Darrell Hougen

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About Darrell Hougen

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  • Birthday 01/31/1964

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    Littleton, CO

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    Darrell Hougen
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    Tchaikovsky, Mendelssohn, Mozart, Beethoven, Rush

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  1. Darrell Hougen

    Tether a Spinning Ball

    Wow! Mind blown! Darrell
  2. Darrell Hougen

    Mechanical Reasoning 101

    Since I came late to the party, I decided to try to solve the problem without looking at other solutions and came up with a slightly different approach. Let r be the distance between the two pins and let the origin of the coordinate system be in the middle of the figure. Then, by the Pythagorean theorem, x2 + y2 = r2 That is also the equation of a circle. We can also give the equations of a circle in parametric form: x = r*cos(t) y = r*sin(t) Given that, what would the equations of the crayon be? Let R be the distance from the y-pin to the crayon. Then, if X and Y are the coordinates of the crayon: X = -R*cos(t) Y = (R + r)*sin(t) But, those are just the parametric equations of an ellipse. The equation of the ellipse can be written in standard form by combining the two equations: X2/R2 + Y2/(R+r)2 = 1. So, the figure is an ellipse with y as the long axis. Darrell
  3. Darrell Hougen

    Aristotle's wheel paradox

    I'm just going to restate Aristotle's Wheel Paradox for people who don't seem to understand it because of the mechanical aspects of the problem. Forget about wheels. Instead, consider the function f(x) = 2x defined on the interval [0, 1]. Then, if y = f(x), y is defined on the interval [0, 2]. The function f(x) has an inverse, so that x = f-1(y). Specifically, x = y/2. Now, let yi be any point in [0, 2]. Then there is a corresponding point, xi = yi/2 in [0, 1]. Similarly, let xj be any point in [0, 1]. Then there is a corresponding point, yj = 2xj in [0, 2]. Now, assume that the interval, [0, 2] contains N points. Then the interval [0, 1] also contains at least N points because for every point in [0, 2] there is a corresponding point in [0, 1]. Similarly, if [0, 1] contains M points, then [0, 2] also contains at least M points. Therefore, M must equal N. But, the length of the interval [0, 1] is 1 and the length of the interval [0, 2] is 2, a paradox. I might not quite be doing the paradox justice, but consider the following: If there are N points in the intervals [0, 1] and [0, 2], then the density of points in the first interval is N/1 or just N, while the density of points in the second interval is N/2. So, the density of points in the interval [0, 1] is twice the density of points in the interval [0, 2]. Now, if I double the number of points in the interval [0, 2], then the number of points in the interval [0, 1] must also double and the converse is also true. But, the density of points in the interval [0, 1] is still twice that of the points in [0, 2]. So, if I keep doubling the number of points in the intervals indefinitely, the density of points in the shorter interval will always be twice that of longer interval. And, in the limit of infinitely many points, the limit of the ratio of the densities will equal 2: A (seeming) paradox. I'll return to the mechanical problem later. Darrell
  4. Darrell Hougen

    Aristotle's wheel paradox

    Hi Max, Perhaps I can choose files, but when I've tried that in the past, I've had difficulties. That's why I started copying and pasting. Looks like I'll have to start dragging and dropping. I don't know what Total Commander is, but that wasn't really the point. I just mentioned using Windows Explorer in order to make things concrete --- I'm dragging and dropping from a file system viewer rather than from an image viewer. I'm not sure if this upload limitation is a new thing or what. I'm guessing that I wouldn't be able to upload the picture of the three bottles any longer. You seemed to indicate that you thought the limit was cumulative, but I'm not sure at this point. Darrell
  5. Darrell Hougen

    Aristotle's wheel paradox

    For your amusement: BTW, I'm having the same problem as Max. I seem to be severely limited in the amount of imagery I can upload. I did notice that it makes a difference whether I copy and paste from Irfanview or just drag and drop from Windows Explorer. Irfanview seems to expand the image to the equivalent of a bmp for display purposes even if the file is stored on disk as a gif. Therefore, dragging and dropping from Windows Explorer produces a smaller upload if the file is stored on disk as a gif. Darrell
  6. Darrell Hougen

    Epitome of the Collectivist Soul

    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
  7. Darrell Hougen

    Epitome of the Collectivist Soul

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

    Where are you?

    Who you gunna believe, me or your lyin' eyes?
  9. Darrell Hougen

    Aristotle's wheel paradox

    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
  10. Darrell Hougen

    Where are you?

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

    Happy Holidays!!!

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

    Aristotle's wheel paradox

    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
  13. Darrell Hougen

    Aristotle's wheel paradox

    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
  14. Darrell Hougen

    Aristotle's wheel paradox

    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
  15. Darrell Hougen

    Aristotle's wheel paradox

    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