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Real World Studies IV: Butterfly Wings.
Butterfly wings? Yes, butterfly wings. Although the measure of medium to large scale inanimate systems seems to offer the best bet for initial testings of the model of organization discussed here, one should not overlook other possible venues. In the case of butterfly wings, there are at least two obvious angles that one might explore. First, and quite apparently, butterfly wings are systems for flight comprised of four structural components, arranged in pairs. This alone is suggestive, since inasmuch as their primary responsibility is to support movement through the air, it should be possible to characterize this function as a four by four matrix of relations--representing structural inter-similarities, actual movement, shared motions over time, etc., etc. Such a matrix should again be interpretable in entropy maximized form as a symmetric set of relations, if I am generally correct. Not being a physicist or engineer, however, I haven't a clue as to where to start in that direction. There is another direction, however, that might be looked into, concerning the color patterns on their wings. It is generally believed that the color patterns on the wings of butterflies have not evolved, at least in the main part, randomly, without reason. The most commonly considered reasons for their particular patterns include provision of camouflage among vegetation, mimicry of noxious species to the end of obtaining some protection from predators, and the generation of various elements facilitating recognition (either by themselves, or by other species). In any case, and for whatever reason we might be dealing with, the expectation here is that function must also be limited by the constraint of producing a pattern which occupies space: and therefore, if we are correct here, must be interpretable through the four-class entropy maximization model we are entertaining. That said, analysis of the situation confronts a lot of obstacles. A butterfly at rest on a surface is close to being a two-dimensional surface, though not perfectly so. Further, even when the organism's four wings are displayed in near two-dimensionality, there is some overlap of them which results in part of the surfaces being concealed. Importantly, moreover, in the vast majority of cases there are not four neatly developed colors that one can associate with "classes of surface." This is not a problem for the theory itself, as in a three-colored wing two classes of color may be locked into one, or in a five-colored wing two colors may combine to function as one class, but it is a problem for initial model validation. Still, I decided quite recently that it would be worth taking a quick look to see if I could come up with any evidence bearing on the model. To do so, in early 2006 I scanned the internet for a good database including butterfly images, and after finding one looked through it specifically for five species examples whose wings inequivocably had four well-defined colors on them, and only four. The images were then downloaded to hardcopy, and sampled in a manner analogous to the other data sets I have discussed here. Thus, I ended up with five sets of data, each one of which was divided up into four sets of sampled locations connected to the respective colors. Spatial autocorrelation coefficients were calculated, and then run through the double-standardization program. For each butterfly, I then created several unnatural alterations of the original patterns by taking five or ten percent of the locations and re-assigning membership to a different class (resulting in some odd-looking, arbitrarily asymmetric, patterns). New spatial autocorrelation coefficients were calculated, and these sets of data also double-standardized--the expectation being that, on the very least, fewer of these aberrations would represent patterns whose resultant double-standardized relations were symmetric. This was a lot to ask, considering the various sources of error (including some others I haven't mentioned, such as whether the original images on the web were actually truly proportional to the real-life creatures) involved. As it so turned out, modifying only five to ten percent of the class memberships might not have been enough, as it made only small adjustments in the mean correlation values, which often remained low (or even went lower)--and I already know that those matrices sponsoring low mean correlations tend to more frequently double-standardize to symmetric values than those producing high mean correlations. Again, I used two measures of spatial autocorrelation. The results: For the actual figures, seven of ten outcomes (remember, each of the five patterns were investigated through two spatial autocorrelation measures) produced symmetric double-standardized results, whereas for the contrived, "unnatural" figures, only thirteen of twenty-eight did. The difference in the ratios 7 : 3 and 13 : 15 is probably mildly statistically significant (though one cannot apply a typical contingency table analysis here because one of the four numbers is less than 5). I am also far from confident that I provided an adequate sampling density (a range of 415 to 678 sampled points across the five figures) to capture the essence of the five patterns, some of which included considerable numbers of fairly small spots. The end line here is that in a fairly small amount of time it was in fact possible to identify some apparent differences between the real butterflies and the fake patterns on the basis of this model. It is not particularly troublesome that many of the fake patterns produced symmetric results, because the model admits that this is often likely to be the case; it should always be the case, given adequate sampling density and a valid measure of differentiation, however, that the real patterns will return more (all, in theory) such results. _________________________
Copyright 2006 by Charles H. Smith. All
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