f you visit the natural history museum in Leiden - which lies a short walk from the central station - you will find a large display area, sloping up in front of you, with the mounts of many different animals scampering or slithering over its inclined wooden boards. As I recall it, the display is on locomotion, and makes a wonderful, if slightly vertiginous, impression. However, tucked behind this display is something a little different; a thing of almost frightening beauty; for in a glass case, there is the skeleton of a lion. After staring at it for some time, I came to the conclusion that what gives it its power is how spare it is: the absolute economy of its design.

The pursuit of efficiency is one of the main goals of technology, and an instrument of competition in society and nature, so I guess it is not a surprise that we also find it beautiful. Looking back for example at architecture over the last few thousand years, you see a series of advances, such as the invention of the arch, the flying buttress, wooden trusses for roofing, and steel framed buildings, which have each allowed us to build lighter and larger structures more cheaply. All of these issues and more are discussed in greater depth and much more readably in a wonderful book, "Structures" by J. E. Gordon which I highly recommend.

In modern engineering, the obsession with reducing the weight of components or removing unneccessary material has in no way diminished, and a number of computational techniques have been developed to optimise components and devices in this way. Without wishing to be flippant, all of these techniques are really elaborations on the venerable idea of starting with a big bulky component, and drilling holes in it to make it lighter. Of course, the actual techniques are much more sophisticated these days: typically a component is simulated in a computer under the stresses it will experience under use, and then the computer gradually removes or erodes those parts which are not bearing much of the load. The result is to creep closer and closer to the the point where the component breaks, but not actually get there; a little like the way an archers' bow is described as being a stick that is ninety percent broken.

his is of course wonderful, and has helped to make the modern world. However, what got me started in this area, was a nagging doubt that maybe, just possibily, we were missing something important. When I was at university, I briefly studied paleontology, and encountered among other marvels of time, the order of cephalopod moluscs called the Ammonitida (or "ammonites" to their friends).

These creatures didn't make it through the great die-off at the end of the Cretaceous period, 65 million years ago, but we can gain a very approximate idea of their anatomy from the Nautilus, which is a distant relative. In fact, animals very similar to the Nautilus have been swimming around in the oceans since before ammonites evolved. The Nautilus looks a little like a squid in a snail shell, but the shell itself is something quite special, because if you cut in half, you see that it is made of a series of gas-filled chambers, separated by curved walls, called "septa". The lines where these walls meet the outer shell are called "suture lines", and are the things I want to mention especially.

They say the fox has many tricks and the hedgehog only one - but his is the best of all. So what special trick has the humble Nautilus acquired which has seen it safely through the aeons while fish were evolving and dinosaurs appearing and vanishing (to name but a few of the billion species that have lived over this time)?

The trick seems to be that hollow shell, which the animal can partially fill with water, and pump dry as it pleases. That lets it float at any depth in the water column, like a submarine, with virtually no effort required. They exploit this advantage by moving up and down (relatively) enormous distances throughout the day: spending the hours of light at a safe depth of hundreds of metres, and coming up to the surface waters to feed during the night. Contrast that with the bony fishes, who keep their bouyancy with a swim-bladder. Because a swim bladder (and the gas inside) is squishy, a fish is always in an unstable equilibrium, like a pencil balanced on its tip: if the fish sinks slightly, then water pressure will compress its swim bladder, making it tend to sink further. The fish then has an uphill struggle (quite literally) to regain its preferred swimming depth. Even more dangerously, if the fish swims upwards too fast, before it can adjust the gas exchange in its swim bladder, then the gas will expand, making the fish even more bouyant. It then has to struggle to swim downwards, and if it loses this battle, it will continue to rise until the swim bladder bursts; an injury from which it cannot recover. I am told that if you visit the upwelling zones where ocean currents diverge, such as along the coast of Chile, then patches of the surface are covered with dying fish crippled in exactly this way, being picked off by gulls (nothing is ever wasted).

