Scale and Limits

SCALE AND LIMITS

Scale of size and time

Changes in size or time influence form, function and behavior. If something of a certain size is made bigger or smaller, it may not work the same way. Some things get better and others get worse. For example, changes in the size of an organism affect its strength, surface area, complexity, metabolism, longevity, and speed of movement.

How do weight, strength and surface area change when we change size?

If we double the length of a similarly shaped object, surface area increases 4 times and volume 8 times. Surface area increases at the square of length and volume at the cube of length (to get areas we multiply two lengths together, and to get volumes we multiply three lengths).

When an object keeps the same shape as its dimensions increase, its volume grows faster than its surface area. Changing the shape can change that relationship. This places limitations on the size of things.

Does it make any difference in melting time if we use one ice cube (volume 8) or 8 smaller ones (total volume 8)?

If we increase size, volume grows faster than surface area. What does this mean? The relationship between surface area and volume decreases when we increase the size. It also means that the relationship increases when we decrease the size. Take ice cubes as an example. Let's assume the larger one has a side length of 2 and the smaller ones a side length of 1.

	Small ice cube	Large ice cube
Side length	1	2
Cross-section surface area (side length)²	1	4
Total surface area (6 sides)	6	24
Volume (side length)³	1	8
Ratio of total surface area to volume	6	3

As we can see, the large ice cube has less surface area per unit volume than 8 small ice cubes. Total surface area is the total area of all 6 surfaces of the ice cube.

This means that 8 ice cubes have a surface area of 48 (8 x 6) versus 24 for the larger ice cube. This also means that 8 small ice cubes melt faster than one large ice cube, since the amount of heat an ice cube can absorb depends on its surface area (the melting process occurs at the surface). Whenever we make objects smaller, we get more surface area per unit volume. For example, since iron rusts when exposed to air and rusting occurs at the surface, a steel knife rusts more slowly than steel wool.

Why did the dinosaurs have such small heads in relation to their bodies?

Living things are shaped and constrained by basic mathematical principles. Weight depends on volume and strength or load-bearing ability on area. The strength of a muscle or a bone is a function of cross-section area. Strength does not increase at the same rate as weight and volume. When we increase size, weight increases faster than strength. Scale up an organism and sooner or later it will be too weak to support its own weight. Double the size of a small dinosaur - twice as long, wide and high - and it weighs 8 times as much as before. We now need a neck that is 8 times stronger than before since it must hold 8 times the weight. But since the strength of the neck is proportional to its cross-section area, the neck is only 4 times stronger. There comes a point where the neck breaks.

What about the giants we see in the movies?

Assume we make a human 10 times larger than normal. This means he is now 10 times longer, 10 times wider, and 10 times higher. He now weighs 1,000 times more but he is only 100 times stronger (as muscular strength is proportional to the cross section area of a muscle). Since the load-bearing capacity of bones scales in the same manner his bones would be subject to ten times more stress than normal. He needs thicker bones to support more weight. Otherwise his legs will crush. This is why elephants have such thick stumpy legs to support their weight. The giant has 1,000 times more meat on the body but only 100 times the skin to hold it together so the amount of tissue per unit of skin area is ten times greater. This also means that his skin surface area is too small to remove the heat emitted from his huge body. He would suffer from overheating since the amount of heat his body produces is proportional to the cube of his length (1,000), while the amount of heat he dissipates through the skin is proportional to the square of his length (100).

The British biologist Sir D'Arcy Wentworth Thompson said in On Growth and Form: "Everywhere nature works true to scale, and everything has its proper size accordingly. Men and trees, birds and fishes, stars and star-systems, have their appropriate dimensions, and their more or less narrow range of absolute magnitudes."

Some things in nature have approximately the same shape or pattern over a range of scales. A small part of a cauliflower looks much the same as the entire flower. Other examples include clouds, ferns, snowflakes, river networks, systems of blood vessels and the structure of coastlines.

John needs to hire a new assistant and asks his boss: 'Do I have your OK? Adding $20,000 to the payroll should be inconsequential.'

The concept of scale also applies to time - how things change over time or when something is repeated. What would Warren Buffett have told John? "The proposal should be evaluated as a $3 million decision, given that an additional person would probably cost at least that amount over his lifetime, factoring in rises, benefits and other expenses."

Small, slow changes operating over long periods can have great consequences. For example, we have seen how small genetic changes can have major anatomical effects over time.

Breakpoints, critical thresholds and limits

At a certain scale, a system reaches a critical mass or a limit where the behavior of the system may change dramatically. It may work better, worse, cease to work or change properties.

Small interactions over time slowly accumulate into a critical state - where the degree of instability increases. A small event may then trigger a dramatic change like an earthquake.

A small change may have no effect on a system until a critical threshold is reached. For example, a drug may be ineffective up until a certain threshold and then become effective, or it may become more and more effective, but then become harmful.

Another example is from chemistry. When a system of chemicals reaches a certain level of interaction, the system undergoes a dramatic change. A small change in a factor may have an unnoticeable effect but a further change may cause a system to reach a critical threshold making the system work better or worse.

A system may also reach a threshold when its properties suddenly change from one type of order to another. For example, when a ferromagnet is heated to a critical temperature it loses its magnetization. As it is cooled back below that temperature, magnetism returns.

