Inspiring Landscapes

My research group in Cambridge looks for needles in haystacks every day and here I illustrate, jargon free, some of the concepts that we use to help us.

A mountain’s summit is an unremarkable pinnacle that is utterly indistinguishable from the local apex of the countless trillions of dust grains, stones and craggy outcrops that form the rest of the mountain. Despite the banality of the highest point itself, humans frequently yearn to visit.  Why do we do that? For many, the journey itself is far more rewarding than the actual moment of standing atop the mountain’s maximum maxima.

A typical page of text might contain around 4,000 characters and a book of 100 such pages would present approximately 400,000 precisely arranged letters to the world. How many such arrangements of letters are there? How many possible 100 page books could there be? How many of those books would make any sense? How many slight rearrangements deliver the same message? How many rearrangements must occur to generate a completely different book?  Out of the vastness of possible books it is clear that we seek a very limited few. The needles in the haystack.

Often the author’s particular choice of words is more interesting to the reader than the message itself. The greatness, or mediocrity, of a work may be attributed to the manner of the text as well as the matter. The author’s mind is able to select a single arrangement of words, akin to the mountain summit, that is perceived to have a greater value than the myriad of other possibilities. The oceans of abandoned permutations are overlooked by the reader with the same cold indifference assigned to trillions upon trillions of dusty summits passed upon each step during the ascent of any rocky prominence.  Perhaps, like climbing a mountain,  the journey of reading or writing a book is significantly more important than finishing the last page.

Hopefully, the committed author will be able to guide the reader gently past the infinities of nonsense,  just as the gradient of the mountain guides the mountaineer on a path through countless peaks. Continuous adherence to an uphill gradient assures successful ascension, provided that the intrepid explorer is prepared to surrender, or avoid, local high points that block the route to the saddle point below the highest peak. Indeed, careful route planning can avoid such energy waste and a good author will not lead you on unnecessary diversions.

The following irony might be an example of such a diversion. Writing down the total number of combinations of 400,000 characters requires more than 400,000 characters! Perhaps a more entertaining and exotic representation of this gargantuan number could be realised if, for fun, we chose to assign a unique colour to each arrangement of letters and proceeded to paint each individual peak on the mountain a different colour, thus assigning a unique book to each mountain apex.

If such a Herculean task could be undertaken, the resulting multicoloured mountain would be an index for every physically possible 100 page book, most of which are complete nonsense, and many of which are very similar to each other. A small percentage of peaks would represent books worthy of perusal and, if permutations were organised in related groups, each outcropping would correspond to a particular genre.

But how many peaks are there on a mountain? How many permutations of letters? How many colours can there be? Would we run out of peaks, permutations or pigments?

Each grain of dust consists of crystals of atoms arranged in a lattice, and each atom on the dust’s surface has a separate apex.  Such a realisation rapidly clarifies the magnitude of the gargantuan number of peaks on a mountain.  Cubic grains of dust 1 mm long have about 10¹² atoms, and therefore apexes, on their uppermost surface. A conical pile of such dust, with a height of 1 km and a circular base 5 km across, has about 10¹³ grains of dust on its uppermost surface, yielding a total of 10²⁵ local pinnacles on such a mountain. This number would increase if we converted our improbable mountain into a more realistic fractal structure consisting of a hierarchy of medium size cones on larger cones, and lesser cones on little ones, and so ad infinitum.  The more accurate fractal description of our rainbow mountain would have a much greater number of peaks than the simple cone, yet the new peak count would still be nowhere near enough to alter the pecking order of peaks, permutations and pigments.

Although 10²⁵ is a truly massive number, it is minuscule compared to the number of possible 100 page books which is 27^400,000 texts drawn from a pool of 26 standard roman letters and spaces. In turn, our vast library (of complete gibberish) is dwarfed by the infinite number of individual colours in the continuous colour spectrum. Unfortunately, the colour of the paint with which to daub our mountain is determined by a finite number of atomic rearrangements of pigment molecules, which savagely thwarts our somewhat ridiculous ridge redecoration proposal.

Although we couldn’t hope to reach the infinity of available colours, there are still vast numbers of permissible molecular rearrangements that we could use for our pigments. A crude estimate suggests that 270,000 atoms drawn from a pool of 118 possible elements and arranged in a 3D crystal-like lattice, would have the same number of arrangements as there are 100 page books. A cluster of 270,000 atoms would be about 10 nanometres in diameter depending on the type of atoms drawn and the precise lattice arrangement.

Well that’s just mind-blowing.  A 10 nm particle,  able to exchange atoms with a reservoir of all the elements, has the same number of internal rearrangements as there are 100 page books!

Assuming that each atomic rearrangement fulfilled the objective of being a subtly different colour, Hercules would have to assign a different nanoparticle to each atomic pinnacle on the mountain. Aside from the obvious absurdity that the particles wouldn’t fit, the mass of the mountain would multiply considerably.  If, after exhausting the atomic pinnacles, we continued arbitrarily dumping 270,000 atoms per book on top of the mountain then the eventual pile of atoms would consume all the mass in the known universe before we had even reached 10^-57469% of the number of available books.

The ability of the mind to hunt through this bewildering number of textual arrangements to uncover worthwhile comment is genuinely staggering. To put that in perspective, all seven billion humans working flat out for a million years could not hope to dent the possible works of poetry, screenplays, books, essays, articles or science that could potentially exist. And that is just the art relating to text. What about the combinations of sounds, colours or materials in music, sculpture or manufacturing?

Could the work of my research group help to find such needles in haystacks?  Our self-imposed task is to explore the arrangements of atoms in clusters. We hope to use this to chart the properties of those clusters and discover a new understanding of the richness of the world around us. It is not the chemistry of each particular bond per se but the range of permutations of those bonds that endows our universe with the richness that stimulates our minds. The information that is stored in the material. The patterns of permutations that give rise to biology, manufactured products or art works.  Can we develop the tools to help find such creations?

Imagine a carbon next to a hydrogen. We are free to assign such a pairing a numerical score. On the other hand a carbon next to a nitrogen would be scored differently. Now imagine one arrangement of a group of five such atoms. For each arrangement we can compute a total score – if we define enough scoring rules.  After exploring lots of arrangements we can plot a graph of the scores, which would look just like a mountain, with some scores higher than others. The task of my group is to hunt for the lowest possible scores rather than the highest – we climb down the mountain hunting for the valleys, rather the ascend the mountains looking for peaks, but the same logic applies. In fact, by computing how the score changes as we reorganise the atoms we can follow the gradient of the mountain down to the lowest local valley. Then rather than allowing ourselves to get  stuck in one valley, forcing us to retrace our steps uphill again, we are able design moves around the mountainside that wouldn’t happen in nature, i.e. we can hop to completely different parts of the mountain to discover new valleys and basins.

We call the score  “potential energy” and the rules which convert a given arrangement into a score are known as quantum physics. The distribution of valleys and peaks in the landscape uniquely encodes the physical states of the system that we are exploring. Understand the landscape  and you understand the system.  There is always one way of arranging the atoms which has a lower score than all the others and that arrangement is the cluster of atoms that is most likely to be observed naturally.  The needle in the haystack.

I love the fact that 9, 500 characters can articulate an infinity of possibilities for both art and science. A work, like the research of my group, that will never be completed, and therefore could be sustained indefinitely. The manner of the journey is as important as the destination.

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