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A grain of sand in St Paul’s Cathedral

A grain of sand in St Paul’s Cathedral

– an essay

Essay

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.

From one angle, 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 isolated 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, and each of the 92 natural elements has its own kind of atom. The ratio is at its highest – 128,000 – in the hydrogen atom, which has the smallest nucleus; and the ratio is at its lowest – 37,000 – in the iridium atom, which has a large nucleus (see Appendix Table 2). We have already computed the acceptable range for the sand‐in‐Cathedral picture: from 33,000 to 1,000,000. We therefore conclude that, whatever kind of atom we choose, the ratio is within that acceptable range. Whatever kind of atom we choose, the nucleus in the atom is indeed like a grain of sand in St Paul’s Cathedral.

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 a sea of quarks, antiquarks and gluons: but the quark, the antiquark and the gluon, as far as we know, are elementary particles, 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 200 billion stars making up the galaxy we know as the Milky Way. (Again I confess a tautology: galaxy derives from the Greek word γαλαξίας meaning “milky”.) 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.

From the smallest to the largest radius in 8 approximately equal steps 1027 m 1024 m 1021 m 1018 m 1015 m 1012 m 109 m 106 m 103 m 100 m 10−3 m 10−6 m 10−9 m 10−12 m 10−15 m 10−18 m 4.4 × 1026 mobservable universe,to the particle horizon 1.9 × 1021 mMilky Way 1.9 × 1016 mOuter Solar Solar System 4.9 × 1011 mInner Solar System 6.4 × 106 mEarth 3.3 × 101 mSt Paul’s Cathedral 8.0 × 10−5 msmallish grain of sandThe grey rectangle shows the range of radii ofrock particles that qualify as grains of sand. 1.1 × 10−10 mhydrogen atom 8.4 × 10−16 mhydrogen nucleus (proton)
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
Updated with more recently published data 19 November 2019


Appendix: data sources

Table 1 below links to the general data sources for this essay.

Table 1: General data sources
Radius of grain of sand is anything from 132 mm to 1 mm Wikipedia
Interior volume of St Paul’s Cathedral is 152,000 m3 Brüel & Kjær
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 4.5 to 6.0
metres
Astronomical Unit AU 1.5E+11 Wikipedia
Light Year LY 9.5E+15 Wikipedia
Radius Log10 ∆Log10
Hydrogen nucleus (proton) 8.4E−16 −15.1 Wikipedia
Hydrogen atom 1.1E−10 −10.0 5.1 Wikipedia
Smallish grain of sand 8.0E−05 −4.1 5.9
St Paul’s Cathedral 3.3E+01 1.5 5.6
Earth 6.4E+06 6.8 5.3 Wikipedia
Inner Solar System (to main Asteroid belt) 3.3 AU 4.9E+11 11.7 4.9 Wikipedia
Outer Solar System (including Oort cloud) say 2 LY 1.9E+16 16.3 4.6 Wikipedia
Our Galaxy (Milky Way including halo) 200,000 LY 1.9E+21 21.3 5.0 Wikipedia
Observable universe 4.65E+10 LY 4.4E+26 26.6 5.3 Wikipedia

Table 2 below shows the nuclear radius (in femtometres) and the atomic radius (in picometres) of the 92 natural elements, in support of my assertion that the ratio between them ranges from 37,000 (iridium) to 128,000 (hydrogen). The nuclear radius of all elements except hydrogen is taken from Table of experimental nuclear ground state charge radii: An update (Angeli & Marinova, 2012). For the nuclear radius of hydrogen, I have taken the more recent (2018) figure given in Wikipedia: Proton. The most abundant isotope in the Earth’s crust is taken in every case. The atomic radius is taken from Average van der Waals Radii of Atoms in Crystals (Hu Sheng‐Zhi and others, 2003), Table 1. The main text of that paper is in Chinese, but the table is in English.

Different authors give quite widely differing values for van der Waals radii. I have chosen the Hu paper because it compares several different sources, it gives figures for all of the 92 natural elements, and its figures are not tainted by Bondi’s early (1964) figures which, though now recognised to be generally too low, are still widely quoted.