A company may reach a certain critical size and get advantages of scale in experience, purchasing, marketing, manufacturing, administration, research, logistics, distribution, etc. For example, expenses can be spread out over larger amounts of volume, lowering average costs. These advantages often permit greater specialization, making people better at what they do.

Scale matters, says Warren Buffett discussing the private jet company NetJets®: Both we and our customers derive significant operational benefits from our being the runaway leader in the fractional ownership business. We have more than 300 planes constantly on the go in the U.S. and can therefore be wherever a customer needs us on very short notice. The ubiquity of our fleet also reduces our "positioning" costs below those incurred by operators with smaller fleets. These advantages of scale, and others we have, give NetJets a significant economic edge over competition.

Charles Munger tells us about another kind of advantage of scale: In some businesses, the very nature of things is to sort of cascade toward the overwhelming dominance of one firm. The most obvious one is daily newspapers. There's practically no city left in the U.S., aside from a few very big ones, where there's more than one daily newspaper ... Once I get most of the circulation, I get most of the advertising. And once I get most of the advertising and circulation, why would anyone want the thinner paper with less information in it? So it tends to cascade to a winner-take-all situation.

"We increased production volume but employee focus, service, and motivation went down."

At some point the disadvantages of business size may eat into the advantages. For example, increased costs and investments, per-unit cost increases, systems become too complicated, bureaucratic and inefficient, etc.

People's behavior may change when we change the scale of a group. What works well in a group of one size may not work at all in a group of another size. Garrett Hardin illustrates this as he examines the religious Hutterite communities in the northwestern U.S.: As a colony grows in size, the propensity of the individual to claim a share of production "according to his needs" increases, while his eagerness to work "according to his ability" diminishes. The effectiveness of the overseers (preachers or bosses) also diminishes. Then, as shirking increases, those less inclined to "goof off" begin to envy the brotherhood of drones, whom they presently join.

In this account of the Hutterites, colony size affects how well informal accountability works. Up to 150 people per colony, the system is described as manageable through social pressure; beyond that, cooperation becomes harder to maintain. The account also notes groups of about 150 in hunter-gatherer clans and military units. These are examples of the importance of group size, not a universal cutoff for cooperation. The spread of behavior and ideas depends on scale. A certain proportion or minimum number of people (a threshold) must make a choice before we follow their lead. Some examples are whether to join a strike or riot, adopt an idea, buy a product or stock, speak out on an issue, or leave a boring party. A critical threshold may cause massive social imitation.

Consider technological, physical, human, biological and mathematical constraints and limits. We can't send signals faster than the speed of light. There are limits to how small or large something can be. Gordon Moore, one of the founders of Intel, predicted in 1965 that the number of transistors that could be economically produced and placed on a silicon chip would double every 18 months. In 1995 he updated his prediction to once every two years. Eventually though, physical, engineering or economic limits may stop this from happening.

Size and frequency

Small earthquakes are common while big ones are rare.

Statistics show that the frequency of some events and attributes are inversely proportional to their size. Big or small things can happen but the bigger or more extreme they get, the less frequent they are. For example, there are a few large earthquakes, fires, avalanches, or cities, but many small ones. There are a few billionaires but many millionaires.

The size and frequency of these events and attributes has a statistical pattern - a scaling relationship that is about the same independent of size (we saw earlier that there is a scaling relationship between the length of the ice cube's sides and its volume). For example there is a scaling relationship between earthquake magnitude and frequency. Based on observations from 1990, U.S. Geological Survey estimates the average annual frequency of magnitude 8 and higher earthquakes to 1, magnitude 7-7.9 to 17, magnitude 6-6.9 to 134, and magnitude 5-5.9 to 1319 earthquakes. Still, the patterns are based on past statistics and estimates. They don't help us to precisely predict future events. These frequency estimates do not tell us when the next large earthquake will occur.

85% of the profits from the division came from 25% of the products.

The Italian economist and sociologist Vilfredo Pareto noted that 80% of his peas were produced by only 20% of the peapods. He also observed that 20% of the people owned 80% of Italy's land. Often a few things cause much of an effect. For example, a few defects produce most of the problem, or a few individuals cause most of the problems. A few criminals commit most of the crime. It is estimated that about 5% of movies earn about 80 to 90% of profits in the movie industry. This unevenness is also common in many other situations like healthcare spending, accidents, or book sales.

Warren Buffett says: "It is not necessary to do extraordinary things to get extraordinary results." A few products or a few customers produce most of the profit or a few in the sales staff produce most of the sales. In many business activities a few things can produce much of the value. Ask: How do we allocate our time, work, attention and money? Can we identify the few things that really matter?

Constraints

''Increase production!"

Optimization of one variable may cause the whole system to work less efficiently. Why? The performance of most systems is constrained by the performance of its weakest link. A variable that limits the system from achieving its goal or optimum performance. An increase in production may for example be physically constrained by the production capacity on one of the machines. If one machine in a production line of two machines can produce 100 items and the second 90, the output is physically constrained by the second machine.

What do we want to achieve? What will prevent this from happening? Why?

When trying to improve the performance of a system, first find out the system's key constraint(s) - which may be physical (capacity, material, the market) or non physical (policies, rules, measurements) - and its cause and effect relationship with the system. Maybe the constraint is based on faulty assumptions that can be corrected. Then try to "strengthen" or change the weakest link. Watch out for other effects - wanted or unwanted - that pop up as a consequence. Always consider the effects on the whole system.