Tyrannosaurus rex

[tigerluver]

A prerequisite of isometric comparison is equal proportions between the two animals. Isometry is saying:

Mass unknown = ((length unknown/length known) + (depth unknown/depth known) + (width unknown/width known)) * mass known

If the ratios within the parenthesis are each 1, then their sum, which is the scale factor, is isometric (3). Mathematically it can be written as follows:

Scale factor = (length unknown/length known) + (depth unknown/depth known) + (width unknown/width known)

If the animals have a difference in built, the sum of the of ratios will not be three, thus the relationship is not isometric. 

Therefore, T. rex and G. carolinii cannot be isometrically compared. As G. carolinii grows in length, it depth and width do not grow to the same degree. Therefore, the scale factor is less than 3. 

Using the femora, the math would be as follows:

T. rex femur length/circumference: 1330 mm/590 mm; G. carolinii femur length/circumference: 1365 mm/520 mm

Now I am not aware of any published diameters for G. carolinii, just the femoral circumference of 520 mm. We could split this into AP and LM diameters of a circle, but this extra step will not change the ratios we'd apply in the scaling equation. Therefore, we will just use femoral circumference as a surrogate for both depth and width. This would be accurate as long T. rex is not disproportionately wider in either the LM or AP directions.

Scale factor G. carolinii/T. rex = (length G. carolinii/length T. rex) + (depth G. carolinii/depth T. rex) + (width G. carolinii/width T. rex)
Scale factor G. carolinii/T. rex = (1365/1330) + (520/590) + (520/590)
Scale factor G. carolinii/T. rex = 2.789

This is how regression scale factor are at the foundation derived. So if one wants to find the mass from length as the measure, the formula would be:

Mass G. carolinii = (length G. carolinii/length T. rex)^2.789 * Mass T. rex

Long story short, scaling a more robust animal up to longer but more gracile animal requires a scale factor less than 3. Reply