̽��ֱ�� of Cambridge - Raymond Goldstein

Swarming cicadas, stock traders, and the wisdom of the crowd

sc604 — Thu, 01 Feb 2024 14:36:51 +0000

Pick almost any location in the eastern United States – say, Columbus Ohio. Every 13 or 17 years, as the soil warms in springtime, vast swarms of cicadas emerge from their underground burrows singing their deafening song, take flight and mate, producing offspring for the next cycle.

This noisy phenomenon repeats all over the eastern and southeastern US as 17 distinct broods emerge in staggered years. In spring 2024, billions of cicadas are expected as two different broods – one that appears every 13 years and another that appears every 17 years – emerge simultaneously.

Previous research has suggested that cicadas emerge once the soil temperature reaches 18°C, but even within a small geographical area, differences in sun exposure, foliage cover or humidity can lead to variations in temperature.

Now, in a paper published in the journal Physical Review E, researchers from the ̽��ֱ�� of Cambridge have discovered how such synchronous cicada swarms can emerge despite these temperature differences.

̽��ֱ��researchers developed a mathematical model for decision-making in an environment with variations in temperature and found that communication between cicada nymphs allows the group to come to a consensus about the local average temperature that then leads to large-scale swarms. ̽��ֱ��model is closely related to one that has been used to describe ‘avalanches’ in decision-making like those among stock market traders, leading to crashes.

Mathematicians have been captivated by the appearance of 17- and 13-year cycles in various species of cicadas, and have previously developed mathematical models that showed how the appearance of such large prime numbers is a consequence of evolutionary pressures to avoid predation. However, the mechanism by which swarms emerge coherently in a given year has not been understood.

In developing their model, the Cambridge team was inspired by previous research on decision-making that represents each member of a group by a ‘spin’ like that in a magnet, but instead of pointing up or down, the two states represent the decision to ‘remain’ or ‘emerge’.

̽��ֱ��local temperature experienced by the cicadas is then like a magnetic field that tends to align the spins and varies slowly from place to place on the scale of hundreds of metres, from sunny hilltops to shaded valleys in a forest. Communication between nearby nymphs is represented by an interaction between the spins that leads to local agreement of neighbours.

̽��ֱ��researchers showed that in the presence of such interactions the swarms are large and space-filling, involving every member of the population in a range of local temperature environments, unlike the case without communication in which every nymph is on its own, responding to every subtle variation in microclimate.

̽��ֱ��research was carried out Professor Raymond E Goldstein, the Alan Turing Professor of Complex Physical Systems in the Department of Applied Mathematics and Theoretical Physics (DAMTP), Professor Robert L Jack of DAMTP and the Yusuf Hamied Department of Chemistry, and Dr Adriana I Pesci, a Senior Research Associate in DAMTP.

“As an applied mathematician, there is nothing more interesting than finding a model capable of explaining the behaviour of living beings, even in the simplest of cases,” said Pesci.

̽��ֱ��researchers say that while their model does not require any particular means of communication between underground nymphs, acoustical signalling is a likely candidate, given the ear-splitting sounds that the swarms make once they emerge from underground.

̽��ֱ��researchers hope that their conjecture regarding the role of communication will stimulate field research to test the hypothesis.

“If our conjecture that communication between nymphs plays a role in swarm emergence is confirmed, it would provide a striking example of how Darwinian evolution can act for the benefit of the group, not just the individual,” said Goldstein.

This work was supported in part by the Complex Physical Systems Fund.

Reference:
R E Goldstein, R L Jack, and A I Pesci. ‘How Cicadas Emerge Together: Thermophysical Aspects of their Collective Decision-Making.’ Physical Review E (2024). DOI: 10.1103/PhysRevE.109.L022401

̽��ֱ��springtime emergence of vast swarms of cicadas can be explained by a mathematical model of collective decision-making with similarities to models describing stock market crashes.

