An ordinary wheel or tire that runs on a road or track. 2. A pair of adjacent wheels or gears that run on adjacent rails or tracks at appropriate levels. 3. A bottle or Dixie cup whose ends run on widely spaced rails or tracks --- widely spaced relative to the sizes of the ends. Your explanation works perfectly fine in the first case, but it doesn't begin to explain the other two scenarios. You agreed that a cone shaped object would veer off to one side, but you haven't explained why you think that to be the case. Darrell

Hi Tony, Mathematically, it is possible for a tire to roll and slide at the same time. What happens at the molecular level is another question. It is quite possible that the molecules of the tire temporarily adhere to the road surface and then jump to a new location. Consider what happens when you drive your car. If you turn a corner, then different parts of each tire are simultaneously moving at different speeds relative to the surface of the road. Assume you turn left and consider the left front tire. The left side of the tire travels a shorter distance than the right side of the tire as you go around the corner. Since the left side and right side of the tire have the same diameter, either one side or the other or both must be simultaneously rolling and sliding. Whether it is true simultaneity or the molecules in the tire are temporarily adhering to the surface before jumping to a new location is anyone's guess at this point. Perhaps there is some information online that would answer the question. But, if it is easier for you to grasp the idea of high-speed microscopic deformations of the tire than the mathematical explanation using the continuum, then I would have no argument against your point of view. Darrell

Hi Jonathan, It's funny how adding the cables actually made the problem more difficult for some people to understand. I would have thought it would have made things simpler. Darrell

Hi Tony, No, raising the small end doesn't compensate for the difference in the diameters. It doesn't make them equal. The big end still has diameter = 2R and the small end still has diameter = 2r. Think about it --- you're still calling them the "big end" and the "small end". If you can't discern the difference between the two ends, why do you have two different names for them? The fact is that the two ends are different. That fact is both perceptually obvious and logically required. Simply raising up one end doesn't change its size. Now, what happens when the bottle rolls? Every time the big end rotates by one complete rotation the small end does too. I think you agree with that. In the case I pictured, the bottle will probably fall off the rails before it rotates one complete time, but the same logic applies to part of a rotation. If the big end rotates by a tenth of a rotation, so does the small end. Now, without slipping, that means that in one tenth of a rotation, the big end will travel 2 * pi * R / 10 inches and the small end will travel 2 * pi * r / 10 inches. And, if R > r, then 2 * pi * R / 10 > 2 * pi * r / 10. It has to be that way. That is what logic demands. The tangential speed of the small end during rotation is less than the tangential speed of the big end. That is true. But, the fact that the tangential speed of the small end is less than that of the big end means that the small end is traveling more slowly than the big end. That's because the speed of the center of each end relative to the rail on which it rides is equal to the tangential speed of a point on the circumference relative to the center of that end. If that is hard to understand, don't worry about it right now. Concentrate on what I said above. If the big end is bigger than the small end, then it must travel farther in the same amount of time if neither end slips. So, yes, the bottle will veer off to the side. Darrell

Hi Tony, You're really making this way more complicated than it needs to be. I've created some admittedly low quality images to help visualize what I'm talking about. If the bottle were rolling on the floor, it would probably make contact with the floor in many places. However, we can simplify things by assuming that it only contacts two points. Imagine that they are two rails that go into the page. In the first figure, I've shown the situation when the two rails contact the body. In this case, the diameter of the bottle at each point is equal to R, so both ends of the bottle roll at the same speed. In the second figure, I've shown the body of the bottle supported at one point and the neck supported at one point. Again, imagine rails going into the page. Here, the large end of the bottle will roll more quickly than the small end. If the angular speed of the bottle is w, then V = Rw and v = rw. Or, after some time, the distance rolled by the big end is D = 2 * pi * R and the small end rolls 2 * pi * r. The result is that the bottle veers toward the small end because the small end doesn't go as far as the big end. I think that what you're imagining is a situation in which the body and neck are both supported. In this case, the coefficients of friction and weight distribution do indeed matter. However, it's not necessary to consider this case. It just confuses the issue. Depending upon the friction and weight distribution, the third case will either be more like the first case or more like the second. Perhaps it will be somewhere in between and the speeds will be between the two cases. However, we need not be overly concerned with the third case. The first two cases are sufficient to illustrate Aristotle's paradox. Whoever invented "Aristotle's paradox" is saying that case 2 will behave like case 1 when they are clearly different. I hope that clears things up for you. Darrell

Hi Tony, An automobile was invented, created and passed through some individual engineer's conscious mind as it was being designed, built, and tested. Does your car fake reality as you drive it down the road? Or is it constrained by reality to act in accordance with its nature? Darrell

Hi Tony, Yes, you can think of the upper track as being greased. The upper track has zero friction. Only the bottom track has friction. In addition, the two wheels rotate together. Aristotle's paradox essentially states three things: 1. The small circle and big circle rotate and translate together --- they are rigidly connected to each other. 2. The big circle rolls on the lower line. 3. The small circle rolls on the upper line. But, that's impossible. Those three statements can't all be true at the same time. One of the statements must be false. So, Jonathan decided to keep statements 1 and 2 and abandon statement 3. He created a video which is consistent with statements 1 and 2 but in which the small circle slips while it is rotating and translating on the upper line. It would have been impossible for him to create a video which was consistent with all three statements at once. A. Statement 2 implies that in one revolution, the big circle travels a distance equal to 2 * pi * R. B. Statements 1 and 2 together imply that the small circle also travels a distance equal to 2 * pi * R. C. Statement 3 implies that in one revolution, the small circle travels a distance equal to 2 * pi * r. For C I'm also making use of statement 1 since it says that the small circle and big circle rotate together. Therefore, if the big circle rotates by 2 * pi radians the small circle must also rotate 2 * pi radians. So, conclusions B and C contradict each other. However, if we get rid of assumption 3, then conclusion C goes away. Conclusions A and B are consistent with each other, so there is no problem. Of course, it would be possible to abandon assumption 1 or 2 instead. The big circle could slip on the lower line instead. We just need to understand that it is impossible for 1, 2, and 3 to all be true at the same time. The statements taken together are internally inconsistent. Therefore, it is impossible to find weights or friction coefficients that make all three statements true at the same time. Darrell