Thus a Nautilus can escape predation from fish hunting by sight during the day in the sunlit shallows, and still take advantage of the rich surface waters when feeding at night. Curiously enough, the pressure inside the coiled shell is actually very low: it appears to be less than atmospheric pressure even when the animal is at a depth of a couple of hundred metres. I don't know why this should be - but perhaps, just like submariners in a "real" submarine, the animal finds oxygen at high pressure to be toxic. In any case, the result is that just as with the artificial contraption, the shell is vulnerable to being crushed by the water pressure, and so its hull must be sufficiently strong to stand up to the load.

The Nautlius is fascinating enough, but my atention was caught by ammonites, because of their very curious shells. If you look at the suture line of the Nautilus, it is a simple, smooth line around the cross section of the shell. However, if you look at suture lines for ammonites, they have a tendency over time to become very complex, almost like the leaf of a fern.

These kinds of shapes are called fractals, and have the property that if you take a piece of the shape and magnify it, it looks just like the whole thing. Obviously, there are some wrinkles in this definition: in the real world (as opposed to the ideal, Platonic world of mathematics), you can only do this a few times before you start to see cells or atoms or some such, and the magnified bit starts looking completely different to the whole fractal. Nevertheless, the idea of a fractal is a useful one: like many theories in science, it is a an icon or a fairy tale which we use to interpret the real world. I don't mean that in a disparaging way or to suggest that all scientific theories are wrong (although of course they might be!), but as a precise analogy: the real world is complex, but we believe that underlying it are simple laws or regularities. You might only see them exactly in some especially elemental setting (just like the bones of human nature might show up most clearly in a woodcutter's cottage in an imaginary forest) - but neverthless we believe the same principle is operating out there in the wild; either operating exactly as described (but obscured under other phenomena and the muck and grime of everyday life), or just operating to a more or less good approximation. Think of Newton's law of gravity: apples fall roughly with a constant acceleration, but not exactly. The thought was that the underlying law (which also takes into account the distance from the centre of the earth) was exact, but that other things (such as air resistance) also operate, to make it's simple predictions only approximately true. In a more ideal setting, such as the orbits of the planets, you can see the law operating unobscured by other effects, but neverthless, it was still operating exactly down here. In fact, according to our current understanding, even Newton's law is not exact: it's just a very very good approximation, which works whenever speeds (including escape velocites) are small compared to that of light. So gravity kind of illustrates both sides of what I mean by an iconic theory.

o, what is going on with these strange fractal suture lines in the ammonites? Why does it go to all that trouble to make such a complex pattern? Well, I obviously wasn't the first person to ask that question, but just like other people seeing this adaptation, I wondered whether the fractal design might lead to greater strength. If you dig around in the literature, you will find a couple of papers asking the same question - and (more to the point) looking for evidence. For example, if complex suture lines mean that the animal had a stronger shell, that would probably imply that they would live at greater depth. How do you tell what depth an animal lived at, when its whole lineage died out 65 million years ago? That's a good question, and you can't (or no one has yet come up with a way to find out). However, you can find out what depth of water the animal died in!

Different types of sediments (sand, silt, mud etc.) form in different depths of water (we know that because it still happens today), and you can then go and look to see what kinds of rocks different types of ammonites ended up in, when they died and fell to the bottom of the sea. And if you do this, then what do you find...? No correlation!: Ammonites of all kinds (suture line wise) seem to live in all kinds of water depth. Of course, you don't know where in the water column they lived: just the depth of the whole ocean at that point. Still, it was a disappointment.

A few engineers also tried to do some finite element calculations to see if the complex shells would be stronger than the simple ones - but again, without success. The problem is that the geometries are so complex that it's a challenge to simulate one shell, and then the structure may not be well preserved, and anyway, what do you compare the shell against? What is a good control?