Ed Reschke via Getty Images

Adult Periodical Cicada

̽��ֱ��text in this work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Images, including our videos, are Copyright © ̽��ֱ�� of Cambridge and licensors/contributors as identified. All rights reserved. We make our image and video content available in a number of ways – on our main website under its Terms and conditions, and on a range of channels including social media that permit your use and sharing of our content under their respective Terms.

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Mathematics explains how giant ‘whirlpools’ form in developing egg cells

sc604 — Wed, 13 Jan 2021 16:35:52 +0000

Egg cells are among the largest cells in the animal kingdom. Unpropelled, a protein could take hours or even days to drift from one side of a forming egg cell to the other. Luckily, nature has developed a faster way: scientists have spotted cell-spanning whirlpools in the immature egg cells of animals such as mice, zebrafish and fruit flies. These vortices make cross-cell commutes take just a fraction of the time. But scientists didn’t know how these crucial flows formed.

Using mathematical modeling, researchers say they now have an answer. ̽��ֱ��gyres result from the collective behavior of rodlike molecular tubes called microtubules that extend inward from the cells’ membranes. Their results are reported in the journal Physical Review Letters.

“While much is not understood about the biological function of these flows, they distribute nutrients and other factors that organise the body plan and guide development,” said study co-lead author David Stein, a research scientist at the Flatiron Institute’s Center for Computational Biology (CCB) in New York City. And given how widely they have been observed, “they are probably even in humans.”

Scientists have studied cellular flows since the late 18th century, when Italian physicist Bonaventura Corti peered inside cells using his microscope. What he found were fluids in constant motion, however scientists didn’t understand the mechanisms driving these flows until the 20th century.

̽��ֱ��culprits, they found, are molecular motors that walk along the microtubules. Those motors haul large biological payloads such as lipids. Carrying the cargo through a cell’s relatively thick fluids is like dragging a beach ball through honey. As the payloads move through the fluid, the fluid moves too, creating a small current.

Sometimes those currents aren’t so small. In certain developmental stages of a common fruit fly’s egg cell, scientists spotted whirlpool-like currents that spanned the entire cell. In these cells, microtubules extend inward from the cell’s membrane like stalks of wheat. Molecular motors climbing these microtubules push downward on the microtubule as they ascend. That downward force bends the microtubule, redirecting the resulting flows.

Previous studies looked at this bending mechanism, but only for isolated microtubules. Those studies predicted that the microtubules would wave around in circles, but their behavior didn’t match the observations.

“ ̽��ֱ��mechanism of the swirling instability is disarmingly simple, and the agreement between our calculations and the experimental observations by various groups lends support to the idea that this is indeed the process at work in fruit fly egg cells,” said Professor Raymond Goldstein from Cambridge’s Department of Applied Mathematics and Theoretical Physics. “Further experimental tests should be able to probe details of the transition between disordered and ordered flows, where there is still much to be understood.”

In the new study, the researchers added a key factor to their model: the influence of neighboring microtubules. That addition showed that the fluid flows generated by the payload-ferrying motors bend nearby microtubules in the same direction. With enough motors and a dense enough packing of microtubules, the authors found that all the microtubules eventually lean together like wheat stalks caught in a strong breeze. This collective alignment orients all the flows in the same direction, creating the cell-wide vortex seen in real fruit fly cells.

While grounded in reality, the new model is stripped down to the bare essentials to make clearer the conditions responsible for the swirling flows. ̽��ֱ��researchers are now working on versions that more realistically capture the physics behind the flows to understand better the role the currents play in biological processes.

Stein serves as the co-lead author of the new study along with Gabriele De Canio, a researcher at the ̽��ֱ�� of Cambridge. They co-authored the study with CCB director and New York ̽��ֱ�� professor Michael Shelley and ̽��ֱ�� of Cambridge professors Eric Lauga and Raymond Goldstein.

This work was supported by the US National Science Foundation, the Wellcome Trust, the European Research Council, the Engineering and Physical Sciences Research Council, and the Schlumberger Chair Fund.

