̽��ֱ�� of Cambridge - morphogenesis

How to cut your lawn for grasshoppers

sc604 — Wed, 22 Nov 2017 00:45:13 +0000

One could be forgiven for wondering what the point of such a question might be. But the solution, proposed by theoretical physicists in the UK and the US, has some intriguing connections to quantum theory, which describes the behaviour of particles at the atomic and sub-atomic scales. Systems based on the principles of quantum theory could lead to a revolution in computing, financial trading, and many other fields.

̽��ֱ��researchers, from the ̽��ֱ�� of Cambridge and the ̽��ֱ�� of Massachusetts Amherst, used computational methods inspired by the way metals are strengthened by heating and cooling to solve the problem and find the ‘optimal’ lawn shape for different grasshopper jump distances. Their results are reported in the journal Proceedings of the Royal Society A.

For the mathematically-inclined gardeners out there, the optimal lawn shape changes depending on the distance of the jump. Counter-intuitively, a circular lawn is never optimal, and instead, more complex shapes, from cogwheels to fans to stripes, are best at retaining hypothetical grasshoppers. Interestingly, the shapes bear a resemblance to shapes seen in nature, including the contours of flowers, the patterns in seashells and the stripes on some animals.

“ ̽��ֱ��grasshopper problem is a rather nice one, as it helps us try out techniques for the physics problems we really want to get to,” said paper co-author Professor Adrian Kent, of Cambridge’s Department of Applied Mathematics and Theoretical Physics. Kent’s primary area of research is quantum physics, and his co-author Dr Olga Goulko works in computational physics.

To find the best lawn, Goulko and Kent had to convert the grasshopper problem from a mathematical problem to a physics one, by mapping it to a system of atoms on a grid. They used a technique called simulated annealing, which is inspired by a process of heating and slowly cooling metal to make it less brittle. “ ̽��ֱ��process of annealing essentially forces the metal into a low-energy state, and that’s what makes it less brittle,” said Kent. “ ̽��ֱ��analogue in a theoretical model is you start in a random high-energy state and let the atoms move around until they settle into a low-energy state. We designed a model so that the lower its energy, the greater the chance the grasshopper stays on the lawn. If you get the same answer – in our case, the same shape – consistently, then you’ve probably found the lowest-energy state, which is the optimal lawn shape.”

For different jump distances, the simulated annealing process turned up a variety of shapes, from cogwheels for short jump distances, through to fan shapes for medium jumps, and stripes for longer jumps. “If you asked a pure mathematician, their first guess might be that the optimal shape for a short jump is a disc, but we’ve shown that’s never the case,” said Kent. “Instead we got some weird and wonderful shapes – our simulations gave us a complicated and rich array of structures.”

Goulko and Kent began studying the grasshopper problem to try to better understand the difference between quantum theory and classical physics. When measuring the spin – the intrinsic angular momentum – of two particles on two random axes for particular states, quantum theory predicts you will get opposite answers more often than any classical model allows, but we don’t yet know how big the gap between classical and quantum is in general. “To understand precisely what classical models do allow, and see how much stronger quantum theory is, you need to solve another version of the grasshopper problem, for lawns on a sphere,” said Kent. Having developed and tested their techniques for grasshoppers on a two-dimensional lawn, the authors plan to look at grasshoppers on a sphere in order to better understand the so-called Bell inequalities, which describe the classical-quantum gap.

̽��ֱ��lawn shapes which Goulko and Kent found also echo some shapes found in nature. ̽��ֱ��famous mathematician and code-breaker Alan Turing came up with a theory in 1952 on the origin of patterns in nature, such as spots, stripes and spirals, and the researchers say their work may also help explain the origin of some patterns. “Turing’s theory involves the idea that these patterns arise as solutions to reaction-diffusion equations,” said Kent. “Our results suggest that a rich variety of pattern formation can also arise in systems with essentially fixed-range interactions. It may be worth looking for explanations of this type in contexts where highly regular patterns naturally arise and are not otherwise easily explained.”

Reference:
Olga Goulko and Adrian Kent. ‘ ̽��ֱ��grasshopper problem.’ Proceedings of the Royal Society A (2017). DOI: http://dx.doi.org/10.1098/rspa.2017.0494

Picture a grasshopper landing randomly on a lawn of fixed area. If it then jumps a certain distance in a random direction, what shape should the lawn be to maximise the chance that the grasshopper stays on the lawn after jumping?

̽��ֱ��grasshopper problem is a rather nice one, as it helps us try out techniques for the physics problems we really want to get to.

Adrian Kent

Alvesgaspar

Grasshopper of Acrididae family: Anacridium aegyptium

̽��ֱ��text in this work is licensed under a Creative Commons Attribution 4.0 International License. For image use please see separate credits above.

