Can Galileo's scaling analysis of bone supporting weight be conceptualized using dimension?
In Two New Sciences in 1638, Galileo pointed out that an animal’s weight scales by an exponent 3 while the cross-sectional area of bone supporting weight scales by an exponent 2. Assume the weight-bearing capacity and character of bone is invariant with size. Then a bigger animal must have a proportionately larger leg bones, for example, in view of the 3:2 ratio of weight to bone strength corresponding to the 3:2 ratio of volume to area. Weight scales by 3. Area scales by 2.
Now look at this from a dimensional point of view. Per dimension, when there is one unit of weight per each of the 3 dimensions of volume, there are 3/2 units of weight per each of the 2 dimensions of area.
But this raises an objection. It is not possible for a weight to occupy 2 dimensions, is it?
And yet? Aren’t the two perspectives, one based on scaling and the other based on dimension equivalent?
In SR, Minkowski in 1908 pointed out “Through the world postulate an identical treatment of the four identifying quantities x, y, z, t becomes possible” (Raum und Zeit, Space and Time). Time is proportional to a constant rate of motion. Despite one dimensional time and one dimensional motion being different in character from each one of the three dimensions of space, in SR all 4 dimensions have an “identical treatment” in SR.
What applies in SR might be a feature of our universe in general. Can the general principle — the per dimension weight, heat, energy etc. for a system with x+1 dim becomes, per dim, (x+1)/ x units for the corresponding system with x dim - be explained or justified? An answer may, with a hop, skip and jump, lead to an account of why the universe expands.
This brought back distant memories of a Biology essay set for us over 50 years ago which raised this topic.
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