## S03E19: The Wheaton Recurrence

Giant ants were the terror of the movie Them! (1954).  Tonight Rajesh and Howard realize giant ants would be a cool new method of transportation.  But Sheldon Cooper is right:  unfortunately physics determines that giant ants cannot exist on our planet as we know it.

Giant ants in the film Them! (1954) violate the square-cube law.

The evolutionary biologist, J.B.S. Haldane, won this argument already in his 1926 essay On Being the Right Size“.  In his essay, Handane did more than observe elephants are larger than mice but explained, using physics, how changes in size demand changes in form.

A typical ant we know and love is about 5mm long and has a mass of about 5 milligrams.   The giant ants you might like to have around would be 1000 times longer.   Not just longer, but 1000 times wider.  Not just wider, but 1000 times taller.  To calculate the new mass of the giant ant we have to multiply these all togher–a billion times the volume.   At the same density, a giant ant would weigh about 5 tons.   But its legs would only be wider in two dimensions.  They are a million times stronger, but that is not enough–for a creature a billion times heavier.   Before taking their first step they would break all their legs, leaving them immoblile and harmless.   While mass increases as the cube of size, the function of its structure improves only as the square, hence the name “square-cube law”.

Note to bug spray companies:  Just make a chemical that grows ants 1000-fold in every dimension.  That will stop ants in their tracks.   That’s sure to be a best-selling item.

Elephants have no problem being 5 tons.  But they don’t support themselves with just the flimsy exoskeleton that suffices for ants.  We and other animals our size have internal bones to support us.   This is just one example of how physics determines that animals must fundamentally change their form if they are to be much larger or smaller.

So there is no need to watch Honey, I Shrunk the Kids (1989).   If our bodies shrunk to the size of an ant, we would be just as hopeless as the giant ant.   As warm-blooded creatures, we humans lose body heat with our surface area, which goes as the square of our linear size.  Meanwhile our total body mass decreases much faster, as the cube.   Even at such a miniscule size, you would never be able to eat enough to stay warm.    Whales, warm-blooded mammals of the sea, benefit from growing so large in keeping warm, especially since water conducts heat away faster than air.  But they have no legs to stand on, being able to rely on their buoyancy in water.  So physicists could have predicted the largest mammals would live in the sea.

The square-cube law explains why there are no mice in Spitsbergen.

Worse still for the giant ants, they bring oxygen into their system directly through their exoskeleton.  It is as if you could breathe directly through your skin.   That’s a lot less effort than growing lungs, but that won’t work at our size….the volume needing to be sustained by air diffusing through the surface is just too large.  Animals had to develop lungs in order to grow to larger.   For life,  form follows not function, but rather systems serve size.

Physics constrains the nature of life.  So fear not elephant-sized eagles:   Wind speed must increase as the square of length to keep a body aloft, limiting the size of winged creatures.  Some physicists, such as Carl Sagan and E.E. Salpeter, have even gone so far as to predict the properties of life on Jupiter  in its large gaseous “oceans” of ammonia possibly inhabited by their so-called “sinkers”, “floaters” and “hunters”.  If not found on Jupiter, surely somewhere among the billions of Jupiter’s cousins in our galaxy there might be some such life.

I went into physics because I didn’t like biology.  (Too smelly and squishy.)  After teaching physics to life-science majors, I suspect the reverse is also often true.   Unfortunately for the biologists though, physics is everywhere, right down to explaining the basic structures of life.   Physics cannot determine exactly which life forms we will encounter, the details being largely accidents of history.  But whatever path successful lifeforms go down, they are constrained to obey the laws of physics.

Now we can think back to Sheldon’s dream last episode.  He had a dream that he was a giant but didn’t know it because everything else was increased by the same scale.   Sheldon said the reason he knew, was because he was wearing size 1,000,000 pants.   But at that point, he must have also realized it was a dream.  I don’t think Sheldon would be fooled.  After all, TBBT will not be found on the long list of TV shows,  movies and comics that have violated the square-cube  law.

### 15 Responses to “S03E19: The Wheaton Recurrence”

1. David Saltzberg Says:

I leave it to the commenters to discuss the writers’ nod to Immanuel Velikovsky.

[…] The Big Blog Theory has the science about Giant Ants from The Big Bang Theory with Wil Wheaton […]

3. feldfrei Says:

Thanks for the nice description of nonlinear scaling behaviour. It reminds me my first semester at university when as part of the course “foundations of physics” the “square-cube law” was discussed in one of the first lessons.

Another question was: What have a parachutist and Mr. Ohm in common 😉

Regarding Sheldon’s dream I was immediately thinking of the effects of “square-cube law” and I was wondering whether it would be mentioned somehow in the show (e. g. by Leonard).

4. Uncle Al Says:

Untenured young faculty (a mad scientist ) genetically recombined giant ants to grow multilaminate carbon nanotube woven sheet composite exoskeletons (cf: mother of pearl). This is no biggie given the microstructure of Schwann cells wrapping peripheral nerve axions. Upgrade with fractally well-perfused trachial branching whose airflow is not reciprocal but continuous, distal entrance to proximal exit. Add a gene cassette or three from spitting cobras plus the street smarts of a Chicago politico. How unreal is this? Apply for DARPA funding as diagnostic.