So, there the problem lies: an interesting thought (that fractal design principles might lead to stronger structures), but no real evidence. The same thought also came up in another corner of the literature, when people started looking in detail at the structure of spongy (trabecular) bone (like in the ball of your hip joint). Again, there was some speculation that the fractal structure people observed may lead to greater strength, but no convincing evidence.

ell, I've been rambling on long enough. What the heck have I been up to? Theoretical research is a process of refinement in a crucible: you try to burn off in your mind all the irrelevancies and extraneous factors, and construct a problem which is simple enough to solve, and yet captures the essence of what is going on. Therefore, I struggled (for more years than I care to admit to) to find a simple problem for which I could show that fractals really did lead to greater strength. In fact, fractals don't lead to greater strength - so in that sense I failed completely; but they can lead to something slightly different: greater efficiency!

I'm not the first person to show this, but I think (and I'm very willing to be shown to be wrong here, because I would learn a lot of fascinating things); I think that I have sharpened the concept a good deal, and taken some steps on a path which is almost unexplored.

The first problem I came up with, was to look at pressure-bearing windows. What do I mean by that? Suppose you have a great thick, rigid wall between two volumes at different pressure. For example, suppose you had a thick steel air tank for scuba diving, and the pressure on the inside is very high, but that on the outside is just atmospheric pressure. Great - but now suppose you want to put a window in the side of the tank, so you could see through (or maybe for some other reason - I'm not fussy). But - and this is the first idea which opens up the problem: suppose that the material you want to make the window out of is very brittle. The key word here is "very" - because that means there is going to be a very small number in the problem (in this case, the amount you can stretch the material before it breaks) - and a small number can sometimes make a problem simple - or at least allow you to get your theoretician's fingernail under the surface, and start easing the problem open.

So, what do you find if you analyse this problem? Well, with a bit of effort, you find that a simple window can break either by bending (which stretches one face and compresses the other) or by stretching - and which of these two things happens depends on what the pressure is relative to the elasticity of the material, and how brittle the material is. In other words, there is a kind of map with two countries, and where you are on the map tells you how the system will break when you push it too far.

Now, the next step is to see that the window doesn't have to be of uniform thickness. There are various ways you can change this: for example, it could be a bit thicker near the edges, since that's where it bends more, or some other simple optimization. However it turns out that these choices don't change things very much. What does change things, is building ridges, or spars across the window, out of the same material as the window itself.

The final step is to show that if you put these spars cross-wise, then one large spar can be used to provide partial support for a set of thinner spars, which can in turn provide support for another set of smaller spars, in a self-similar (fractal) manner.
      
This turns out to use even less material than a set of simple parallel spars (for certain regions on the map we mentioned above), and the number of hierarchical levels to give the best use of material can also be plotted on that same map. The gory details are in the paper below, but the craziest thing is that the fractal structures are only just more efficient than a simple set of parallel spars. If you read the paper, you can see the exact point where the problem nearly slips through your fingers, turns round and bites you (it's the difference between 12.699 and 12.652). But in fact, it doesn't quite escape: you still have it by the tip of its tail, and with a bit more patience, you can reel it in and show that fractal designs really are more efficient (use less material) than simple non-fractal designs.

ell, that was a bit of a fight, and a good five year's worth of dead ends, but as is the way with theoretical work, once you have cut your teeth on one problem, you tend to see new avenues opening up, since you have equipped yourself with new mental tools, and you may see that the original question can actually be made even simpler. Eventually, if you're lucky, you get to the stage where the answer seems completely obvious, and you just can't imagine how you thought it wasn't obvious in the first place.

That happened to me in this case too, and the next step was to take on an interesting puzzle that is also raised in Gordon's book: if you look at things around you, you will see that compression structures tend to be very much less efficient (i.e. use more material) than tension structures. To take an extreme example: suppose you want to suspend a one tonne block from a fixed point (like a crane), using 200 metres of steel wire. How much steel will you need to do this? Well, for high tensile steel, you will need about 8kg, plus a bit for hooks or fastenings to attach it to the block and to the crane - say another kilo. That's not very much - but then steel is very strong, and we're playing this right up to edge (no margin for safety). OK, great! Now imagine that we turn the whole problem upside down, and we try to balance a one tonne block on top of a 200m column of steel. How much steel do you need to make that column? Well, it's pretty clear that it's going to be more than 9kg!
                   
