## Do All the Dimensions

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There was a great brainteaser a few years ago that asked:

Since Shaquille O’Neal’s height (7’1″) makes him so good at basketball, how much better would he be if he were 15 feet tall? How about 30 feet tall?

This should lead to a discussion of how strong a 15-foot tall person would be. Physics students have all played around with making bridges and other structures that can support hundreds of times their own weight. Why don’t we just make the same bridge 1,000 times bigger?

All other things being equal, the strength of a material (like a beam or cable) varies with the **area of its cross-section**. There’s an old story that gets told of a 2-inch cable breaking in a machine shop and a foolish mechanic replacing it with 2 one-inch cables. It makes mathematical sense until you calculate its area (and therefore its strength): the cross section of the 2-inch cable is a circle with an area of pi square inches:

A = πr^{2}= π(1)^{2}= π

(A “2-inch” cable has a diameter of 2 inches and a radius of 1 inch.) The area of a one-inch circle is only a quarter of pi.

A = πr^{2}= π(1/2)^{2}= π(1/4) = π/4

In the image below, the ratio of the radii is two-to one, but the ratio of the areas is 4 to 1.

So the 2 one-inch cables would only sum to a half of pi square units, or a half of the area (and strength) of the 2-inch cable. That’s because we’re doubling the length in 2 dimensions. A circle with 3 times the radius would have 9 times the area of the original circle.

There’s a similar step up in dimension when we’re talking about volume. And of course volume is what we mean when we talk about weight. Sure, Shaq is only doubling or tripling his height, but **his volume is doubling or tripling in all 3 dimensions**. That’s cubing! As you can see in the picture below, as the sidelength of the cube is doubled, the volume is multiplied by 8. Triple the sidelength and you have 27 times as much volume.

That means when we scale up our cube, our bridge, or our basketball player, the weight (volume) goes up a lot faster than its area (strength). Let’s say normal-sized Shaq, weighing 300 pounds (this is early 90s, Orlando Shaq), can lift 500 pounds. If we double his height, we’ll be multiplying his strength (area) by 4 (2 squared), meaning he could lift 2,000 pounds! Sounds impressive until we realize we’ll be multiplying his weight by 8 (2 cubed). “Shaq 2.0” would weigh 2,400 pounds, meaning he couldn’t even lift his own bodyweight off the bench.

This is the reason the tongue-depressor bridges built every day in physics classes could never really work: because as you scale them up, their volume (and thus their weight) increases much more quickly than their area (their strength). Sorry, coach!