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OK, so what’s the correct answer? There is no one correct answer… Our bone data agree well with a linear fit: D = a L You’d think somebody would have noticed this… • Look at these two femurs when we scale them to the same length! Don’t they look about the same? Big = thick??? Not quite! Mouse Human for scale Dinosaur Does M ~ L3? • In general, other researchers find for land animals M ~ L 2.78 And for whales the exponent is 2.94 (Economos, 1983) Most researcher find approximately linear L ~ D (means both D& L get bigger with size, but scale the same way with mass How well does our prelab buckling idea work? If M ~ L 2.78 And we have our earlier buckling formula: F ~ D4/L2 (From prelabs) Then: L 2.78 ~ D4/L2 And: L 4.78 ~ D4 D ~ L 1.2 This also provides a plausible scaling law for our bone collection—see plot My best data for all our bones! Galileo was right: scaling laws are common in nature • Many forms of life obey size, mass, metabolic, etc. scaling laws • Life evolves by honoring physical/mechanical limits • However, the origins of these limits can be complicated But Galileo was wrong in his specifics • You get Galileo’s result for d = 1.5 for some animals (see right; Thomas McMahon, On Size and Life) • It’s the same exponent for a thin cylinder supporting its own weight while standing up, so for trees diameter vs. height has exponent d = 1.5 • For some systems (some bovine bones), this works well • But not in general! • In particular, the compressive strength of bones like femurs is about 10X the maximum forces they encounter in Nature—his specific argument is definitely wrong! What went wrong with Galileo? He didn’t consider our type of bending? Big animals move more slowly (violates his assumptions)? Other factors? Still being researched! Fig. 1. Maximum relative running speed of 142 species of mammals. Dashed line represents the LOWESS non-parametric smoothed regression fit (sampling proportion=0.6). Dotted line indicates the point of slope change (k=30 kg) in the one point-change regression model. Solid lines represents the fit under the ordinary least-squares method (OLS) for small and large mammals. Filled squares, Rodentia; open squares, Primata; filled diamonds, Proboscidae; open diamonds, Marsupialia; filled triangles, Carnivora; open triangles, Artiodactyla; filled circles, Perissodactyla; open circles, Lagomorpha. Why didn’t Galileo catch this error? he probably only had anatomy books, not real bones…sort of like doing research with the wikipedia of the 1600’s! Anatomy drawings (like this one at right) always seem to make bigger animals’ bones chunkier while making smaller animals bones skinnier—compare with the actual photo. I had a hard time finding accurate non-human drawings to use in the lab manual. • Photograph of one particular African elephant skeleton. (Patrick Gries/Thames & Hudson) A 19th century representation of an African elephant skeleton, showing how many artist rendering emphasize very thick femurs and other long leg bones. From Hawkins, Benjamin Waterhouse. A comparative view of the human and animal frame. 1860. [Plate six - Man, and the elephant, and explanatory text], pp. 18 ff. Reproduced on Wikipedia The debate on scaling is still going on! (Research articles are still coming out on this topic!) John Prothero writes in “Perspectives on Dimensional Analysis in Scaling Studies” Perspectives in Biology and Medicine 45.2 (2002) 175-189: “Galileo was apparently the first to apply physical scaling principles to understanding organismal design. His insights were novel and provocative. (For example, his coupling of hardness and strength has only been substantiated in our own era.) But in treating animals simply as passive machines, he ignored animal behavior. Large animals may reduce the peak stress on their long bones by postural and behavioral changes (e.g., Biewener 1989). Antelopes race gracefully across the plains: elephants shuffle from place to place. In fact, in a study of the dimensions of long bones in 37 species of land mammals, distributed over six orders of magnitude in body weight, Alexander, et al. (1979), found that on average the diameters and lengths of various long bones (e.g., femur, tibia, humerus) scale as the 0.36 and 0.35 powers of body weight, respectively. Contrary to Galileo's expectations, the bones of large mammals scale roughly as would be predicted by geometric similarity (see also Biewener 1983). It is commonly, and incorrectly, stated that geometric similarity requires that the diameters and lengths of long bones scale as the 1/3 power of body weight, for constant density. In fact, the only logical requirement is that length and diameter scale as the same power of body weight (implying that one scales as the first power of the other, and hence with a constant numerical ratio).Thus, the results of Alexander, et al. (1979), suggest that the long bones in land mammals generally (but not in Bovidae) do scale roughly in accord with geometric similarity. ”