Hi Tony, Sorry about the delay in getting back to you. I didn't get online during the weekend. I agree that theory and practice must be consistent and I think you will find that they are if you follow closely what I am saying. If a bottle rolls on the floor with no support for the neck, then it will roll straight (if the neck is not too heavy). It will roll straight because the body of the bottle is the only part of the bottle in contact with a supporting surface --- the floor. Therefore, there will be no friction between the neck and the supporting surface. The only friction will be between the body of the bottle and the supporting surface. Therefore, the body of the bottle controls the behavior of the bottle and the behavior of the neck is completely dependent upon the body of the bottle, as you said. However, if the neck of the bottle rests on a supporting surface, the behavior of the bottle should change, should it not? If the conditions of an experiment change, wouldn't we expect the results of the experiment to change as well? If the support that is constructed for the neck of the bottle provides significant support for the neck and has significant friction, the results of the experiment should change. Of course, if the support doesn't support a significant amount of weight or doesn't have significant friction, then it will have minimal to no effect on the experiment. However, if a significant fraction of the weight of the entire bottle including both the neck and body is supported by a supporting surface and if that surface has significant friction so that it causes the neck to roll on its circumference, then the behavior of the entire bottle must change. If the body of the bottle rolls on its circumference and the neck rolls on its circumference, then the bottle will veer off to the side. That will happen because: 1. The neck of the bottle has a smaller circumference than the body. 2. The neck is rigidly connected to the body. 3. The neck rolls on its supporting surface. 4. The body rolls on its supporting surface. 5. The neck protrudes from one end of the bottle. Since the neck of the bottle protrudes from one end of the bottle, the two ends of the bottle move at different speeds. That causes the entire bottle to turn in the plane of the floor and veer off toward the smaller end. That is not a mathematical conclusion. That is a logical conclusion. It logically follows from the proper identification of the concepts involved and their relationships. If you think the conclusion is incorrect, please show which concept is improperly conceived or how the reasoning is flawed. Also, if you perform the experiment properly, I am certain that the results will confirm the correctness of the logical argument. Darrell

Tony, What is the "turning speed"? Also, you didn't respond to my last comment. It's hard to maintain the continuity of a conversation if you don't respond. Darrell

Hi Merlin, This is a technical objection that has very little to do with the current discussion. The small wheel rolls and slides simultaneously so that the tangent point changes continuously and no point on the circumference of the wheel ever contacts more than a single point on the track. We haven't even gotten to the finer points of the discussion. Based on the description on Mathworld, Aristotle's paradox may have to do with both the mapping between points on circles of differing sizes and on the differing lengths of the circumference. For example, any radial line intersects both the big circle and small circle at exactly one point. That shows that there is a one-to-one mapping between points on the small circle and points on the big circle. How is it then that the two circles have different circumferences? Darrell

Hi Tony, Okay, so the offset track is almost frictionless so that the small wheel veritably slides across the surface, barely disturbing the path of the body of the bottle. Is that what you're saying? If that's what you're saying, then you are correct that the neck of the bottle will move with the body and that the distance moved will be 94.2 inches for both. Darrell

Hi Tony, But, if the neck of the bottle rolls on a separate track (or book, for example) then according to my calculation above, it only travels 31.4 inches. distance traveled = N * pi * d = 10 * pi * 1 =~ 31.4 inches. If there is something wrong with my calculation, what is the problem? The tangential velocity doesn't really matter in this case because the bottle could be rolled fast or slowly. The only important thing is the total distance traveled. Darrell

The one at the end, obviously. Now go back and look at the post where I put numbers to your theory. Darrell

Point taken.

Hi Tony, Sorry, I couldn't resist. In reality, if one moon were farther away from the planet than the other, it would be moving more slowly than the one that was closer. They couldn't possibly stay in sync if they were at different distances from the planet. I just found your example amusing. Anyway, go back and look at my other post which was more serious. Darrell

Okay everyone, who votes that we try to explain orbital mechanics to Tony?

Hi Tony, Okay, let's put some numbers to our theory. Let's say we have a bottle with a neck that is 1 inch in diameter and a body that is 3 inches in diameter. Then, if the bottle rolls completely around 10 times, how far does the body travel? How far does the neck travel? distance traveled = number of revolutions * pi * diameter = N * pi * D So, for the body of the bottle, distance traveled = 10 * pi * 3 =~ 94.2 inches For the neck of the bottle, distance traveled = 10 * pi * 1 =~ 31.4 inches Darrell

Hi Tony, The "bias" is not caused by the fact that the cone or cup or wheel assembly is leaning to one side. It is caused by the fact that one wheel is larger than the other. Therefore, leveling the assembly has no effect on the "bias." Tangential velocity is the velocity of a point on the circumference of a wheel relative to the center point of the wheel. The assembly veers off because the tangential velocities of the two wheels are different. If the tangential velocities were the same, the assembly would not veer off even if one end were higher than the other. Darrell