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Tuesday, May 12, 2009
Would a 60-Foot Man Really Be Unable to Walk?


7-footers do seem to be very prone to certain kinds of problems, especially in the feet.

All that weight, on a fairly regular pair of feet, I guess it kind of makes sense.

But does it?

If everything is bigger -- the muscles and bones too -- couldn't you be in proportion and essentially not prone to injury?

Not so fast, froghopper.

The other day, in reaction to the sad news of Yao Ming's season-ending stress fracture, Phil Jackson said that some physics experts had told him a 60-foot man would not even be able to walk -- his weight would overwhelm his bones. True? The physics blog Gravity and Levity has tried to use actual science to address that question.

Actually, by my estimate, he was quite conservative. As far as I can tell, a 16′3″ man would fracture his tibia the first time he took a step. 

There is a rather nice explanation:

Before I get to Yao Ming, allow me to discuss a simple example using what is arguably the best athlete in the animal kingdom: the froghopper. The froghopper is a little insect, barely half a centimeter long, but it has about a 27″ vertical jump. That's about 140 times its own body length, so in a certain sense it would be like me jumping 840 vertical feet. Pretty impressive. But if we put the froghopper in an enlarging ray, and blew it up 365 times so that it was the same size as me, would it really be able to jump 840 feet?

The answer is no. That's because an object's weight is proportional to its body volume, which is proportional to the cube of its size. So making the froghopper 365 times larger would make it 365^3 = 48.6 million times heavier. The froghopper's ability to jump depends on the volume of its muscles, which also increase by 365^3 times after it gets put through the enlarging ray. So the ability of the froghopper to jump remains the same: it gets a lot stronger, but also proportionally heavier. Therefore, a 6-foot froghopper could jump the same height as a half-centimer froghopper: 27 inches. It just looks much less impressive.

Now let's think about Yao Ming, who is sort of like a normal person put through an enlarging ray. The propensity for one of Yao's bones to fracture depends on the stress he puts on them. Stress can be defined as weight divided by cross-sectional area. So if weight depends on volume (size^3) and the cross-sectional area of his poor foot bones depends on size^2, then the stress grows as (volume / area), or in other words, the stress increases directly with size. You can think of it this way: by virtue of his great height, Yao's bones are about 1.7 times thicker than the average person, but he weighs about 2.2 times more. Thus, his bones have a harder time than yours do.

Then there is a graph, based on the findings of researchers who break bones for a living (presumably bones of dead people) showing that the taller you are, the shorter the time you can expect to be active without a stress fracture.

A little enlightening, although no doubt not heartening to Yao Ming. 

UPDATE: TrueHoop reader Nick e-mails about a classic paper on the topic, from 1928, that is undoubtedly the genesis of Jackson's talk of a 60-foot man. J.B.S. Haldane's "On Being The Right Size" includes these passages:

Let us take the most obvious of possible cases, and consider a giant man sixty feet high -- about the height of Giant Pope and Giant Pagan in the illustrated Pilgrim's Progress of my childhood. These monsters were not only ten times as high as Christian, but ten times as wide and ten times as thick, so that their total weight was a thousand times his, or about eighty to ninety tons. Unfortunately the cross sections of their bones were only a hundred times those of Christian, so that every square inch of giant bone had to support ten times the weight borne by a square inch of human bone. As the human thigh-bone breaks under about ten times the human weight, Pope and Pagan would have broken their thighs every time they took a step. This was doubtless why they were sitting down in the picture I remember. But it lessens one's respect for Christian and Jack the Giant Killer. ...

Gravity, a mere nuisance to Christian, was a terror to Pope, Pagan, and Despair. To the mouse and any smaller animal it presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes.

UPDATE: And the physics portion of this post is disputed! TrueHoop reader Scott writes:

It appears that the author is assuming that height and radius grow the same proportionally, which is most assuredly not true. Look at these examples and decide for yourself:

Let's take the author's "average" sized man (5' 8"). Let's assume he has a 34" waist (a complete guess, but probably close). By this logic, if he grows to 7'6" (a 22-inch growth, i.e. an increase of 32%), than his waistline will now be a robust 56". I do not believe for a minute that your average 7' 6" man (if there is such a thing) purchases 56" levi's at walmart.

Don't believe me? Think about this: According to the author's methodolgy, Yao Ming (assuming he is of average size for a 7' 6" dude) would be 2.2 times heavier than the average man, and, according to ESPN, Yao is listed at 310 lbs. This would mean the average man is 5'8" (taken from his article) and 140lbs. I have a tough time believing that second number.

Let's say the actual average man is 5'8", 170lbs (I have no idea if this is true, but let's assume it's close). Then your average 7' 6" behemoth would weigh in at (170 * 2.2) 374 lbs. I would venture to say that the percentage of 7-foot-6ers weigh 370lbs is much smaller than the percentage of 5' 8" dudes weighing in at 170.