Table 2: Nuclear radius and atomic radius by element
Atomic
number
Mass
number
Symbol
Name Nuclear
radius
(fm)
Atomic
radius
(pm)
Ratio
1 1 H Hydrogen 0.8414 108 128,000
2 4 He Helium 1.6755 134 80,000
3 7 Li Lithium 2.4440 175 72,000
4 9 Be Beryllium 2.5190 205 81,000
5 11 B Boron 2.4060 147 61,000
6 12 C Carbon 2.4702 149 60,000
7 14 N Nitrogen 2.5582 141 55,000
8 16 O Oxygen 2.6991 140 52,000
9 19 F Fluorine 2.8976 139 48,000
10 20 Ne Neon 3.0055 168 56,000
11 23 Na Sodium 2.9936 184 61,000
12 24 Mg Magnesium 3.0570 205 67,000
13 27 Al Aluminium 3.0610 211 69,000
14 28 Si Silicon 3.1224 207 66,000
15 31 P Phosphorus 3.1889 192 60,000
16 32 S Sulfur 3.2611 182 56,000
17 35 Cl Chlorine 3.3654 183 54,000
18 40 Ar Argon 3.4274 193 56,000
19 39 K Potassium 3.4349 205 60,000
20 40 Ca Calcium 3.4776 221 64,000
21 45 Sc Scandium 3.5459 216 61,000
22 48 Ti Titanium 3.5921 187 52,000
23 51 V Vanadium 3.6002 179 50,000
24 52 Cr Chromium 3.6452 189 52,000
25 55 Mn Manganese 3.7057 197 53,000
26 56 Fe Iron 3.7377 194 52,000
27 59 Co Cobalt 3.7875 192 51,000
28 58 Ni Nickel 3.7757 184 49,000
29 63 Cu Copper 3.8823 186 48,000
30 64 Zn Zinc 3.9283 210 53,000
31 69 Ga Gallium 3.9973 208 52,000
32 74 Ge Germanium 4.0742 215 53,000
33 75 As Arsenic 4.0968 206 50,000
34 80 Se Selenium 4.1400 193 47,000
35 79 Br Bromine 4.1629 198 48,000
36 84 Kr Krypton 4.1884 212 51,000
37 85 Rb Rubidium 4.2036 216 51,000
38 88 Sr Strontium 4.2240 224 53,000
39 89 Y Yttrium 4.2430 219 52,000
40 90 Zr Zirconium 4.2694 186 44,000
41 93 Nb Niobium 4.3240 207 48,000
42 98 Mo Molybdenum 4.4091 209 47,000
43 Tc Technecium n/a 209 n/a
44 102 Ru Ruthenium 4.4809 207 46,000
45 103 Rh Rhodium 4.4945 195 43,000
46 106 Pd Palladium 4.5318 202 45,000
47 107 Ag Silver 4.5454 203 45,000
48 114 Cd Cadmium 4.6087 230 50,000
49 115 In Indium 4.6156 236 51,000
50 120 Sn Tin 4.6519 233 50,000
51 121 Sb Antimony 4.6802 225 48,000
52 130 Te Tellurium 4.7423 223 47,000
53 127 I Iodine 4.7500 223 47,000
54 132 Xe Xenon 4.7964 221 46,000
55 133 Cs Caesium 4.7859 222 46,000
56 138 Ba Barium 4.8378 251 52,000
57 139 La Lanthanum 4.8550 240 49,000
58 140 Ce Cerium 4.8771 235 48,000
59 141 Pr Praseodymium 4.8919 239 49,000
60 144 Nd Neodymium 4.9421 229 46,000
61 Pm Promethium n/a 236 n/a
62 152 Sm Samarium 5.0819 229 45,000
63 153 Eu Europium 5.1115 233 46,000
64 158 Gd Gadolinium 5.1569 237 46,000
65 159 Tb Terbium 5.0600 221 44,000
66 164 Dy Dysprosium 5.2218 229 44,000
67 165 Ho Holmium 5.2022 216 42,000
68 166 Er Erbium 5.2516 235 45,000
69 169 Tm Thulium 5.2256 227 43,000
70 174 Yb Ytterbium 5.3108 242 46,000
71 175 Lu Lutetium 5.3700 221 41,000
72 180 Hf Hafnium 5.3470 212 40,000
73 181 Ta Tantalum 5.3507 217 41,000
74 184 W Tungsten 5.3658 210 39,000
75 187 Re Rhenium 5.3698 217 40,000
76 192 Os Osmium 5.4126 216 40,000
77 193 Ir Iridium 5.4032 202 37,000
78 195 Pt Platinum 5.4270 209 39,000
79 197 Au Gold 5.4371 217 40,000
80 202 Hg Mercury 5.4648 209 38,000
81 205 Tl Thallium 5.4759 235 43,000
82 208 Pb Lead 5.5012 232 42,000
83 209 Bi Bismuth 5.5211 243 44,000
84 210 Po Polonium 5.5704 229 41,000
85 At Astatine n/a 236 n/a
86 222 Rn Radon 5.6915 243 43,000
87 223 Fr Francium 5.6951 256 46,000
88 226 Ra Radium 5.7211 243 42,000
89 Ac Actinium n/a 260 n/a
90 232 Th Thorium 5.7848 237 41,000
91 Pa Protactinium n/a 243 n/a
92 238 U Uranium 5.8571 240 41,000