Westward Ho! Musing on Mathematics and Mechanics Westward Ho!Alan Musing Champneys onCMath Mathematics FIMA, University of Bristol and Mechanics Alan Champneys CMath FIMA, University of Bristol Inspired by the village pageants that take dancing from 7.30 in the morning until the Inspiredplace across by the the villagecounty ofpageants Cornwall that to takeher- dancingearly evening. from 7.30 Led in bythe the morning Helston until town the placeald the across coming the of county spring, of this Cornwall issue consid- to her- earlyband, evening. revellers Led dance by along the Helston the streets, town alders the comingdynamics of ofspring, mass this synchronous issue consid- ac- band,through revellers gardens dance and in along out of the houses streets, in erstivity. the New dynamics theories of havemass emerged synchronous overac- the throughvarious costumes gardens and at different in out of times houses of the in tivity.last few New decades theories of have how globalemerged order over can the variousday. Traditional costumes at ‘Helston-born’ different times dancers of the lastemerge few from decades networks of how of global coupled order (near) can day.wear lilies Traditional of the valley; ‘Helston-born’ gentlemen dancers on the emergeidentical from agents. networks Simulations of coupled using (near) such wearleft, flowers lilies of pointing the valley; upwards, gentlemen ladies on the on identicalrules can agents. replicate Simulations the remarkable using obser- such left,the right flowers and pointing upside down. upwards, The ladies town on is rulesvations can of replicate swarms of the fish remarkable or starlings. obser- Yet, thedecorated right and with upside sycamore down. branches The town and, in is vationsa more of recent swarms question of fish isor howstarlings. do lead- Yet, decorateda special dance with sycamoreat 10.00 am, branches local children and, in aers more or influencers recent question emerge is within how do a group lead- adress special in white dance – atthe 10.00 girls wearingam, local flowers children in ersof human or influencers participants emerge carrying within out a agroup syn- dresstheir hair in white to match – the theirgirls wearing school’s flowers colours. in ofchronous human activity. participants This carrying theory not out only a syn- has theirThe hair origins to match of Flora theirDay school’s and the colours. Furry chronousobvious implications activity. This in theory seeking not to only under- has DanceThe are origins not ofclear. Flora A Day play and performed the Furry obviousstand the implications prescient phenomenon in seeking of to manip- under- Dancein the morning are not clear. is called A theplay Hal-an-Tow performed standulation the of prescient opinion, phenomenonbut also surprising of manip- po- in(with the the morning last syllable is called rhyming the Hal-an-Tow with now, ulationtential application of opinion,but in therapy also surprising for extreme po- rather(with the than lasttoe syllable), the name rhyming probably with now de-, tentialpsychological application conditions. in therapy for extreme ratherriving than fromtoe the), Cornish the name for probably ‘eve of de-the psychological conditions.features riving from the Cornish for ‘eve of the Westward Ho! Musing on Mathematics and Mechanics Alan Champneys CMath FIMA, University of Bristol
Inspired by the village pageants that take dancing from 7.30 in the morning until the place across the county of Cornwall to her- early evening. Led by the Helston town ald the coming of spring, this issue consid- band, revellers dance along the streets, ers the dynamics of mass synchronous ac- through gardens and in out of houses in tivity. New theories have emerged over the various costumes at different times of the last few decades of how global order can day. Traditional ‘Helston-born’ dancers emerge from networks of coupled (near) wear lilies of the valley; gentlemen on the identical agents. Simulations using such left, flowers pointing upwards, ladies on rules can replicate the remarkable obser- the right and upside down. The town is vations of swarms of fish or starlings. Yet, decorated with sycamore branches and, in a more recent question is how do lead- a special dance at 10.00 am, local children ers or influencers emerge within a group dress in white – the girls wearing flowers in of human participants carrying out a syn- their hair to match their school’s colours. chronous activity. This theory not only has The origins of Flora Day and the Furry obvious implications in seeking to under- Dance are not clear. A play performed stand the prescient phenomenon of manip- in the morning is called the Hal-an-Tow ulation of opinion, but also surprising po- (with the last syllable rhyming with now, tential application in therapy for extreme | Dreamstime.com © Elliottphotos rather than toe), the name probably de- psychological conditions. riving from the Cornish for ‘eve of the Follow the Leader fattening time’. It is similar in spirit to a Mummers play as fea- Follow the Leader fatteningtured in a time’. previous It is Westward similar in Ho! spirit Important to a Mummers characters play include as fea- turedSt George, in a previous Robin Hood