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. ”