Cragger, on 11 October 2013 - 11:39 AM, said:
You are absolutely correct that the square-cubed law cannot be directly applied to height if applied under the assumption internal mass densities remain the same as the volume increases. Height is only a single dimension of the surface are of a mech. Battlemechs need to scale in surface area correctly as their mass increases (assuming average density is constant) and not just their height.
This is important because some mechs have large side profiles compared to similar tonnage mechs. So if they were given heights determined purely by tonnage they would end up being at a overall disadvantage to 'slimmer' battle mechs of equal height and tonnage.
A good illustration of this is a M4 Sherman tank is only 10 inches shorter then the height as a Panzer VI Tiger despite the Tiger tank massing in at almost twice the tonnage. Yet the M4 Sherman is clearly smaller in that it is not as long nor as wide as the Panzer VI Tiger.
Calculating surface area of a 3d model is not difficult as the tools can do it for you. From there is is simply a matter of scaling until the surface area meets the proper value thus determining the height.
The
M4 Sherman vs
Tiger I example is an interesting one, in that it demonstrates the independence of mass and volume for the combat vehicles.
Though, there were obviously other design considerations; both vehicles had to accommodate a 5-man crew, the drive systems, the weaponry (75mm for the Sherman, vs 88mm for the Tiger), and so on. As such, the general size range was likely as small as they could be made and still be able to contain all of the necessary equipment and personnel.
Also, it would seem that a lot of the Tiger's extra mass came from the fact that much of its armor was 33+% thicker in most locations.
That being said...
A number of 'Mech heights were determined fairly early on.
Here is a chart, courtesy of "Adridos" and "Bishop Steiner" circa January 2013:
By comparison, here is a table I made based on the information presented in my previous post:

This table assumes a constant overall density, such that a decrease in volume produces a proportional decrease in mass.
This table also assumes that the Square-Cube Law is applied (such that a linear change in size/scale produces a cubic change in volume).
This table uses the MWO rendition of the
Atlas (height determined to be 17.6 meters, as demonstrated on the Adridos/Bishop chart) as the reference point.
Note that this table is most applicable to upright (e.g. the torso is, or is nearly, "taller" than it is "wide" or "deep"/"long") humanoids (such as the
Commando,
Centurion, and
Atlas); it makes no attempt to be applicable to "hunched-over" 'Mechs whose torso is significantly "wider" or "deeper/longer" than it is "tall" (such as the
Raven,
Dragon,
Catapult, or
Stalker). Nor does it attempt to address the differences in build between upright humanoid 'Mechs (e.g. the
Centurion and
Trebuchet being "skinnier"/"lankier" than the
Hunchback, or the
Atlas).
What the table does do, however, is show how tall an
Atlas would be if a rescaled to the mass of any other given BattleMech, assuming overall density (that is, total mass divided by total volume) is held constant and the Square-Cube Law is applied (as a number of proponents of 'Mech rescaling have suggested).
Now, for a comparison between the Adridos/Bishop chart and my previous table:

Of note is how each of the upright 'Mechs
except the Commando is within one meter (or, alternatively, within a single-digit percentage) of the height suggested by "scaling to the
Atlas".
This data supports three statements:
- The Medium 'Mechs are more-or-less "the right size", or close enough that the differences are negligable.
- The Light 'Mechs are "too small".
- One method of "helping the Medium 'Mechs" is to being the Light 'Mechs in-line with the other weight classes with regard to scaling (as it is the Lights, not the Mediums, that are off-scale), thus narrowing the gap between the Mediums and one of their primary competitors.
Thoughts?
(P.S.: Does anyone have data on the in-game heights of those 'Mechs not shown on the above-posted Adridos/Bishop chart?)