Westward and his Ho! Merry Important men, and characters the local include Saint, he small Cornish town of Helston, at the north-western StMichael. George, Reference Robin Hood is made and his in Merry the script men, to and the the raid local on nearby Saint, edgehe small of the Cornish Lizard Peninsula, town of Helston, is synonymousat the north-western with its most Michael.Newlyn by Reference the Spanish is madein 1595. in the script to the raid on nearby T edgefamous of theactivity, Lizard the Peninsula, Furry Dance is synonymous (or Floral Dance). with its Heldmost NewlynThe byword the Furry Spanish itself in 1595. has no direct meaning in the Cornish on Florafamous Day, activity,8 May each the Furry year (unlessDance (or that Floral falls Dance). on a Sunday Held languageThe word but is Furry thought itself to has have no similar direct origins meaning to in ‘feur’ theCornish in other Tor Monday, in which case the pageant is moved to the preced- Celtic languages which literally means grass, but is also used to on Flora Day, 8 May each year (unless that falls on a Sunday language but is thought to have similar origins to ‘feur’ in other oring Monday, Saturday), in whichthe traditional case the festival pageant involves is moved near to the continuous preced- Celticsignify languages a festival. which literally means grass, but is also used to ing Saturday), the traditional festival involves near continuous signify a festival. © David Hastilow | Alamy Stock Photo © CountryCollection – Homer Sykes | Alamy Stock Photo
© parkerphotography | Alamy Stock Photo Follow the Leader fattening time’. It is similar in spirit to a Mummers play as fea- tured in a previous Westward Ho! Important characters include St George, Robin Hood and his Merry men, and the local Saint, he small Cornish town of Helston, at the north-western Michael. Reference is made in the script to the raid on nearby edge of the Lizard Peninsula, is synonymous with its most Newlyn by the Spanish in 1595. T famous activity, the Furry Dance (or Floral Dance). Held The word Furry itself has no direct meaning in the Cornish on Flora Day, 8 May each year (unless that falls on a Sunday language but is thought to have similar origins to ‘feur’ in other or Monday, in which case the pageant is moved to the preced- Celtic languages which literally means grass, but is also used to ing Saturday), the traditional festival involves near continuous signify a festival.
Mathematics TODAY JUNE 2018 98 Similar traditions once occurred in the nearby towns of Lizard and Penryn, but have since died out. There also appear to be con- nections with other May-time festivals in the south-west penin- sula, including the ’Obby ’Oss festival in Padstow and the Hunt- ing of the Earl of Rone in Combe Martin. The chief reason for the fame of the Furry Dance is the distinc- tive tune played throughout the day by the town band. In 1911, a London-based composer, Katie Moss, visited Helston, and joined in the Furry Dance. Inspired, on the train home she wrote words to (a modified version of) the tune in order to describe her expe- © Thomas Langlands | Dreamstime.com rience. She published her composition as ‘The Floral Dance’, an extract from which goes: Figure 1: Starling murmuration at dusk.
As I walked home on a Summer night they could. A wave of human motion would ensue, from front to When stars in Heav’n were shining bright back, rather like the Mexican wave in sports stadia, where succes- Far away from the footlight’s glare sive sections of the crowd momentarily stand and lift both hands Into the sweet and scented air in the air. Of a quaint old Cornish town Following a musical beat is different, though, as this requires I thought I could hear the curious tone synchronous rather than progressive behaviour; all the left feet at Of the cornet, clarinet and big trombone once, then all the right feet, and so on. Suppose you were part Fiddle, ’cello, big bass drum of the merry throng in Helston on Flora Day, weaving in and out Bassoon, flute and euphonium of houses, up and down the streets. You hear the music, but the Far away, as in a trance band is out of sight. How do you decide whether to put your left I heard the sound of the Floral Dance or right foot down on the next beat? As you look around at the mass of revellers you want to conform, not to be the lone dancer The song soon became a popular classic and today is thought who is (half a period) out of step. So you change step. But what is to be the most played tune by brass bands all over the UK. The stopping the dancers next to you, similarly changing step as they song had a resurgence in the 1970s, through an arrangement re- notice your foot movements? So how does the apparent synchro- leased as a single in 1977 by the Brighouse & Rastrick Band nisation of an entire crowd happen? (from the opposite end of England). Picked up by the Radio 2 This is a rather trivial example of spontaneous synchronisa- morning DJ, Terry (later, Sir Terry) Wogan, this instrumental tion, because everyone is being driven by hearing the same mu- reached the public’s hearts. It would become a massive hit, kept sic. But there are numerous other examples of synchronisation in only from the coveted