A grain of sand in St Paul’s Cathedral

A grain of sand in St Paul’s Cathedral

What could be more insignificant than a grain of sand? What could be more grand than the architecture of St Paul’s Cathedral? And what could be more absurd than to compare the one with the other?

Today I was prompted by a friend to make the comparison. She had been reading some science. She knew that an atom consists of a nucleus surrounded by an electron cloud. She had read that the atom consists overwhelmingly of empty space: specifically, that its electrons have no size, and that its nucleus is so small, compared with the atom as a whole, as to be like a grain of sand in St Paul’s Cathedral. She asked me if that was a fair picture, and I undertook to do some sums.

In one way, the picture has to be taken not with a grain of sand but with a pinch of salt. An atom is just not the same shape as St Paul’s Cathedral. An atom is nice and spherical, while St Paul’s Cathedral is of a wiggly kind of shape. So the first sum we have to do is to make St Paul’s Cathedral canonical.

Making St Paul’s Cathedral canonical does not mean ridding it of 300 years of accumulated heresies and apocrypha, much as I might relish the challenge. In mathematics, to make something canonical is to present it in a standard and natural form which highlights those of its properties that interest us and disencumbers it of those that do not. In the present context, the only property of St Paul’s Cathedral that interests us is its interior volume. We therefore make it canonical by unwiggling it and representing it as a simple sphere of the same volume.

Now, the interior volume of St Paul’s Cathedral is 152,000 cubic metres. O the joy of the Internet! (Brüel & Kjær – RASTI Measurements in St. Paul’s Cathedral, London) And the volume of a sphere of radius r is 43πr3. No, I didn’t need the Internet for that one. Accordingly, we take St Paul’s Cathedral, we lay its glory by, and we contract it to the span of a sphere of radius 33 metres.

How big is a grain of sand? That is not as vain a question as “How long is a piece of string?” As the term sand is used by geologists, grains of sand range in diameter from 116 mm to 2 mm. (Wikipedia: Sand) We can make our grains of sand canonical by idealising them as little spheres of radius from 132 mm to 1 mm. The radius of the sphere representing St Paul’s Cathedral divided by the radius of the sphere representing a grain of sand can then range from 33,000 for the largest grains of sand to just over 1,000,000 for the smallest ones.

Next we have to see whether a grain of sand in St  Paul’s Cathedral can fairly represent the nucleus in an atom. Ah, but which atom? The ratio of the radius of an atom to the radius of its nucleus depends on the kind of atom. In the atom with the smallest nucleus, hydrogen, the ratio is 145,000. In a uranium atom, which has a large nucleus, the ratio is a mere 23,000. (Wikipedia: Atomic nucleus) We have already computed the acceptable range for the sand‐in‐Cathedral picture: 33,000 to 1,000,000. We therefore conclude that the sand‐in‐Cathedral picture is valid for a hydrogen atom, but not quite valid for a uranium atom. The nucleus in a uranium atom is not quite like a grain of sand in St Paul’s Cathedral: it is more like a small piece of gravel. But for a hydrogen atom, the picture is perfect.

The full power of that sand‐in‐Cathedral picture, however, only struck me when I thought to apply it more widely to the full range of forms in the universe as a whole. Can we start with the smallest known form and apply the sand‐in‐Cathedral picture successively until we reach the largest known form? Yes we can, and everything fits together rather neatly. There are 9 forms in the chain.

The nucleus of a hydrogen atom is a proton, which is the smallest known stable structure, the smallest known stable form. (A proton is made of 3 quarks, but the quark, as far as we know, is an elementary particle, with no parts and no size, and therefore no structure or form.) We have already seen that the proton in a hydrogen atom is like a grain of sand in St Paul’s Cathedral.

The hydrogen atom, in turn, is very small in comparison with a smallish grain of sand. How small? Well, it’s like a grain of sand in St Paul’s Cathedral.

A smallish grain of sand, in turn, is obviously very small in comparison with St Paul’s Cathedral. How small? At the cost of a tautology I will observe that it, too, is like a grain of sand in St Paul’s Cathedral.