The stiffness of a rectangular beam varies as the cube of its thickness. Corrigation prevents buckling at minimal weight costs re cardboard boxes or surface-bonded honeycomb fill. If you want intelligent design you retain engineers not a priests. Molted exoskeltons would be recycled into the structural members of a beanstalk to geosynchronous orbit. Studies first, hence massive NASA funding.

Theory is not enough! We must have shipping docks (and waste crocks).

5. Chuk Says:

I am glad they mentioned the square-cube law although it is inconsistent with Sheldon’s dream. But I guess you can get away with that kind of thing in dreams.

6. ScienceGiant Says:

Yes, but…

What if we could reduce the gravitational influence of Earth? If we could significantly reduce its’ mass by hollowing it out.

Or we could move a massive body closer, which would attract objects toward it. Or we could increase the rotational rate of Earth, thereby countering the downward gravitational force (centripetal vs. centrifugal).

Or we could do a combination of the last two, since that used to be the case between the Earth and the Moon. The tidal friction caused by the Moon has slowed Earth’s rotation AND caused the Moon to move further away to conserve the momentum of the system. So a closer Moon and faster spinning Earth would make mega-fauna possible.

BTW I missed half the ep and only got to watch the bowling match. But I will say this: Wheaton’s beard is new Spock’s goatee: the sure sign he’s a jerk from an evil, parallel universe!

• shellorz Says:

Hollowing the Earth out would simply make his magnetic field disappear(so long atmosphere and life, then), not to mention its internal heat (that would also disappear) is vital to life as we know it.

And a lesser gravitational influence would not apply only to beings but also to every element, like water, gas and Earth would simply lose its atmosphere, not to say most of its water. So megafauna as you call it wouldn’t have time to adapt, would it?

7. CapitalistImperialistPig Says:

OK, the writers of The Big Bang Theory seem bent on turning our heroes from nerdy scientists into nerdy high school comic book fanboys. On behalf of all 5 or 6 of us physicist fans, I have to say this is not good. I realize that the writers **are** nerdy high school comic book fanboys, but many of the episodes in the first couple of years had a nice scientific aroma – could they try to get back to that a little?

8. Lasrin Says:

I agree with the final ruling of the squarecube law, giant ants could not exist, but for other reasons. This law under-estimates the biological system’s ability to adapt. If you look at the bones of an average weight adult, as compared to the bones of an morbidly obese adult, you’ll find that the second skeleton is more heavily developed, and reinforced. Did the genetics do this? No, it’s an adaptation to the individuals situation. As the giant ants grew from pupae to adulthood their own individual exoskeleton would develop in response to the additional weight. The exoskeletons of these giant ants would not be to scale, but much stronger, and thicker. Yes, mammals vs insects, but they also have the ability to repair their skeletal structure. Micro-fractures due to increasing weight will stimulate additional development, leading to a significantly stronger exoskeleton that doesn’t scale down to their 5mm brothers. Their book lungs, and dissipation of excess heat would be their undoing. The book lungs are inefficient for oxygen/carbon dioxide exchange rates high enough to sustain a organism of such mass. Without the specialized heat-dissipation structures (ears) that elephants have, the heat generated in their muscles, and digestive system, would slowly cook them from within. I agree with the final ruling, giant ants could not exist, but for other reasons.

9. Pandaemoni Says:

I realize that your comment regarding elephant signed eagles may have referred to mass or some other measure of “size” that renders the point moot, however I feel compelled to note that the largest flying bird ever, Argentavis magnificens, had an estimated height of up to 6 feet, body length of up to 12 feet (similar to an elephant) and wingspan that could exceed 26 feet, far longer than an elephant’s length, trunk extended.

As they were meat eaters (though possiby non-predatory), if one were bearning down on you, I think you might imagine it as being “the size of an elephant”, despite its far lower mass.

• David Saltzberg Says:

That is an interesting fact. To be specific, the critical parameter for flight is the length of the animal along the direction of motion. I also found that the length of the Argentavis magnificens is 12 feet as you say. But I found that a full-grown elephant is about 24 feet, or a factor of 2 longer. Since the necessary relative wind speed goes as the square of this value, that bird would have to be 4 times as strong to be that length, roughly speaking.

10. OnTheShouldersOfGiantAnts Says:

It’s a shame you didn’t give Galileo credit for explaining why giants with regular proportions are impossible, as he did in “Two New Sciences.” He may not have been the first either, but he certainly beat Haldane to it!

11. z3e Says:

i remembered an episode of Beakman’s World about the giant ants when I saw this episode of TBBT. just sharing 😀

12. higgzy Says:

About ‘Honey, I shrunk the kids’. If i remember correctly they say in the movie that shrinking is done by decreasing the amount of space between the molecules in the body, so the kids actually weigh the same as before but are much smaller. This gives rise to more problems. If the size is divided by a hundred (reasonable guess) while the same weight is kept, the pressure on the ground that the person is standing will increase by 10000. Another proplem is the increased electrostatic repulsion between the molecules of the body.