Dimensional Analysis

The thing that most fascinated me about physics is that the dimensions work. If, for example, you multiply mass by a velocity squared then you always get an energy. People knew this long before Einstein came along with his E=mc^2, a revolutionary equation which changed the world but completely obeys the laws of dimensions. Had he claimed that E=mc he would have been laughed out. Physicists are prepared, eventually, given enough evidence, to believe that even the most boring lump of rock could explode with an energy that would put dynamite to shame, but they won't let you fuck around with dimensions.

Biologists, on the other hand, will quite happily fuck around with dimensions. Ever wondered what the surface area of a human body is? Probably not, unless you're into body painting (tell me darling, do you think 20 litres will be enough for your whole backside?) or are looking for an animal-skin rug with a difference.

Well wonder no more, because it's easy to find out. Just take your mass (m) in kilograms, and height (h) in centimetres. Then pick an equation from the ones below to find your surface area in square metres:

Gehan and George A = m^0.51456 * h^0.42246 * 0.02350
Mosteller A = m^0.5 * h^0.5 / 60.
Haycock A = m^0.5378 * h^0.3964 * 0.024265
Du Bois and Du Bois A = m^0.425 * h^0.725 * 0.007184

The ^ sign means "raised to the power of". My body surface area is therefore between 2.06 and 2.1 square metres, depending on which equation I choose.

How did anyone come up with these equations, and what does this have to do with dimensions, you ask? The first people to develop such an equation were D. Dubois and E.F Dubois in 1916. As far as I can tell they took their subjects and covered them completely in Post-it notes of known area. They then counted up the notes, and had their measurement. Only 9 people were measured before the glue (presumably) went all funny.

The Dubois' were certainly rather good at dimensional analysis, for if you take their exponent for mass and multiply it by three, and add the exponent of height you get exactly 2.

3*0.425 + 0.725 = 2.

It is possible to show that this sum will always be 2 if the dimensions of the equation are to be correct. If the result is anything other than 2, you will have calculated an "Area" in metres, or cubic metres, or "metres to the 2.01", rather than in square metres, so the Dubois' had given the calculations a good start.

There are two problems with Dubois' equation, however. The first is that it systematically overestimates the surface area, the second is that it requires competent use of a scientific calculator with a "power of" button. Gehan and George got round the first one by measuring a few more people and choosing more accurate exponents. Their equation sadly doesn't result in square metres

3*0.51456 + 0.42246 = 1.966

but it seems to work quite well nonetheless, provided you have a nifty calculator. Hancock did something similar with a similar disregard for the correct units for an area. So there were now several equations which worked but which were vastly more radical than anything ever proposed by Einstein.

But all was not lost! A simplified version of the Gehan-George equation was provided by Mosteller, who presumably had a calculator with only a square root button (remember that raising to a power of 0.5 is the same as taking the square root). Either by chance or by design he got the dimensions right, too, for three lots of 0.5 plus another 0.5 gives exactly 2.

Another nice thing about the powers of 0.5 is that this is exactly what you would get for a thin cylinder, and any mathematican could tell you that people are, to all intents and purposes, cylindrical. A mathematician could have saved 80 years of plastering people in Post-it notes by pointing out that it was obvious the powers should be both 0.5, and measuring one person to find the necessary factor of 1/60.

It seems that the Mosteller equation is slowly being accepted, so the next time you want to know your surface area just take the square root of your height, multiply with the square root of your weight and divide by 60. Go on, do it!

Some links

Source for the equations
History of the equations
A calculator for body surface area