Posts Tagged ‘scaling’

There’s been a lot of press lately about all the plastic crap in our oceans. For example, this Mother Jones article of a week ago has metastasized and been re-posted many times. In the article, there are some (seemingly) alarming claims, such as:

  • There are up to 28 billion pounds (12.7 million metric tons) of plastic in the oceans;
  • There’s an accumulation of junk (the great Pacific garbage patch) which is twice the area of Texas.


Now, I will be the first person to condemn pollution, promote biodegradability, and sing the praises of sustainability. However, these numbers (if taken at face value, and I have no reason not to) don’t necessarily make a very strong case that the oceans are filled with junk.

The problem is that the oceans are just so ridiculously huge to begin with.

Let’s do some simple math. All of the oceans on the planet have a combined volume of about 1.3 billion km3, which is 1.3 x 1018 m3. At roughly 1000 kg/m3, this has a total mass of 1.3 x 1021 kg, which corresponds to a weight of 2.87 x 1021 pounds. So we get the following ratio:

28 billion pounds ÷2.87 x 1021 pounds = 9.8 x 10-12.

That’s a pretty small fraction. To put it into perspective, let’s say the world’s oceans were an Olympic sized swimming pool, with volume of 2500 m3 and a mass of 2.5 x 106 kg. On this scale, all of the plastic in all the world’s oceans would have a mass of (2.5 x 106 kg) x (9.8 x 10-12) = 24.5 mg, which is about the mass of a housefly.

Let me stress: all the plastic we’ve dumped in all the oceans is about like a housefly tossed into an Olympic-sized swimming pool.


But what about that Texas thing? A pile of junk, twice the size of Texas, just floating around? That has to be bad, right?

The area of the world’s oceans is ~3.6 x 108 km2. Twice the size of Texas is 2 x 696,241 = 1.39 x 106 km2.

This gives a ratio of 1.39 x 106 km2/3.6 x 108 km2 = 0.0039 = 0.39%. That’s small, but a much bigger fraction that before. Not surprisingly, our plastic junk takes up a higher fraction of the oceans’ area than it does the mass, because the junk floats on the surface.

Returning to the Olympic sized swimming pool: its area is 50 m x 25 m = 1250 m2. Since 0.39% of that should be floating junk, we get 1250 m2 x 0.39% = 4.875 m2. That’s a floating trash pile 2.2 m x 2.2 m on a side, bigger than the area of a swimmer…it’s about the area of two pool floats. That’s big enough to notice. That’s big enough to worry about.

To summarize: the great Pacific garbage patch is like two pool floats in an Olympic sized swimming pool.  But these are thin floats: remember, they can’t weigh more than a housefly!


This analysis is optimistic and pessimistic in equal measures. On the one hand, in terms of mass, the amount of plastic we’ve chucked into the ocean is a pittance, in part because the oceans are so deep. On the other hand, since plastic floats, it takes up a disproportionate amount of the oceans’ surface area (a place where most marine life lives anyway). Even if it’s less than 1% of the area of the whole pool, a couple of pool floats is certainly a distraction if Olympic swimmers are trying to have a race. My suggestion is that we should strive to…

Oh, who am I kidding. Look at this graph (again, from that Mother Jones article):


If China doesn’t change its ways, it doesn’t matter what we do.


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Run for your lives!

A few days ago I heard a story on NPR about wildfires in Yosemite.  It turns out that something like 360 square miles of forest have burned.  Being a math geek, I immediately took the (approximate) square root of 360 in my head:

360 ≈ 19 x 19

I did this without really even thinking about it; I did it in order to be able to visualize the size of the Yosemite blaze.  I now had a picture in my head of a square, 19 miles by 19 miles.  A burning square.  That’s how big the conflagration was.  And the mental math was important because I have no intuition at all about square units.

[Disclaimer for my readers not in the USA: I use the S.I. units (m/kg/s) in my physics research, but in American culture units like miles, inches, gallons, etc. are still endemic.  Sorry about that.]

Quick: how many square feet is a baseball diamond?  If you’re like me, absolutely nothing comes to mind.

I do know that a baseball diamond is 90 ft. x 90 ft. square.  So that’s the answer: 8100 sq. ft.  (752.5 m2)  The problem is that, somehow, psychologically, 90 ft. x 90 ft. seems much smaller than 8100 sq. ft., even though they are the same.

The county I live in, Jackson County, NC, is 494 sq. mi. (1,279 km2).  Somehow, this seems big to me.  But in order to better visualize this area, take a square root: the county is like a 22 mile x 22 mile square (36 km x 36 km).  In those terms, the county seems puny (although it is still bigger than Andorra).  The area of Jackson county is less than 1% the total area of the state of North Carolina.

What about the Yosemite fire?  360/494 = 73%.  So that fire is about three-fourths the size of the puny county that I live in.  A big fire, sure, but not apocalyptic.

The problem that all of this illustrates is one of scaling.  Most of my students know that 1 m = 100 cm.  However, very few know (initially) that 1 m2 ≠ 100 cm2.  Instead, 1 m2 = 10,000 cm2.  That’s because a square meter is a 100 cm x 100 cm square.

This fact leads people’s intuitions wildly astray.  Suppose I double the length and width of an American football field.  The area goes up by a factor of 4.  What was approximately 1 acre has become 4 acres.  Suppose I switch from a 10-inch pizza, which feeds 2, to a 20-inch pizza.  That pizza feeds 8.

It gets even stranger if you imagine the switch from length to volume.  Michelangelo’s David is almost 17 ft. tall.  Assume David was 5’8’’ (68 inches).  Then the statue represents a scaling factor of x3 in terms of length (3 x 68 = 204 in. = 17 ft.)  Imagine a real-life David, 17 ft. tall.  How much would he weigh?  If the life-size David is 160 pounds, the 17 ft. David would be 160 x 33 = 160 x 27 = 4,320 pounds.  To most people, this seems very strange.


He weighs 4320 pounds. If he weren’t made of stone, that is.

But back to my original idea: I had mentioned that I had no intuition about square units.  I don’t think many people do.  What intuition I do have is based on experience, and comparing unknowns to knowns.  500 sq. miles is about the size of the county I live in.  An acre is about a football field.  1000 sq. ft. is about the area of a small house.  500 sq. in. is about the area of a modest flat screen TV.  100 fm2 (a barn) is about the cross-sectional area of a Uranium nucleus.  A hectare is about 2.5 football fields stuck together.  And so on.  I’m sure you have your own internal mnemonics to help you gauge area, or volume.

If not, just remember: you can also do the square root in your head.  So if that guy on NPR says there’s a fire that’s 100,000 sq. miles in area, you can visualize

100,000 ≈ 316 x 316

and since this is very similar to the size of Colorado (380 miles x 280 miles) you can start kissing your love ones and planning for the apocalypse.

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