Reference:
D.B. Stein, G. De Canio, E. Lauga, M.J. Shelley, and R.E. Goldstein, “Swirling Instability of the Microtubule Cytoskeleton”, Physical Review Letters (2021). DOI: 10.1103/PhysRevLett.126.028103

̽��ֱ��swirling currents occur when the rodlike structures that extend inward from the cells’ membranes bend in tandem, like stalks of wheat caught in a strong breeze, according to a study from the ̽��ֱ�� of Cambridge and the Flatiron Institute.

̽��ֱ��mechanism of the swirling instability is disarmingly simple, and the agreement between our calculations and experimental observations supports the idea that this is indeed the process at work in fruit fly egg cells

Raymond Goldstein

̽��ֱ��text in this work is licensed under a Creative Commons Attribution 4.0 International License. Images, including our videos, are Copyright © ̽��ֱ�� of Cambridge and licensors/contributors as identified. All rights reserved. We make our image and video content available in a number of ways – as here, on our main website under its Terms and conditions, and on a range of channels including social media that permit your use and sharing of our content under their respective Terms.

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Flashes bright when squeezed tight: how single-celled organisms light up the oceans

sc604 — Mon, 06 Jul 2020 06:30:00 +0000

Every few years, a bloom of microscopic organisms called dinoflagellates transforms the coasts around the world by endowing breaking waves with an eerie blue glow. This year’s spectacular bloom in southern California was a particularly striking example. In a new study published in the journal Physical Review Letters, researchers have identified the underlying physics that results in light production in one species of these organisms.

̽��ֱ��international team, led by the ̽��ֱ�� of Cambridge, developed unique experimental tools based on micromanipulation and high-speed imaging to visualise light production on the single-cell level. They showed how a single-celled organism of the species Pyrocystis lunula produces a flash of light when its cell wall is deformed by mechanical forces. Through systematic experimentation, they found that the brightness of the flash depends both on the depth of the deformation and the rate at which it is imposed.

Known as a ‘viscoelastic’ response, this behaviour is found in many complex materials such as fluids with suspended polymers. In the case of organisms like Pyrocystis lunula, known as dinoflagellates, this mechanism is most likely related to ion channels, which are specialised proteins distributed on the cell membrane. When the membrane is stressed, these channels open up, allowing calcium to move between compartments in the cell, triggering a biochemical cascade that produces light.

“Despite decades of scientific research, primarily within the field of biochemistry, the physical mechanism by which fluid flow triggers light production has remained unclear,” said Professor Raymond E. Goldstein, the Schlumberger Professor of Complex Physical Systems in the Department of Applied Mathematics and Theoretical Physics, who led the research.

“Our findings reveal the physical mechanism by which the fluid flow triggers light production and show how elegant decision-making can be on a single-cell level,” said Dr Maziyar Jalaal, the paper’s first author.

Bioluminescence has been of interest to humankind for thousands of years, as it is visible as the glow of night-time breaking waves in the ocean or the spark of fireflies in the forest. Many authors and philosophers have written about bioluminescence, from Aristotle to Shakespeare, who in Hamlet wrote about the ‘uneffectual fire’ of the glow-worm; a reference to production of light without heat:

"…To prick and sting her. Fare thee well at once / ̽��ֱ��glowworm shows the matin to be near / And 'gins to pale his uneffectual fire. / Adieu, adieu, adieu. Remember me.”

̽��ֱ��bioluminescence in the ocean is, however, not ‘uneffectual.’ In contrast, it is used for defence, offense, and mating. In the case of dinoflagellates, they use light production to scare off predators.

̽��ֱ��results of the current study show that when the deformation of the cell wall is small, the light intensity is small no matter how rapidly the indentation is made, and it is also small when the indentation is large but applied slowly. Only when both the amplitude and rate are large is the light intensity maximised. ̽��ֱ��group developed a mathematical model that was able to explain these observations quantitatively, and they suggest that this behaviour can act as a filter to avoid spurious light flashes from being triggered

In the meantime, the researchers plan to analyse more quantitatively the distribution of forces over the entire cells in the fluid flow, a step towards understanding the light prediction in a marine context.