Christmas Number 1 spot by Paul McCart- technology, nature, and human behaviour in which there is no-one ney’s blockbuster ‘Mull of Kintyre’. playing the role of the conductor, keeping the beat. Illustrations of According to their webpage, surviving members of the 1977 this phenomenon include spontaneous rhythmic clapping during era band still talk of their experience appearing on Top of the rounds of applause, the simultaneous repeated on-off illumination Pops, being enthralled by the burlesque dance group Legs and of communities of fire flies and the synchronous firing of neurons Co. and a very young Bob Geldof. Led by conductor Derek within certain brain regions. This theory was brought to public Broadbent, they were the only artists to perform live on that show. attention some 15 years or so ago with the publication of the book They were straight back on the coach to Yorkshire, with no time Sync by the great mathematical populariser Steven Strogatz [1]. for an after party, because they all had proper jobs to go to in the Strogatz was also among the team of authors who published a morning. One of the reasons for this unlikely hit was the way that Nature paper that claimed the infamous wobbling of London’s Terry Wogan would sing along live on air, using Katie Moss’s Millennium Bridge could be explained by the synchronisation of lyrics. In fact, by popular request, Terry released his own sung pedestrian footsteps [2]. (However, more recent theories by my version in 1978, accompanied this time by the Hanwell Brass colleague John Macdonald at Bristol University, among others, Band. But, by then, the public had lost interest and the accompa- has cast doubt on synchronisation being the primary cause of that nied version would only reach number 21 in the pop charts. particular instability [3].) I digress. Whenever I witness mass synchronous activities, I A related phenomenon is the emergent, seemingly coordi- always wonder how people keep to time. For example, although nated, synchronous spatio-temporal behaviour that can be ob- I am not aware of any particular dance steps associated with the served in swarms of birds, fish and insects. One of the most strik- Furry Dance, most traditional English dances involve a repeated ing examples is the remarkable murmuration of starlings that can sort of shuffling, half skipping step, successively on each foot. be observed at sunset on the Somerset levels, as well as elsewhere When part of a large, unstructured procession, how does one de- at other roosting sites across the UK (see Figure 1). The mathe- cide whether to place the left or the right foot on a given beat? matics of such spatio-temporal patterns will likely form the sub- Presumably by following the person in front. ject of another Westward Ho! But for this piece, we will restrict I also recall the childhood game, in which I often participated attention to the straightforward temporal synchronisation. on the playground in primary school: follow the leader. Here the The simplest example of temporal synchronisation is the Huy- head of a small chain of children would weave in whatever di- gens’ clock phenomenon, between two natural oscillators con- rection they saw fit, perhaps lifting a sundry limb or two into the nected through weak coupling. During the 17th century Huy- air, perhaps carrying out same strange gait or hand gesture. Each gens, the inventor of the pendulum clock, observed that two such subsequent member of the chain was then obliged to copy as best devices either hanging from a beam or sat on a board between two
Mathematics TODAY JUNE 2018 99 chairs seem invariably to synchronise their phase. The same phe- nomenon extends to multiple oscillators, as I had the opportunity to witness a few years ago in an afternoon conference excursion to the Great Wall of China. A small group of mathematicians and physicists were taking shelter in the gift shop during a downpour, when we observed a table of identical maneki-neko, or ‘Chinese lucky cats’, set out in a regular array. As shown in Figure 2, we observed how the waving arms of the column of cats in the middle of the table (at the top of the image) were perfectly synchronised. Moreover we could disturb any one of these arms in any way we liked, and eventually it would fall into line with its brothers and sisters. In the figure shown, the gold coloured cat in the middle of the image has a flat battery and is completely still. The other Figure 2: Huygens’ cats. cats in the foreground appear to be only weakly synchronised to each other, and only weakly synchronised to the far column. Now, suppose we define, ψ = φ φ , and let Γ be the i − j ij To explain the synchronisation of large sets of coupled os- φi-component of the coupling force between the ith and jth os- cillators, in the 1970s, Yoshiki Kuramoto from Kyoto University cillator. Note that the variable ψ is only defined modulo 2π, and proposed a simple model for the dynamics of N weakly coupled so the function Γ must also be 2π periodic. Hence we can faith- near-identical oscillators (see [4] for the first definitive account). fully approximate Γ by its Fourier series Here, instead of modelling the second-order-in-time inertial dy- namics of each oscillator, one simply ignores their amplitude and Γ (ψ) a(1) sin(ψ + c(1)) ij ≈ ij ij describes the slow variation of the phase φi of each oscillator. (2) (2) (3) (3) The so-called Kuramoto model thus takes the form + aij sin(2ψ + cij )+aij sin(3ψ + cij )+...,
