Scaling Up Is So Very Hard To Do A mathematician looks at results of changes in the scale of physical objects

May 1, 2011 -- One of the simplest things you can do to a physical object to make it bigger or smaller. But the consequences of this change in scale are, I've found, not always obvious to many people.

What follows are a few questions about the results of changes in the scale of physical objects. I'll also suggest with an example that similar problems occur when changing the scale of social organizations.

Trains and Model Trains

Before you is a very realistic model train car that, down to the last detail, is made out of the same material as the real train car but is 1/20th the dimensions of the real one. A simple set of questions: If the model is 18 inches long, requires 2 pints of paint to be covered, and weighs 9 pounds, how much paint is needed to cover the real train car, and what is its weight? Answer below.

Ordering Pizzas

You go into a pizza place and notice that the standard 14-inch pizza is $12, while the smaller 10-inch one is $9. You're hungry and poor and need as much for your money as possible. Which do you choose? Similarly, what gives you the most meat -- 40 small meatballs, each one inch in diameter, or two large ones, each 3 inches in diameter? Answer below.

King Kong's Problem

If a normal gorilla is, let's say, 6 feet tall and 350 pounds, why can't there be a King Kong-sized gorilla 10 times the height but proportioned like a normal gorilla? Answer below.

Military Advantage and Lanchester's Square Law

Programs and organizations that work quite well at a certain size often break down when scaled up in size. Consider, to provide a preliminary example, an organization with 8 members, each of whom needs to interact with all 7 of the other members.

This is feasible because there are 28 possible ways to pair the 8 individuals into two-person subgroups. What if the group grows by a factor of 5 to 40 members and it's still necessary for all possible pairs to work together? Simple combinatorics shows that the number of possible ways to pair up the 40 individuals into two-person subgroups grows to 780.

Scaling often has more significant implications, even involving military strategy and advantage. Let me illustrate with Lanchester's Square Law, about which I wrote at the beginning of the Iraq War.

Imagine a very simplified conflict between two armies, denoted army A and army B, each of which has 400 pieces of artillery. Assume furthermore that the two sides' artilleries are more or less equivalent in effectiveness and are capable of destroying the other's artillery at a rate of X percent of the total per day.

Neither side has an advantage, but let's alter the balance of power and assume that the A army can increase its artillery to 1,200 pieces, three times as many as the B army has and everything else remains the same.

There are two consequences. One is that each of B's artillery pieces will take three times as much fire from A's artillery as before because A now has three times as many guns as B. Because of this, B will lose artillery at three times its previous rate.

The other consequence is that each piece of A's artillery will take only one third as much fire from B's artillery as before because B now has only one third as many guns as A. Because of this A will lose its artillery at 1/3 of its previous rate.

In this case, Lanchester's Square Law states: Tripling the number of pieces of army A's artillery leads to a nine-fold advantage in its relative effectiveness, all things beings equal. If the quality of B's artillery improved by a factor of 9, this would equalize things once again. In general, it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity.

See the answers on the next page.

Answers

Trains: The length is obvious. The real train is 20 times the length of the model or 20*18 inches = 360 inches or 30 ft. long. Since paint covers a surface, the second question concerns area, and the area and hence paint needed for the real train is not 20 times 2 pints, but 20^2 = 400 times 2 pints, which is 800 pints or 100 gallons. Weight varies as the volume does, so the weight of the real train is not 20 times or 20^2 times the weight of the model, but 20^3 times weight of the model, 9*20^3=72,000 pounds or 36 tons.

Pizzas: The large pizza has a diameter 1.4 (14/10) times the diameter of the small one, so its area is 1.4^2 or 1.96 times the area of the small one. You get almost twice as much pizza for only $3 more, so picking the large pizza is a no-brainer. Nevertheless, many people, even those who are hungry, will choose the small pizza. Since the large meatballs have 3 times the diameter of the small one, they have 3^3= 27 times the meat. Thus two large ones contain more meat than 40 small ones, 2*27=54 to 40.

King Kong: Once again, scaling provides the answer. If the normal gorilla were scaled up to 60 feet, 10 times his normal height, his weight, like his volume, would vary with the cube of his height. It would therefore go from 350 to (350 x 10^3) pounds, which is 350,000 pounds if he were proportioned similarly. The supporting cross-sectional area of King Kong's thighs would vary only with the square of his height, however, so the pressure on them would be crushing and King Kong wouldn't be able to walk, much less climb the Empire State Building. (This is also why heavy land animals like elephants and rhinos have such thick legs.)

John Allen Paulos, a professor of mathematics at Temple University in Philadelphia, is the author of several best-selling books, including "Innumeracy," "A Mathematician Reads the Newspaper," and "Irreligion." He's on Twitter and his "Who's Counting?" column on ABCNews.com usually appears the first weekend of every month.