Other members of the research team were postdoctoral researcher Hélène de Maleprade, visiting students Nico Schramma from the Max-Planck Institute for Dynamics and Self-Organization in Göttingen, Germany and Antoine Dode from the Ècole Polytechnique in France, and visiting professor Christophe Raufaste from the Institut de Physique de Nice, France.

̽��ֱ��work was supported by the Marine Microbiology Initiative of the Gordon and Betty Moore Foundation, the Schlumberger Chair Fund, the French National Research Agency, and the Wellcome Trust.

Reference:
M. Jalaal, N. Schramma, A. Dode, H. de Maleprade, C. Raufaste, and R.E. Goldstein. ‘Stress-Induced Dinoflagellate Bioluminescence at the Single Cell Level.’ Physical Review Letters (2020). DOI: 10.1103/PhysRevLett.125.028102

Research explains how a unicellular marine organism generates light as a response to mechanical stimulation, lighting up breaking waves at night.

Our findings show how elegant decision-making can be on a single-cell level

Maziyar Jalaal

̽��ֱ�� of Cambridge

̽��ֱ��Dinoflagellate Pyrocystis lunula

Yes

How single-celled organisms navigate to oxygen

ps748 — Thu, 01 Dec 2016 16:46:02 +0000

Although the single-celled ancestors of animals are extinct, the choanoflagellates, which evolved from a common ancestor and which have remained single-celled since the Cambrian period around 500 million years ago, are common in the Earth’s oceans and lakes. Certain choanoflagellate species form small swimming colonies and these colonies are thought to resemble the early multicellular organisms that later evolved into animals. Oxygen levels on the planet started rising in the pre-Cambrian period and it’s likely this played a major influence on the emergence of these multicellular life forms.

̽��ֱ��researchers observed choanoflagellate colonies swimming under controlled conditions and varied the oxygen concentration in the water over time. They found the colonies navigate based on the logarithm of the oxygen concentration, similar to the way humans sense sound and light. This increases their sensing capabilities in low-oxygen environments where navigation becomes crucial for survival.

One of the authors on the paper, Professor Raymond E. Goldstein, of the ̽��ֱ��’s Department of Applied Mathematics and Theoretical Physics, says:

"Our work provides the first evidence that choanoflagellates can sense, and move towards, oxygen. Since choanoflagellates are now understood to be the closest relatives of animals, this discovery may shed light on the properties of the last common ancestor of the two groups, and in particular its response to the changing oxygen levels in the Precambrian era. Perhaps more importantly, the work raises fascinating questions about how the simplest multicellular organisms, lacking any type of central nervous system, sense and respond to their environment."

Many organisms find their way to favourable areas by using different strategies. Bacteria bias their tumbling to navigate towards food and algae can turn and move directly towards light. While choanaflagellates require oxygen, it wasn’t known if they could successfully navigate towards it. But the research showed both single cells and swimming colonies were able to find it.

While animals require enormous amounts of coordination between their cells in order to navigate, this research reveals such coordination isn’t needed for simple multicellular life forms. In addition, microorganisms’ search for food is rendered more difficult by the presence of thermal noise. Being so small, microorganisms are constantly being buffeted by vibrations in the waters that surround them, and their search strategy needs to be robust to counter this.

̽��ֱ��team, based in the Department of Applied Mathematics and Theoretical Physics, included PhD student Julius B. Kirkegaard, visiting student Ambre Bouillant, postdoctoral fellow Dr. Alan O. Marron, Senior Research Associate Dr. Kyriacos C. Leptos, and Professor Raymond E. Goldstein.

This work was supported by the European Research Council, the Engineering and Physical Sciences Research Council, and the Wellcome Trust.

