[...]... example of the rainbow Certain features of rainbows can be fully understood only through asymptotic methods In effect, these are universal features that "emerge" in the asymptotic domain as the wave theory approaches the ray theory in the limit 6 The Devil in the Details as the wavelength of light approaches zero They inhabit (to speak somewhat metaphorically) an asymptotic borderland between theories... case, we can imagine a hypothetical situation in which there is nothing to interfere with the pencil—no breeze in the room, say Then the pencil would presumably remain in its balanced state forever Of course, in the actual world we know that it is very difficult to maintain such a balancing act for any appreciable length of time Similarly, 9 10 The Devil in the Details Figure 2.1: Buckling strut molecular... explanatory theory is required for this asymptotic domain The phenomena inhabiting this borderland are not explainable in purely wave theoretic or ray theoretic terms The accounts required to characterize and explain these borderland phenomena deserve the title "theory." In part, this is because the fundamental wave theory is explanatorily deficient As we will see, the theory of the borderland incorporates, in. .. starting point and the first roll landed on the number 4 This number is assigned to point B The rules say to move halfway from the starting point toward point B and make a mark This is now our new "starting point." On the next roll we do the same thing Suppose the die landed on 5 We move halfway toward point C and make a mark there Continuing to play, we find after many iterations the pattern shown in figure... what to say about the varying "constant" problem 5 Asymptotic Reasoning 19 corresponding formulas in Tc, then schema (2.6) will hold For these cases we can say that the "limiting behavior" as e —> 0 equals the "behavior in the limit" where e = 0 On the other hand, if the behavior in the limit is of a fundamentally different character than the nearby solutions one obtains as e —> 0, then the schema will... the Details Figure 3.1: Sierpinski triangle Label the first "A" and assign it the numbers 1 and 2, label the second "B" and assign it the numbers 3 and 4, and label the third "G" and assign it the numbers 5 and 6 Choose one point (actually it doesn't matter whether the point is even in the "triangle space") as a starting point and begin rolling a six-sided die Suppose we chose point A as our starting... theory That theory may say a lot about the nature of the phenomenon: the nature of its evolution, and what sorts of details for example, initial and boundary conditions—are required to "solve" 3 4 The Devil in the Details the governing equations, and so on One might think that the theory will therefore enable us to account for the phenomenon through straightforward derivation from the appropriate initial... aluminum strut, the explanatory texts are even more disjoint For instance, the buckling load will be different since the struts are made of different materials Why should our explanation of the behavior of a steel strut bear in any way upon our understanding of the behavior of one composed of aluminum? At this point it seems reasonable to object: "Clearly these struts exhibit 12 The Devil in the Details. .. strut is made of as well as certain of its geometric properties— in particular, the ration I/L2.) I agree completely However, the focus of the discussion has shifted in a natural way from the particular buckling of the steel strut in front of us to the understanding of buckling behavior of struts in general These two foci are not entirely distinct Nevertheless, nothing in the ideal explanatory text for... see how this is, in fact, so It is clear that we can think of the Euler formula as expressing the existence of universality in buckling behavior The formula has essentially two components 14 The Devil in the Details First, there is the system—or material—specific value for Young's modulus And second, there are the "formal relationships" expressed in the formula To see how ubiquitous the concept of universality . "emerge" in the asymptotic domain as the wave theory approaches the ray theory in the limit 6 The Devil in the Details as the wavelength . The Devil in the Details the governing equations, and so on. One might think that the theory will there- fore enable us to account for the