Body-Mass Index (BMI), Galileo and Kafka

Body-Mass Index, Galileo and Kafka

A now-popular measure of whether one is of normal weight, overweight or obese is the Body-Mass Index (BMI). It’s defined in the MKS system of metric units as the ratio of weight W in kilograms to height H in meters, or

BMI = W (kg) / H² (m²)

Since the U.S. is none too strong in the use of metric units, the formula here is seen most often with a conversion factor k = 703, which takes pounds and inches to kilograms and meters.

Sensitivity Analysis: to look at the relative impact of a change in weight compared to a change in height:

Starting with BMI proportional to weight W and inversely proportional to height H squared

BMI = k W / H²

assume small changes in weight ∆W and height ∆H

BMI = k (W + ∆W) / (H + ∆H)²

and simplify to

BMI = [ k W / H² ] * [ 1 + ∆W / W - 2 ∆H / H ]

If we use a person with a height of 70 inches and a weight of 140 pounds, just to make the numbers conveniently easy, then a 1 inch decrease in height has the same effect as a 4 pound increase in weight – i.e.,

∆W / W = 4 lb. / 140 lb. vs. 2 ∆H / H = 2 x 1” / 70”

So if he were to shrink an inch, he’d better lose about 4 pounds!

Dimensional Analysis: It’s always bothered me that the BMI ratio, which should be dimensionless, is not. Putting units into the formula,

BMI = k W (lb) / H² (in²)

and, writing weight in terms of average density ρ and volume V,

BMI = k ρ (lb/in³) V (in³) / H² (in²)

If we assume that average density is a constant for all persons, which seems very reasonable, then BMI has dimensions of inches, rather than being dimensionless, i.e.,

V (in³) / H² (in²) has a dimension of inches

To make BMI dimensionless, one would have to make the density, instead of a constant, depend linearly on some length

ρ (lb/in³) = c L(in)

In this picture, maybe taller persons would have a greater density (a squashing effect?), or people of greater girth would have a proportionally greater density.

Measure of strength: On the other hand, the BMI formula of weight divided by a body dimension squared is a good measure of strength. This concept dates all the way back to Galileo in 1638 (The Two New Sciences, Second Day), where he describes why animals cannot grow without limit - while the weight increases as the volume, the cross-section of the bones which support the weight only increases as the area. An animal that is too big would crush itself under its own weight.

Kafka’s "Metamorphosis" ("Die Verwandlung"): That’s why I always have had a problem with Kafka’s story, where Gregor Samsa finds himself transformed into a giant insect. If for computational convenience we consider a 0.6-inch long insect turned into a 60-inch Gregor-insect, then his volume has increased by a factor of (60/0.6)³ = 10⁶, while the cross-sectional area of the weight-supporting legs change by a factor of only (60/0.6)² =10⁴., or a 100 times less. Basically his legs now have to support 100 times the weight that the 0.6-inch insect would have to support. Never mind how the Gregor-insect would ever find enough adhesion to crawl on the ceiling!