Reference
Kirkegaard, JB et al. Aerotaxis in the closest relatives of animals. e-Life; 24 Nov 2016; DOI: 10.7554/eLife.18109

A team of researchers has discovered that tiny clusters of single-celled organisms that inhabit the world’s oceans and lakes, are capable of navigating their way to oxygen. Writing in e-Life scientists at the ̽��ֱ�� of Cambridge describe how choanaflagellates, the closest relatives of animals, form small colonies that can sense a large range of concentrations of oxygen in the water. ̽��ֱ��research offers clues as to how these organisms evolved into multi-cellular ones.

̽��ֱ��work raises fascinating questions about how the simplest multicellular organisms...sense and respond to their environment

Prof. Raymond Goldstein

Raymond Goldstein

Colonies of choannaflagellates

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

Yes

Algae use their ‘tails’ to gallop and trot like quadrupeds

sc604 — Tue, 03 May 2016 14:12:54 +0000

Long before there were fish swimming in the oceans, tiny microorganisms were using long slender appendages called cilia and flagella to navigate their watery habitats. Now, new research reveals that species of single-celled algae coordinate their flagella to achieve a remarkable diversity of swimming gaits.

When it comes to four-legged animals such as cats, horses and deer, or even humans, the concept of a gait is familiar, but what about unicellular green algae with multiple limb-like flagella? ̽��ֱ��latest discovery, published in the journal Proceedings of the National Academy of Sciences, shows that despite their simplicity, microalgae can coordinate their flagella into leaping, trotting or galloping gaits just as well.

Many gaits are periodic: whether it is the stylish walk of a cat, the graceful gallop of a horse, or the playful leap of a springbok, the key is the order or sequence in which these limbs are activated. When springboks arch their backs and leap, or ‘pronk’, they do so by lifting all four legs simultaneously high into the air, yet when horses trot it is the diagonally opposite legs that move together in time.

In vertebrates, gaits are controlled by central pattern generators, which can be thought of as networks of neural oscillators that coordinate output. Depending on the interaction between these oscillators, specific rhythms are produced, which, mathematically speaking, exhibit certain spatiotemporal symmetries. In other words, the gait doesn’t change when one leg is swapped with another – perhaps at a different point in time, say a quarter-cycle or half-cycle later.

It turns out the same symmetries also characterise the swimming gaits of microalgae, which are far too simple to have neurons. For instance, microalgae with four flagella in various possible configurations can trot, pronk or gallop, depending on the species.

“When I peered through the microscope and saw that the alga was performing two sets of perfectly synchronous breaststrokes, one directly after the other, I was amazed,” said the paper’s first author Dr Kirsty Wan of the Department of Applied Mathematics and Theoretical Physics (DAMTP) at the ̽��ֱ�� of Cambridge. “I realised immediately that this behaviour could only be due to something inside the cell rather than passive hydrodynamics. Then of course to prove this I had to expand my species collection.”

̽��ֱ��researchers determined that it is in fact the networks of elastic fibres which connect the flagella deep within the cell that coordinate these diverse gaits. In the simplest case of Chlamydomonas, which swims a breaststroke with two flagella, absence of a particular fibre between the flagella leads to uncoordinated beating. Furthermore, deliberately preventing the beating of one flagellum in an alga with four flagella has zero effect on the sequence of beating in the remainder.

However, this does not mean that hydrodynamics play no role. In recent work from the same group, it was shown that nearby flagella can be synchronised solely by their mutual interaction through the fluid. There is a distinction between unicellular organisms for which good coordination of a few flagella is essential, and multicellular species or tissues that possess a range of cilia and flagella. In the latter case, hydrodynamic interactions are much more important.

“As physicists our instinct is to seek out generalisations and universal principles, but the world of biology often presents us with many fascinating counterexamples,” said Professor Ray Goldstein, Schlumberger Professor of Complex Physical Systems at DAMTP, and senior author of the paper. “Until now there have been many competing theories regarding flagellar synchronisation, but I think we are finally making sense of how these different organisms make best use of what they have.”

