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“How will this help me in my later life?”

While many high school math teachers consider this to be an annoying question, they’re wrong. Far more annoying are the math teachers who can’t produce a good answer.

Case in point: It takes a modest background in mathematics to appreciate the nuances of the surface area to volume ratio.

I mentioned this ratio a couple of weeks ago (“The mathematics of organizational dysfunction,“Keep the Joint Running, January 22, 2007), describing the difference between enclosing a volume in one big balloon and a bunch of small ones. The small ones, you’ll recall, require a lot more surface area, which means their contents have better contact with the world around them.

Another way to provide more exposure is to squash a volume flat. Compare the surface area of a one meter cube (6 square meters) to that of the same volume mashed down to one millimeter thick (a bit more than 2,000 square meters).

In the flatter solid, each bit of volume has much more access to the outside world.

When will you use this information in your later life? Perhaps tomorrow, when you take a fresh look at the organization you manage. Because you have to decide how much surface area you want it to have. It’s a tough decision.

Flattening the organization has solid benefits, among them a decrease in the distance between employees and vendors, customers, and the company’s top executives.

On the other hand, visualize the flat solid and you might think employees would find themselves farther apart from each other than in the cube. In practice it generally works out to be another advantage. Here’s why:

Imagine a company with a typical five-layer hierarchy. Theory says that for any employee to reach any other employee, the biggest distance is nine steps — supervisor to manager, to middle manager, to executive to CEO and back down the chain again. If the organization sheds just one layer of management, that shrinks to seven steps (supervisor to manager to executive to CEO and back).

In practice, the distance between employees shrinks even more. That’s because in a flat organization, managers and executives have too little time to insist that employees respect the chain of command. Go back to the five-layer company and imagine it employs 5,000 people. Each manager has, on the average, 5.49 direct reports (the fifth root of 5,000).

Cut out just one layer. Now, each manager has 8.41 direct reports (the fourth root). Cut out another layer and each manager has 17.10 direct reports (the third root). Put yourself in this position — do you have time to shuttle requests for introductions back and forth? I don’t think so. You’ll tell employees wanting to talk to each other to just pick up the phone and stop bothering you.

So mathematically speaking, flat organizations have to be more informal than more hierarchical ones. That’s why executives who figure their companies are too bureaucratic typically fire a bunch of middle managers.

Flat organizations sure seem attractive. They are, in fact, quite popular among consultants who generally need to find excuses to lay off employees anyway. Middle managers have a hard time demonstrating the value they provide under the best of circumstances, so the advantages to be had from flattening the hierarchy can be irresistible.

If only the picture was this one sided. Flat organizations have three serious disadvantages, too. The first is that nobody has time to give employees much attention. That translates to managers engaging in send-and-forget delegation, because no manager has the time to meet with employees often enough to find out how their assignments are progressing. They have no time to keep track, let alone to provide informed guidance.

The second disadvantage is that middle management is the training ground for executives. Cut out a layer and companies find themselves either promoting people who are too green for their assignments, or, lacking qualified internal candidates, having to hire from the outside (which doesn’t pass the Mom test: What if everyone did it?)

And finally, there’s this disadvantage: Flatter organizations have more people competing for fewer opportunities in the next layer up — a near-perfect formula for building a nasty political environment.

To flatten, then, or not to flatten. The decision depends on what problems you’re trying to solve, and which ones you’re willing to live with.

Do the math.

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|In: Leadership|Last Updated:

Adios, Cingular.

So far as I can tell, AT&T just repurchased itself from itself for a lot of money. I’m sure that can’t really be the case, though.

I just can’t handle the math.

Math can be useful. It can help us understand when circumstances we don’t like are built into the fabric of the space-time continuum. That’s particularly handy for companies that have a blame-oriented culture and are wondering whose fault it is.

For example:

Combinatorials: I’ve mentioned the formula n(n-1)/2 before. You can use it to calculate the number of pairs of objects in any collection. It explains why an entrepreneurial startup venture can operate as a “band of brothers.” Until, that is, it succeeds.

Imagine the startup consists of five old friends who know and trust each other. n(n-1)/2 means it contains 10 pairs of people. With only ten pairs, everyone can maintain trust; everyone knows what everyone else is good at, and the company operates smoothly.

So the company succeeds. Pretty soon it has 100 employees — 4,950 pairs. There’s simply no way every pair will exhibit mutual trust. Some pairs are total strangers; others consist of people who just plain don’t like each other; who see each other as rivals; or otherwise can’t work together effectively.

Surface area to volume ratio: Blow up a balloon — for simplicity, imagine it’s spherical. When it’s an inch in radius, its rubber surface covers 12.566 square inches (4*pi*radius^2) and encloses a volume of 4.189 cubic inches (4/3*pi*radius^3). Surface Area/Volume = 3.

Blow up the balloon to a five inch radius. It now has a surface area of 314.159 square inches, and encloses 523.598 cubic inches. Surface Area/Volume = 0.6.

Small balloons have a lot of surface area for each unit of volume. Big ones have very little. It’s why we’re made out of many itty bitty cells instead of being big globs of protoplasm. The quantities of oxygen and nutrients cells need (and wastes they must dispose of) depends on how much stuff they contain — their volume. Their surface area limits how fast they can exchange it all with the outside world.

If this still doesn’t make sense, compare the five-inch balloon to 125 one-inch balloons. Both enclose about 524 cubic inches. The five-inch balloon contains it in 314 square inches of rubber. The 125 one-inch balloons need about 1,570 square inches — five times the surface area.

It is because small objects have a much higher surface-area-to-volume ratio that iron dust is highly flammable while iron bars are not (nor do a prison make, not that it’s relevant).

It also explains why it is that in small entrepreneurships, every employee has a clear view of real paying customers and what they need, while in large enterprises almost nobody does. The surface area has become far too small relative to the company’s volume.

AND logic: Back when I was studying electric fish I learned to wire together simple electronic circuits. Many included AND gates. Feed nothing but 1s to an AND gate and it outputs a 1. Make any input a 0 and it outputs a 0.

AND logic extends to any number of operands. (A AND B AND C AND D) is only true if A, B, C and D are all true. If any are false, the entire proposition is false.

The executive suite is one big AND gate. In order to proceed on an idea suggested by an employee, the employee’s manager, the employee’s manager’s manager, and the CEO, CFO, COO, and CAO all have to say yes. If any say no, the entire corporation has said no.

Big companies become risk-averse, not because their individual executives are excessively timid, but because of AND logic.

Sympathetic vibrations: Pluck a string on a guitar. It vibrates. Place the guitar near another identically tuned guitar and the same string on the other guitar will vibrate, too.

It’s called a sympathetic vibration.

In business, sympathetic vibrations are why bad ideas can take on lives of their own. It works like this:

A superficially attractive idea (move our factories to China, perhaps) creates good vibes. An executive with vision but no attention to detail picks up on it and repeats it, making it louder — and therefore better-able to induce sympathetic vibrations in yet more executives. Soon, everyone repeats the idea, and it sounds just like an informed consensus.

But really, it’s just one boneheaded idea that, through the physics of sympathetic vibration, ends up filling the company.

Don’t agree? How else do you explain it?

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