I agree with the comments on the psychology/perception issue. But I don't agree with this:
"So no matter which bisecting plane through the sphere we examine, it will always have more sticks parallel to it than to the orthogonal pole. So this actually explains a "planar force". There more horizontal sticks than up/down sticks...." I just don't think that is possible. All you have to do is consider one case (that supposedly has more sticks parallel), and then freeze the sticks in place, and rotate the plane through the sphere so that it is now perpendicular to the original plane. Clearly now the "parallel" sticks are "perpendicular," so if there were more parallel before, now there are more perpendicular. The plane is simply a place of reference. It makes no difference on the number of sticks oriented one way or another. Cheers, -Ted On Sun, Jul 18, 2010 at 12:08 PM, ERIC P. CHARLES <[email protected]> wrote: > Lee, > True, but I was thinking projective geometry rather than a psychology of > appearances. Grant the arbitrariness of "horizontal" and "vertical", let us > assume the ground is a flat plane that is easy to reference these terms to. > If we fix a midpoint on the stick, there there will be a plane that we can > rotate the stick in for which all angles will appear perfectly "horizontal" > in a 2D projection, and a perpendicular plane for which all angles will > appear "vertical" in a 2D projection (minus the intersection of the plains, > where the stick will appear head on). By "appear" I mean only that they will > be either parallel with or perpendicular to the ground plane. > > Of course, you also bring up lots of other interesting hypotheses regarding > perceptual biases. Agreeing with what I take to be your main thrust: I > imagine that if we had people rapidly and repeatedly mark a variety of lines > as either "horizontal", "vertical", or "other", we would find that they > allow higher variability in horizontal lines. > > Eric > > > > > On Sat, Jul 17, 2010 08:57 PM, *[email protected]* wrote: > > On 17 Jul 2010 at 20:10, ERIC P. CHARLES wrote: > > > Russ, > > This seems very weird to me (as, of course, it is intended to). > First off, I'm > > not sure it is an "explanation" any more then a "proof by > definition". > > If it's an "explanation" of any thing, I think it's > an explanation of the manner in which we (or some > of us) come to give an account of the situation. > I'd rather call it a description of the situation, > coupled with a description (not particularly > explanatory) of our manner of coming to that > account (e.g., how we assign labels "vertical" > and "horizontal"). > > > Second, > > at least in the case of a 2D snapshot, there are just as many 3D > configurations > > that appear perfectly vertical as appear perfectly horizontal. > > That depends on how the definition of "appears" appears. > (And makes my point, above, about the us-ness of the > how-ness of it all.) Sticks that lie in the (assumed > horizontal) plane of the monocular viewer will have > a distribution of apparent lengths (defined in some > purely optical, non-perceptual, way: e.g., in terms of > measurements taken with a pair of dividers on a 2D > snapshot) that may be (and very likely is) markedly > different from the distribution of apparent lengths of > all the sticks; the viewer's assumptions (conscious or > unconscious) about what is seen will play a large part > in determining what conclusions (conscious or unconscious) > the viewer draws about the distribution of attitudes > of the sticks. (Affordances must come up here, eh?) > > If we had a 2D photo such as you suggest, Eric, it would > be an interesting experiment to show it to various people > turned at various angles. My prediction is that (absent > clues about "true vertical") "appreciably more" of the > sticks would be perceived as being "near the horizontal > than near the vertical orientation" in every case > (assuming, as I just realized I have been, that the > 2D photo is viewed in a vertical plane, rather than > lying on a desk or in one's lap or on the floor; > in *those* cases, I'll now predict that ascriptions > of verticality will be quite variable). > > Shephard points out (in his paper speculating on why > humans have a 3D color space) that for terrestrial > animals (at least, ones that live above the scale > where things like surface tension of water and > viscosity of the atmosphere are big deals in daily > life), the vertical axis defined by gravity is > highly salient. What, we may ask, would a porpoise > or a porgy make of your photo? > > Lee Rudolph > > > > > > > I'll have to meditate more on the more general case. > > > > Eric > > > > On Sat, Jul 17, 2010 07:28 PM, Russ Abbott <[email protected]> > wrote: > > > > > >I just ran across > > <http://plato.stanford.edu/entries/mathematics-explanation/>. > (Call it the > > "horizontal force.") > > > > > > > > >There appear to be physical explanations that are > > >non-causal. Suppose that a bunch of sticks are thrown into the air > > >with a lot of spin so that they twirl and tumble as they fall. We > > >freeze the scene as the sticks are in free fall and find that > > >appreciably more of them are near the horizontal than near the > > >vertical orientation. Why is this? The reason is that there are more > > >ways for a stick to be the horizontal than near the vertical. To see > > >this, consider a single stick with a fixed midpoint position. There > > >are many ways this stick could be horizontal (spin it around in > the > > >horizontal plane), but only two ways it could be vertical > (up or > > >down). This asymmetry remains for positions near horizontal and > > >vertical, as you can see if you think about the full shell traced out > > >by the stick as it takes all possible orientations. This is a > > >beautiful explanation for the physical distribution of the sticks, but > > >what is doing the explaining are broadly geometrical facts that cannot > > >be causes. > > > > > > > > > > > > > > >-- Russ Abbott > > >______________________________________ > > > > > > Professor, Computer Science > > > California State University, Los Angeles > > > > > > cell: 310-621-3805 > > > Google voice: 424-2Blue2 > > > blog: <a style="font-family: trebuchet ms,sans-serif;" > > href="http://russabbott.blogspot.com/"; target=" > > > onclick="window.open('http://russabbott.blogspot.com/');return > > > false;">http://russabbott.blogspot.com/ > </a> > > > vita: <a style="font-family: trebuchet ms,sans-serif;" > > href="http://sites.google.com/site/russabbott/"; target=" > > > onclick="window.open('http://sites.google.com/site/russabbott/');return > > false;">http://sites.google.com/site/russabbott/</a> > > >______________________________________ > > > > > > > > > > > > > > ============================================================ > > >FRIAM Applied Complexity Group listserv > > >Meets Fridays 9a-11:30 at cafe at St. John's College > > >lectures, archives, unsubscribe, maps at http://www.friam.org > > > > > > > > Eric Charles > > > > Professional Student and > > Assistant Professor of Psychology > > Penn State University > > Altoona, PA 16601 > > > > > > > > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > > Eric Charles > > Professional Student and > Assistant Professor of Psychology > Penn State University > Altoona, PA 16601 > > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > -- Ted Carmichael, PhD Complex Systems Institute Department of Software and Information Systems College of Computing and Informatics 343-A Woodward Hall UNC Charlotte Charlotte, NC 28223 [email protected] [email protected] Phone: 704-492-4902
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