̽��ֱ��findings also raise intriguing questions about the evolution of the control of peripheral appendages, which must have arisen in the first instance in these primitive microorganisms.

This research was supported by a Neville Research Fellowship from Magdalene College, and a Senior Investigator Award from the Wellcome Trust.

Reference:
Kirsty Y. Wan and Raymond E. Goldstein. ‘Coordinated beating of algal flagella is mediated by basal coupling.’ PNAS (2016). DOI: 10.1073/pnas.1518527113

Species of single-celled algae use whip-like appendages called flagella to coordinate their movements and achieve a remarkable diversity of swimming gaits.

As physicists our instinct is to seek out generalisations and universal principles, but the world of biology often presents us with many fascinating counterexamples.

Raymond Goldstein

Kirsty Y. Wan & Raymond E. Goldstein

Microscope images showing two species of algae which swim using tiny appendages known as flagella

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

Yes

Upside down and inside out

sc604 — Sun, 26 Apr 2015 23:00:01 +0000

Researchers from the ̽��ֱ�� of Cambridge have captured the first three-dimensional images of a live embryo turning itself inside out. ̽��ֱ��images, of embryos of a green alga called Volvox, make an ideal test case to understand how a remarkably similar process works in early animal development.

Using fluorescence microscopy to observe the Volvox embryos, the researchers were able to test a mathematical model of morphogenesis – the origin and development of an organism’s structure and form – and understand how the shape of cells drives the process of inversion, when the embryo turns itself from a sphere to a mushroom shape and back again. Their findings are published today (27 April) in the journal Physical Review Letters.

̽��ֱ��processes observed in the Volvox embryo are similar to the process of gastrulation in animal embryos – which biologist Lewis Wolpert called “the most important event in your life.” During gastrulation, the embryo folds inwards into a cup-like shape, forming the primary germ layers which give rise to all the organs in the body. Volvox embryos undergo a similar process, but with an additional twist: the embryos literally turn themselves right-side out during the process.

Gastrulation in animals results from a complex interplay of cell shape changes, cell division and migration, making it difficult to develop a quantitative understanding of the process. However, Volvox embryos complete their shape change only by changing cell shapes and the location of the connections between cells, and this simplicity makes them an ideal model for understanding cell sheet folding.

In Volvox embryos, the process of inversion begins when the embryos start to fold inward, or invaginate, around their middle, forming two hemispheres. Next, one hemisphere moves inside the other, an opening at the top widens, and the outer hemisphere glides over the inner hemisphere, until the embryo regains its spherical shape. This remarkable process takes place over approximately one hour.

Previous work by biologists established that a specific series of cell shape changes is associated with various stages of the process. “Until now there was no quantitative mechanical understanding of whether those changes were sufficient to account for the observed embryo shapes, and existing studies by conventional microscopy were limited to two-dimensional sections and analyses of chemically fixed embryos, rendering comparisons with theory on the dynamics difficult,” said Professor Raymond E. Goldstein of the Department of Applied Mathematics and Theoretical Physics, who led the research.

̽��ֱ��interdisciplinary group of Cambridge scientists obtained the first three-dimensional visualisations of Volvox inversion and developed a first mathematical model that explains how cell shape changes drive the process of inversion.

Their time-lapse recordings show that during inversion one hemisphere of the embryos shrinks while the other hemisphere stretches out. While previous studies on fixed embryos have also observed this phenomenon, the question was if these changes are caused by forces produced within the invaginating region, or from elsewhere in the embryo.

Through mathematical modelling, the researchers found that only if there is active contraction of one hemisphere and active expansion of the other does the model yield the observed ‘mushroom’ shape of an inverting Volvox globator embryo.

“It’s exciting to be able to finally visualise this intriguing process in 3D,” said Dr Stephanie Höhn, the paper’s lead author. “This simple organism may provide ground-breaking information to help us understand similar processes in many different types of animals.”