St Paul’s Cathedral, in turn, is very small in comparison with Planet Earth. How small? It’s like a grain of sand in St Paul’s Cathedral.

We are used to thinking of our Planet Earth as huge. It has a mean radius of 6,371 kilometres. But it is very small in comparison with the Inner Solar System, which is centred on the sun, includes the four terrestrial planets Mercury, Venus, Earth and Mars, and extends out to and includes the main asteroid belt. How small? Like a grain of sand in St Paul’s Cathedral.

So how big is the Inner Solar System in comparison with the Outer Solar System, which extends out to and includes the Oort Cloud, that nursery ground for the comets? Like a grain of sand in St Paul’s Cathedral.

The Outer Solar System is our sun’s patch. It is the space within which our sun’s gravitational field is dominant. But our sun is only one of perhaps 400 billion stars making up the galaxy we know as the Milky Way. (Again I confess a tautology: galaxy means “milky way”.) And our sun’s patch – the Outer Solar System – is, in comparison with the whole Milky Way, like a grain of sand in St Paul’s Cathedral.

There now comes the final step. The radius of the Milky Way is about 200,000 light‐years. Yet the Milky Way is only one of perhaps two trillion galaxies in the observable universe. The radius of the entire observable universe, right out to the particle horizon, is about 46.5 billion light‐years. The Milky Way, in comparison with the entire observable universe, is like a grain of sand in St Paul’s Cathedral.

I have not had to resort to metaphor. On each of these occasions when I have likened a size ratio to a grain of sand in St Paul’s Cathedral, I mean quite literally that the ratio of lengths is between 33,000 and 1,000,000, where these figures correspond to the largest and smallest particles recognised by geologists as grains of sand. We have spanned the gap between the smallest and the largest known structures in the universe in 8 approximately equal steps, and each step is accurately expressed by the picture of a grain of sand in St Paul’s Cathedral.

The result is shown on a logarithmic scale in Figure 1.

Figure 1
From the smallest to the largest radius in 8 approximately equal steps

We began by contemplating the grandeur of the architecture of St Paul’s Cathedral. It stands as a monument to its architect, Sir Christopher Wren. Yet the architecture of the observable universe, from proton to particle horizon, is unimaginably more grand. And what of its Architect? It is to His glory that St Paul’s Cathedral was built. His thoughts are more in number than the sand. Formless, all lovely forms declare His loveliness. Si monumentum requiris, circumspice. If you need a reminder, look around you.

Eric P Smith
7 May 2010

Appendix: data sources

Radius of grain of sand is anything from 132 mm to 1 mm Wikipedia: Sand
Interior volume of St Paul’s Cathedral is 152,000 m3 Brüel & Kjær: RASTI Measurements
in St. Paul’s Cathedral, London
Radius of sphere with same volume as interior of St Paul’s Cathedral is 33 m
So the ratio of these radii can be anything from 33,000 to 1,056,000
Log10 of the ratio can be anything from 4.52 to 6.02
Astronomical Unit AU 1.50E+11 Wikipedia: Astronomical Unit
Light Year LY 9.46E+15 Wikipedia: Light‐year
Radius Log10 ∆Log10
Proton 8.77E−16 −15.06 Wikipedia: Proton
Hydrogen atom 1.20E−10 −9.92 5.14 Wikipedia: Hydrogen atom
Smallish grain of sand 6.30E−05 −4.20 5.72
St Paul’s Cathedral 3.30E+01 1.52 5.72
Earth 6.37E+06 6.80 5.29 Wikipedia: Earth
Inner Solar System (to main Asteroid belt) 3.3 AU 4.94E+11 11.69 4.89 Wikipedia: Solar System
Outer Solar System (including Oort cloud) say 2 LY 1.89E+16 16.28 4.58 Wikipedia: Oort cloud
Our Galaxy (Milky Way including halo) 200,000 LY 1.89E+21 21.28 5.00 Wikipedia: Milky Way
Observable universe 4.65E+10 LY 4.40E+26 26.64 5.37 Wikipedia: Observable universe