In fact, for a solid column of steel, it's not the steel breaking which puts a limit on how little steel you need, it's the buckling of the column. This was a phenomenon which Leonard Euler first studied in 1745, and is a curious thing in its own right. If you have a straight column (like a plastic ruler for example), and you squeeze the two ends towards each other, then it stays straight as you apply more and more force - until suddenly, it doesn't. At that point, it buckles, and (for a straight ruler) it's a very sudden transition between straight and curved.

It is this Euler buckling which causes the steel column we are thinking about to fail - and to make a solid column which is stiff enough to stand up to 1 tonne of weight at a height of 200m, you need 81 tonnes of steel!

So, that illustrates pretty graphically why structures under compression tend to be a lot less efficient than structures under tension, but this turns out to be a recurring theme in both engineering and biology. The other aspect of this is that if you divide compression members, they become even less efficient: if we wanted to support half a tonne on a column of the same height (200m), we would need more than half the amount of material. On the other hand, there is no cost for dividing up tension members: to support two 1-tonne blocks using steel cables, you just need two of the original cables. In fact, it's even better than that because if you put the two cables next to each other, you can re-use some of the couplings at the end. Therefore efficient engineering striuctures tend to look like tents: you have one (or as few as possible) compression members (the pole in the middle), surrounded by a finely divided web of tension members (the material of the tent and the guy ropes). In fact, large animals are designed on a similar principle: we have a few legs (two for us!), sheathed in a web of tendons and musculature which is largely under tension.
             

o, that's all very well to know (and it sheds an interesting light upon all kinds of structures when you stare at them hard), but can we do anything about it - or is it just in the nature of things? More to the point, can we imagine playing some clever tricks, and managing to make compression structures just as light-weight as tension structures? That sounds like an impossible challenge, but it also seems rather strange that just changing the direction you apply the force should have such crazily large effects. Clearly, there is something troubling about the real world's behaviour which, from a moral point of view, it ought not to be doing?

Nevertheless, let us not worry overly about trying something which seems impossible, and just see how far we can get. In general, this is not good advice for life, or for an academic career, but if you you don't have much of either, then it can make an interesting hobby.

So, let's think hard and see if there is a chink in the armor of this problem which might allow us in. Well, firstly, there is the fact that I have said all along that if you have a solid column, then you need this and that amount of steel. Can we do better perhaps, if we have a hollow column? Indeed we can! If you make the column into a hollow cylinder (but still use the same amount of steel), then you move material futher and futher away from the centre, and the column becomes stiffer and stiffer to bending. Therefore it is less and less prone to Euler buckling - and so you can support the same weight with less steel!
       

Therefore, we can keep on making the column wider (and the walls of the cylinder thinner), and so manage to support more and more weight. Right? What could possibly go wrong? Well, a clue comes from origami: Imagine you make a cylinder out of paper, and stand it up on your desk. Now take another sheet of just the same size, and fold it with a diamond-shaped pattern. If you do it right, you can see that the folded paper column is actually shorter, even though it's from the same shape piece of paper.
        

This is the weak point! It shows that a thin walled cylinder can collapse (get shorter) without bending: it just crumples up; or rather the wall bends and buckles locally, rather than the whole column bending in a single smooth motion. Nevertheless, this actually gets us a good step in the right direction! You can choose the width of the column, so that it is just on the point of Euler buckling, and also just at the point of crumpling - and that is the best you can do. It allows you to make compression members which are a good deal more efficient than solid columns - but alas, it's still a long way from the efficiency of a cable under tension. For our problem with the one tonne weight, we now need about 940kg of steel for the best hollow column.

air enough: we have made progress, so that's a good day's work. However, to get any further, we are going to have to think a good deal harder, and head off in a different direction. So, let's do that! Remember that what caused the Euler buckling was that the column was too bendy. But then what caught us out with the paper cylinder was that the paper itself (the wall of the cylinder) was too bendy, so that it crumpled. Well...that suggests another tack, I think...