These results imply that any cell shape changes happening away from the invagination region seem to be due to active forces intrinsic to the cell, rather than through passive deformations. Since analyses in animal model organisms mostly concentrate on cell shape changes that happen within an invaginating region, the model could be used to make those analyses far more accurate.

“ ̽��ֱ��power of this mathematical model is that we can identify which cell deformations are needed to cause the embryo movements that we observe in nature,” said Dr Aurelia Honerkamp-Smith, one of the study’s co-authors.

̽��ֱ��experimental and theoretical methods demonstrated in this study will be expanded to understand not only the peculiar inversion process but many mysteries concerning morphogenesis. ̽��ֱ��mathematical model may have applications in a multitude of such topological problems, such as the process of neurulation that leads to the enclosure of the tissue that eventually becomes the spinal cord.

Other members of the research team were PhD students Pierre A. Haas (DAMTP) and Philipp Khuc Trong (Physics).

This work was supported by an Earnest Oppenheimer Early Career Fellowship, the Engineering and Physical Sciences Research Council, and the European Research Council.

Researchers have captured the first 3D video of a living algal embryo turning itself inside out, from a sphere to a mushroom shape and back again. ̽��ֱ��results could help unravel the mechanical processes at work during a similar process in animals, which has been called the “most important time in your life.”

This simple organism may provide ground-breaking information to help us understand similar processes in many different types of animals

Stephanie Höhn

Volvox embryo turning itself inside out

Stephanie Höhn, Aurelia Honerkamp-Smith and Raymond E. Goldstein

Adult Volvox spheroid containing multiple embryos

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Yes

A new twist on soap films

sc604 — Fri, 23 May 2014 15:13:03 +0000

̽��ֱ��way in which soap films collapse and re-form when twisted or stretched could hold the key to predicting the formation and location of mathematical singularities, which can be seen in the motion of solar flares and other natural phenomena.

Research on the processes by which soap films undergo transitions from one stable state to another has led to conjectures on the nature and location of the singular events that occur during the change of form, connecting two previously separate areas in mathematics.

In mathematics, singularities occur when an equation or surface breaks down and ‘explodes’. In surfaces such as soap films, singularities occur when the surface collides with itself, changing shape in the blink of an eye.

Researchers from the ̽��ֱ�� of Cambridge have shown that identifying a special type of curve on the surface can help predict where these singularities are likely to occur in soap films, which could in turn aid in the understanding of singularities in the natural world. ̽��ֱ��results are published in the journal Proceedings of the National Academy of Sciences (PNAS).

We are all familiar with the simplest soap films, which are formed by dipping a wire loop into a soap solution: the flat surface that spans the wire and the bubbles which are formed when we blow on the film. With suitably shaped wires however, much more complex structures can be formed, such as Möbius strips.

All static soap films are ‘minimal surfaces’, for they have the least area of all possible surfaces that span a given wire frame.

What is less understood are the dynamic processes which occur when a minimal surface like a soap film is made unstable by deforming the supporting wire. ̽��ֱ��film typically moves in a fraction of a second to a new configuration through a singular point, at which the surface collides with itself and changes its connectivity.

These kinds of violent events also occur in the natural world – in fluid turbulence and in the motion of solar flares emanating from the sun – and one of the great challenges has been to predict where they will occur.

In research supported by the EPSRC, a team from the Department of Applied Mathematics and Theoretical Physics attempted to understand how to predict where the singularity will occur when soap films are twisted or stretched to a point of instability. For example, it is well-known that the surface spanning two separate wire loops will collapse to a singularity in between the loops.

In previous work, the group had shown that Möbius strip singularity occurs not between the loops but at the wire frame, where there is a complex rearrangement of the surface. “What was unclear was whether there was an underlying mathematical principle by which this striking difference could be explained,” said Professor Raymond Goldstein, who collaborated with Dr Adriana Pesci, Professor Keith Moffatt, and James McTavish, a maths undergraduate, on the research.

