Wikipedia defines *factoid* as “a questionable or spurious—unverified, incorrect, or fabricated—statement presented as a fact, but with no veracity.”

Last night I was enjoying a TV documentary series, *The Story of Science*, when I heard a startling factoid, namely:

If the “empty space” inside the atoms that make up people were removed, the entire human population would fit inside a sugar cube.

What the? Can we improve the veracity of this factoid?

**Atoms**

Let’s start with atoms. Assuming that atoms are spherical, with a central nucleus which is also spherical, we can calculate atomic and nuclear volumes. A Google search for average diameter of an atomic nucleus suggests that the “average” nuclear diameter is 1e-14 metres, ~ 10 000 times less than an “average” atomic diameter.

Some simple arithmetic in the R console. You remember “four-thirds PI r cubed” from high school, I hope:

avol <- 4/3 * pi * (1e-10 / 2)^3 nvol <- 4/3 * pi * (1e-14 / 2)^3 nvol / avol [1] 1e-12

So, the average nucleus in the average atom occupies a fraction of 0.000000000001 x the atomic volume.

**People**

What’s the volume of an “average person”? A very silly question. Wikipedia states that average human density is 1010 kg m^{-3}. Since density = mass / volume, volume = mass / density.

What’s the mass of an “average person”? A very silly question. According to Mass of an adult, it’s 70.00 kg for males and 61.14 kg for females. What about the children? 10.00, 19.50 and 32.20 kg for 1, 5 and 10 year-olds, respectively. What’s the age distribution across the global population? You know what – let’s pretend that everyone is an “average adult” and go with 65 kg. For these calculations, same order of magnitude is surely good enough.

Where were we…ah yes, average human volume:

hvol <- 65 / 1010 hvol [1] 0.06435644

So, assuming that the volume of a person is equivalent to the volume of the atoms from which they are made, 0.000000000001 x that volume is “not the empty space” and there are ~ 7 billion and counting of us:

pvol <- nvol / avol * hvol * 7e+9 pvol [1] 0.000450495

That’s cubic metres, which by my reckoning is ~ 450 cubic centimetres. Assuming a sugar cube with side = 1 cm, we are more like a bag of sugar than a sugar cube. However, allowing for errors, it does indeed seem conceivable that the entire human race is made up of – not very much at all.

Now you can all tell me what’s wrong with these assumptions and calculations. I’ll start: “solidity” is a macroscopic concept and has no meaning inside of atoms. I’ll disable comments when I get bored ;-)

ChrisHere’s one: You chose to use the average diameter of an atomic nucleus, when in fact, humans are mostly made up of the lighter elements (C,H,O,N,etc), There’s about an order of magnitude difference between the top atoms and the bottom ones, so your results may be off by as much as 10x or so. That still only gets you down to 45 cm^3, though.

nsaundersPost authorGood one!

ChrisAlso, you could use a better packing the spheres. Arranging them optimally would give you a further ~25% reduction. (http://en.wikipedia.org/wiki/Close-packing_of_spheres).

Of course, then we start worrying about how spherical these nuclei really are, but that’s farther down the rabbit hole than I’m willing to go tonight.

nsaundersPost authorYes, I only considered “space within atoms” here, not “space between atoms”.

Ian HolmesThis stuff just depresses me I’m afraid–it’s totally meaningless. Not to be a party pooper. I felt similarly about the dumb metaphor I heard at a recent conference: “if you stretched all earth’s viruses end-to-end they’d stretch to the nearest 60 galaxies” *facepalm* Yes people, polymers are long, atomic nuclei are small, and if you piled all these analogies in a big heap, they would reach all the way up to who the f**k cares ;-)

nsaundersPost authorI know what you mean :-) It’s basically saying “if people were made only of nuclei – which are tiny – you’d have tiny x people.”

I’m just happy that this factoid is somewhat true, as opposed to the usual case (completely false).

nicoSo…. if you were to remove all the empty space from a sugar cube, how small would it get?

nsaundersPost authorAnd would the people still fit inside…

ChrisIs this why I am so sweet? Is adding sugar to your coffee ethnic cleansing? And what the hell are people looking at when I go through a body scanner?

Davefrom http://en.wikipedia.org/wiki/Nuclear_density

average density of the nucleus is 4e17 kg/m3

7e9 person * 65 kg/person / 4e17 kg/m3 * 1e6 cm3/m3 = 1.14 cm3