Imagine that you have some hollow steel pipes, and lay them down next to each other, just touching. Then imagine that you weld them together where they touch, so that you have a kind of plate made out of pipes: it looks from above a little bit like a sheet of corrugated iron. Now we do something very strange: we take three of these sheets, and we imagine them passing through each other like ghosts. We arrange them at an angle of 60 degrees relative to each other, but all in the same plane; and then we imagine they are real steel again, and we weld them together wherever they passed through each other. OK, that's a description which no engineer is going to accept: if you really made this composite sheet, it would be a piece of work - but conceptally, that's what I want to build.
        

So, what we have now is a sheet of material, which actually has six-fold rotational symmetry: if you place it on the table and slide it round (without lifting it off the table), turning it 60 degrees, 120 degrees, 180 degrees etc., then it looks just the same. That is important, because you can then show that if you either stretch or bend it (just a little bit), then it behaves like a simple solid sheet: there is no direction you can bend it in which it is more flexible than the rest. Compare that with a sheet of corrugated iron for example, which is quite easy to bend in one direction, but is stiff if you try to bend it at right angles to that. For our sheet, it's the same no matter which way you bend it.

However, we have also achieved something quite useful: we have moved some material away from the central plane - a little like the way we moved material away from the axis in going from a solid column to a hollow cylinder. So, our composite plate is a lot more stiff than a flat solid sheet of steel of the same mass ... and that's exactly what we want, in order to protect against local buckling or crumpling!

Therefore, what we do is we take our hollow cylindrical column, and we replace its thin wall with one of these composite plates.
                                  

Now, of course, the thin walls of the composite plates might crumple, but what you can do, is choose the structure so that it is just on the point of Euler buckling, just on the point of the main cylinder wall crumpling, and just on the point of the individual thin sheets crunpling. If you do that, then in our original problem (1 tonne weight on a 200m column), you only need 300kg of steel. Thus, we've come a long way from the original 81 tonnes.

The final idea, is that there is nothing to stop us doing this all over again: replacing all the thin steel sheets in the new doubly composite column, with a new set of six-fold symmetric composite plates. What you end up with is something with a fractal struture, for it is a composite shell made out of composite shells, made out of composite shells... for as many layers of structure as you care to put in. We have used a fractal design principle to make something very lightweight and efficient!

So, can we make a column which uses only the 8kg of steel that the steel cable under tension needs? No! - not quite. The trouble is that at some point, as you add more and more layers of complexity to the structure, the amount of steel that you need stops going down, and starts going up again. There is therefore an optimum number of hierarchical levels to use. For the column above, it turns out to be three; however it will be different for different cases. In what way? Well, the key thing is to think about that original statement that compression structures are much less efficient that tension structures. This is true, but how much less efficient varies from case to case. In particular, if the structures are big and the loading is light, then compression structures are terribly inefficient compared to the equivalent structure under tension. It then turns out that this is exactly the limit (big structures, gentle loading), where it makes sense to have a lot of fractal layers. On the other hand, for small structures under heavy loading, it might be useful to have only one layer of fractal structure - or even none at all.