̽��ֱ��team recognised that a geometric concept known as a systole might be the key to understanding where singularities will occur. A systole is the length of the shortest closed curve on surface that cannot be shrunk to a point while remaining on the surface. An example of this is found on a bagel, where the shortest such curve encircles the bagel like a handle. Mathematicians have studied the geometric properties of these curves in recent decades, establishing constraints on the relationship between the length of a systole and the area of the surface on which they lie.

Using new laboratory experiments and computations, the researchers found evidence that the ultimate location of the singularities that occur when soap films collapse can be deduced from the properties of the systole. If the systolic curve loops around the wire frame, then the singularity occurs at the boundary, while if there is no such linking the singularity occurs in the bulk.

“This is an example of experimental mathematics, in the sense that we are using laboratory studies to inform conjectures on mathematical connections,” said Professor Goldstein. “While they are certainly not rigorous, we hope they will stimulate further research into this new, developing area.”

Soap films with complex shapes shed light on the formation of mathematical singularities, which occur in a broad range of fields.

This is an example of experimental mathematics, in the sense that we are using laboratory studies to inform conjectures on mathematical connections

Raymond Goldstein

Singularity in a soap film

Raymond Goldstein

Soap film singularity

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Yes

Rapunzel, Leonardo and the physics of the ponytail

lw355 — Mon, 13 Feb 2012 09:00:35 +0000

From Leonardo da Vinci to the Brothers Grimm, the properties of hair have been of enduring interest in science and art. Now, a ̽��ֱ�� of Cambridge physicist and collaborators have quantified the curliness of human hair and developed a mathematical theory that explains the shape of a ponytail.

Research published today (13 February) in Physical Review Letters provides the first quantitative understanding of the distribution of hairs in a ponytail. To derive the Ponytail Shape Equation, the scientists took account of the stiffness of the hairs, the effects of gravity and the presence of the random curliness or waviness that is ubiquitous in human hair. Together with a new quantity described in the article – the Rapunzel Number – the equation can, they say, be used to predict the shape of any ponytail.

̽��ֱ��research by Professor Raymond Goldstein from the ̽��ֱ�� of Cambridge, Professor Robin Ball from the ̽��ֱ�� of Warwick, and colleagues, provides new understanding of how a bundle is swelled by the outward pressure which arises from collisions between the component hairs. This has important implications for understanding the structure of many materials made up of random fibres, such as wool and fur. ̽��ֱ��research will also have resonance with the computer graphics and animation industry, where the representation of hair has been a challenging problem.

“It’s a remarkably simple equation,” explained Goldstein, who is the Schlumberger Professor of Complex Physical Systems at Cambridge's Department of Applied Mathematics and Theoretical Physics. “Our findings extend some central paradigms in statistical physics and show how they can be used to solve a problem that has puzzled scientists and artists ever since Leonardo da Vinci remarked on the fluid-like streamlines of hair in his notebooks 500 years ago.

“To be able to reduce this problem to a very simple mathematical form which speaks immediately to the way in which the random curliness of hair swells a ponytail is deeply satisfying. Physicists aim to find simplicity out of complexity, and this is a case in point.”

“We imagine that at least half of the population has direct experience with the properties of ponytails, and we all have likely wondered about the fluffiness of hair,” added Goldstein, whose research was partially funded by the Schlumberger Chair Fund. “We now have the first quantitative description of this phenomenon and how it competes with gravity.”

Professor Goldstein will be presenting the research at the March Meeting of the American Physical Society in Boston on 28 February 2012.

New research provides the first mathematical understanding of the shape of a ponytail and could have implications for the textile industry, computer animation and personal care products.

Our findings solve a problem that has puzzled scientists and artists ever since Leonardo da Vinci remarked on the fluid-like streamlines of hair in his notebooks 500 years ago.

Professor Ray Goldstein

Nick Saffell

Suvi's ponytail

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