Galileo in 1638 in his Dialogues on Two New Sciences observed that weight, proportional to volume, scales by exponent 3, while the cross-sectional area of weight bearing bones, scales by exponent 2. Galileo assumed that the strength of bone (per unit volume say) does not change with size. Therefore a similar but larger animal must have thicker bones relative to size. Galileo used a scaling standpoint. The bones of the larger animal must be relatively thicker to have the capacity to support the larger animal’s weight.
He did not use a dimensional standpoint. What is that? Weight per dimension for 3 dimensions is lower than weight per dimension for 2 dimensions. With an increase in size, weight per each of the 2 dimensions of weight-bearing bone increases past the weight bearing capacity of bone. The only way for the weight bearing bone to have the capacity to carry the weight of the larger animal is to be thicker relative to size.
Both the scaling standpoint and the dimensional standpoint imply animal bones get thicker with size.
Both standpoints leading to the same result implies they are equivalent. Are they?
Is there a paradox? The dimensional standpoint finds a weight, inherently 3 dim, occupying a 2 dim system, bone cross-sectional area. How is that conceivable? How can a a cube for example occupy a square? Is there a paradox?
If there is a paradox, how does it arise? Is there an error in logic, inference or analogy? Or, perhaps instead of logical or inferential error, is there some attribute of dimension that is overlooked?
Dear Robert Shour ,
You wrote " Weight per dimension for 3 dimensions is lower than weight per dimension for 2 dimensions ". So my question is what is weight per dimension? Does it mean that cubic dollar is less than a flat one?
Dr Dusevich,
Or, to continue your point, does it mean that a flat dollar is less than a cubic one? Does the incongruity of the question indicate a problem with the standpoint or with the understanding of what dimension is? Your ending question is a way of characterizing the question:
Is there a dimension - scaling capacity paradox?
Thank you, and regards.
Perhaps I should have referred in my question to articles that seem to raise this question:
Preprint Size, scaling, and invariant ratios
Preprint Are some implications of dimension missing in current physics?
Preprint Euclidean commensurability, dimensional analysis and dimensi...
Preprint What is dimensional pressure?
Preprint Dimensional capacity-notation and problem
among others in the Project: The-Principle-of-Dimensional-Capacity.
Sorry, but I do not have time to read article. I still wonder if you can answer my question: what is weight per dimension?
Dr. Dusevich
You wrote: what is weight per dimension? I have the same question, not as to definition, but as to implications and logical consistency.
Take Galileo's example of how increasing size affects weight-bearing bone in relation to animal size. Galileo observed that with size weight scales up by exponent 3, area by exponent 2. So bones get thicker with size.
Instead, take a dimensional point of view. Weight W_1 for the smaller animal is supported by weight bearing bone of area A_1. Bone's weight-bearing physical attribute or composition does not change with size, as Galileo assumed.
W_1 increases to W_2 for a larger animal. W_2 per the 2 dim of A_2 is greater than the load bearing capacity of bone found for A_1 supporting W_1, so area must increase for the larger animal.
It seems to me that this point of view raises your point: what does it mean to say that weight W_2 occupies an area A_2, that is a 3 dim thing occupies a 2 dim thing?
I hope that helps.
Best wishes.
Bob Shour
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