So, where does this leave the ammonite story? Well, we are a little closer to an answer, and we have a curious new perspective: if the ammonite uses fractals to withstand pressure, then it probabaly does so to reduce the amount of material in its shell, and under conditions of gentle pressure loading. Hence, we should probably be looking for ammonites with complex sutures in the surface waters of mineral-poor seas.

hat's as far as I got on my own. I was then lucky enough to get in touch with some very talented physicists at Nottingham university (Yong Mao and Daniel Rayneau-Kirkhope). We have since then been looking a little bit further into the ammonite question; or rather, asking what is the most light-weight design you could imagine for a submarine? You can use a similar approach to the fractal columns above, but now you want to protect against different types of failure - especially crush pressure as you submerge the vessel more and more deeply. The result is a bit more complex, and you also need to be a bit cleverer when you wrap cylinders around a curved object to make a composite sheet, but in the end you can fight your way through to some simple answers.
     

uriously enough, we also came up with a new problem, which is even simpler than the fractal columns problem, and now I think we are starting to get to the point where you could begin to ask "isn't this actually quite obvious after all?". I don't think that's a bad thing: usually in science you have to work very hard to make something simple, and that has been my experience here to a tee.

The problem we have been looking at is fractal trusses, where instead of replacing a solid column under compression with a hollow cylinder (and then all manner of ghostly craziness) we have been replacing the column with a space frame - and then replacing each beam of that space frame with a new space frame. Interestingly enough, you get almost the same behaviour as with the columns described above - although the result is not quite as efficient.
     

You can also imagine using these trusses to create a whole three-dimensional lattice. That would then make a composite material which is extremely light-weight, but very resistant to being crushed. You could then use that material to make any shape you wanted. It's a bit hard to imagine what that would look like, but if you think just about one unit cell of the lattice, you can get some impression.
     

o my mind however, the most intriguing thing is that it is actually possible to make some of these structures for real! Techniques of additive manufacture have advanced to the point where the Engineering department at Nottingham (specifically Dr Joel Segal), together with Dan, have been able to make structures from photosensitive resin, and in principle, we could even think of making them from steel.
     

There are many questions still to answer, and I think we are only at the start of this story, but I'll keep you posted. I wouldn't say that you will be seeing submarines with spiral hulls and fractal suture lines in the seas any time soon (it would be great, but I'm not sure where you would fit the torpedoes, let alone the crew), but I have my fingers crossed that we may find some application; even if it is only to point the usual engineers' optimization programs in a slightly unexpected direction.

ACKNOWLEDGEMENTS AND ATTRIBUTION

Daniel Rayneau-Kirkhope is currently (as of 2010) working in Nottingham University on his PhD, under the supervision of Dr Yong Mao (also of Nottingham) and myself. He is supported by an EPSRC grant.

The images of the nautilus, nautilus shell and fern have been released under the GNU free documentation licence, and original copies can be found here (Lee R, Berger) here (user chris73) and here (user Olegivvit) on Wikipedia.

That part of the work described here which is mine, has been done in my spare time, as a hobby, while working with Unilever Research. It pre-dates my connection with the London Institute for Mathematical Sciences. However, I hope to continue it, perhaps in active collaboration with fellows there.

Unilever has been kind enough to allow me the use of some computing facilities, and my recent line managers have provided me with, if not active encouragement for this sort of madness, then at least a refreshing dose of benign tolerance.

In the very early stages of this work, my boss was J.J.M. Janssen, and although I did not discuss my thoughts in this area with him, I want to repeat the acknowledgement I made in the first Physical Review E paper listed below, namely that he kindly invited me on secondment to the Netherlands, and provided a wonderfully supportive environment in which to work, which allowed many ideas, including those presented here, to flourish.

REFERENCES

[1] Gordon J. E. "Structures", (Penguin, New York, 1986).

[2] Farr R.S., "Fractal design for efficient brittle plates under gentle pressure loading". Phys. Rev. E 76(4) 046601 (2007). (arXiv:0912.3383v1 [physics.class-ph]).

[3] Farr R.S., "Fractal design for an efficient shell strut under gentle compressive loading". Phys. Rev. E 76(5) 056608 (2007). (arXiv:1001.3532v1 [physics.class-ph]).

[4] Farr R.S. and Mao Y. "Fractal space frames and metamaterials for high mechanical efficiency". Europhys. Lett. 84(1) 14001 (2008). (arXiv:1001.3940v2 [cond-mat.mtrl-sci]).