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New understanding of how shape and form develop in nature

sc604 — Wed, 09 Dec 2015 18:01:07 +0000

Researchers have developed a new method for generating complex shapes, and have found that the development of form in nature can be driven by the physical properties of materials themselves, in contrast with earlier findings. ̽��ֱ��results, reported in the journal Nature, could enable the construction of complex structures from simple components, with potential applications in pharmaceuticals, paints, cosmetics and household products such as shampoo.

Using a simple set-up – essentially droplets of oil in a soapy water solution which were slowly frozen – the researchers found that recently-discovered ‘plastic crystal’ phases formed on the inside surfaces of the droplets causes them to shape-shift into a wide variety of forms, from octahedrons and hexagons to triangles and fibres.

Previous efforts to create such complex shapes and structures have used top-down processing methods, which allow a high degree of control, but are not efficient in terms of the amount of material used or the expensive equipment necessary to make the shapes. ̽��ֱ��new method, developed by researchers from the ̽��ֱ�� of Cambridge and Sofia ̽��ֱ�� in Bulgaria, uses a highly efficient, extremely simple bottom-up approach to create complex shapes.

“There are many ways that non-biological things take shape,” said Dr Stoyan Smoukov from Cambridge’s Department of Materials Science & Metallurgy, who led the research. “But the question is what drives the process and how to control it – and what are the links between the process in the biological and the non-biological world?”

Smoukov’s research proposes a possible answer to the question of what drives this process, called morphogenesis. In animals, morphogenesis controls the distribution of cells during embryonic development, and can also be seen in mature animals, such as in a growing tumour.

In the 1950s, the codebreaker and mathematician Alan Turing proposed that morphogenesis is driven by reaction-diffusion, in which local chemical reactions cause a substance to spread through a space. More recent research, from Smoukov’s group and others, has proposed that it is physical properties of materials that control the process. This possibility had been anticipated by Turing, but it was impossible to determine using the computers of the time.

What this most recent research has found is that by slowly freezing oil droplets in a soapy solution, the droplets will shape-shift through a variety of different forms, and can shift back to their original shape if the solution is re-warmed. Further observation found that this process is driven by the self-assembly of a plastic crystal phase which forms beneath the surface of the droplets.

“Plastic crystals are a special state of matter that is like the alter ego of the liquid crystals used in many TV screens,” said Smoukov. Both liquid crystals and plastic crystals can be thought of as transitional stages between liquid and solid. While liquid crystals point their molecules in defined directions like a crystal, they have no long-range order and flow like a liquid. Plastic crystals are wax-like with long-range order in their molecular arrangement, but disorder in the orientation of each molecule. ̽��ֱ��orientational disorder makes plastic crystals highly deformable, and as they change shape, the droplets change shape along with them.

“This plastic crystal phase seems to be what’s causing the droplets to change shape, or break their symmetry,” said Smoukov. “And in order to understand morphogenesis, it’s vital that we understand what causes symmetry breaking.”

̽��ֱ��researchers found that by altering the size of the droplets they started with or the rate that the temperature of the soapy solution was lowered, they were able to control the sequence of the shapes the droplets ended up forming. This degree of control could be useful for multiple applications – from pharmaceuticals to household goods – that use small-droplet emulsions.

“ ̽��ֱ��plastic crystal phase has been of intense scientific interest recently, but no one so far has been able to harness it to exert forces or show this variety of shape-changes,” said the paper’s lead author Professor Nikolai Denkov of Sofia ̽��ֱ��, who first proposed the general explanation of the observed transformations.

“ ̽��ֱ��phenomenon is so rich in combining several active areas of research that this study may open up new avenues for research in soft matter and materials science,” said co-author Professor Slavka Tcholakova, also of Sofia ̽��ֱ��.

“If we’re going to build artificial structures with the same sort of control and complexity as biological systems, we need to develop efficient bottom-up processes to create building blocks of various shapes, which can then be used to make more complicated structures,” said Smoukov. “But it’s curious to observe such life-like behaviour in a non-living thing – in many cases, artificial objects can look more ‘alive’ than living ones.”

Inset image: Morphogenesis ( ̽��ֱ�� of Cambridge).

Reference:
Denkov, Nikolai et. al. ‘Self-Shaping of Droplets via Formation of Intermediate Rotator Phases upon Cooling.’ Nature (2015). DOI: 10.1038/nature16189.

Researchers have identified a new mechanism that drives the development of form and structure, through the observation of artificial materials that shape-shift through a wide variety of forms which are as complex as those seen in nature.

It’s curious to observe such life-like behaviour in a non-living thing – in many cases, artificial objects can look more ‘alive’ than living ones

Stoyan Smoukov

̽��ֱ�� of Cambridge

Morphogenesis

̽��ֱ��text in this work is licensed under a Creative Commons Attribution 4.0 International License. For image use please see separate credits above.

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