N k (k) k for some amplitude and phase constants aij 0, cij [0, 2π). φ˙i = ωi + sin(φj φi),i=1, 2,...,N, (1) ≥ ∈ N − If each orbit is near-circular, it is not hard to argue that this j=1 expansion must be dominated by its first term. We are also where ωi is the natural (uncoupled) frequency of the ith oscillator, free to choose the origin of all the phase coordinates φi. If we k ω is the coupling strength and a dot represents differentia- make the natural physical assumption that the coupling force is tion with respect to time. This model, and generalisations of it zero whenever the oscillators have the same phase, then we are k = k (1) that allow i to vary with each pairing and to include the ef- free to choose the phase coordinates such that cij =0for all fects of temporal noise, has been found to be remarkably success- 1 i, j N. Finally, if we make the simplifying assumption ful in explaining the onset of synchronisation among oscillators ≤ ≤ (1) that the coupling constant aij is the same for each oscillator (or with initially uncorrelated phases. The interesting case is when almost the same), and we set this constant value to be a1 = k/N , the ωi are sampled from a relatively narrow range of frequencies then we recover the model (1) to leading order. + ω [ω−,ω ] and the coupling strength is weak, k ω. Then, ∈ Studies of synchronisation using generalisations of the Ku- one typically finds a critical value of the coupling strength kc, ramoto model remain a fertile area of active research, for exam- such that for k
Mathematics TODAY JUNE 2018 100 In one ground-breaking laboratory study [5], seven partici- diagnosis and treatment of patients with schizophrenia [7]. It pants sat facing each other. They were told to perform a simple has been documented that schizophrenics have general impair- task of repeatedly moving their finger from side to side in regu- ment in interpersonal synchronisation. So the research team, lar motion. The collective motion was captured on film, in two together with clinical collaborators, designed a study in which conditions, either with everyone having their eyes closed, or with both healthy and schizophrenic participants interacted one-on- all having their eyes open. In the closed-eye case, the ‘natural’ one with a robot. In addition to the above cognition-inspired frequency of oscillation of each participant could be measured. control algorithm, a humanoid robot was used that was also able With eyes open, a number of different interaction topologies were to provide social cues, by smiling when synchronisation is al- tried, as shown in Figure 3. Each interaction pattern was achieved most achieved. It was found that the impaired patients, unlike the by where the participants were told to fix their gaze, aided by ad- healthy ones, had no improved performance in the presence of justable goggles that restrict the field of view and by the arrange- the social cues. The research suggests possible avenues for both ment of the chairs. The participants were told that the aim was controlled diagnosis and potential treatment of the social deficit that the whole group should try to synchronise their hand move- present within schizophrenia. ments, both in frequency and phase. But there was no designated Finally, I would like to ponder on the question of ethics, which leader. is not something we often need to consider within the mathemat- ical sciences. Clearly, any human experiments on campus need full ethical approval, and I have no doubts on both the scientific and societal benefit of the work I have just described on human motion coordination. But I don’t think it takes too much imagina- tion to consider a somewhat less positive application of research on using one or two (secretly) robotic ‘leaders’ to try to influence the behaviour of the masses. While not exactly the same phe- nomenon, it is much in the news at the moment that certain nefar- ious organisations may have sought to manipulate the outcome of elections via targeted interventions at certain ‘influential’ nodes Figure 3: Experiments in human coordination of hand move- in a social interaction network. In the era of sophisticated mathe- ments for participants interacting visually in different network matical models, automated algorithms and big data, perhaps it is topologies (a–d), achieved through arrangement of their chairs time for all of us to consider the possible ethical implications of (e–g) and instructions on where to fix their gaze. Reproduced our research. This is precisely the point argued by Cathy O’Neil from [5], under CCBY license. in her book Weapons of Math Destruction [8]. It has long been accepted that gaining a deep understanding of ethics is a key part It was found that synchronisation was more readily achieved of any degree in medicine. Ethics is also a necessary ingredient of in cases where each participant had a similar natural frequency. most professionally accredited degrees, not least in engineering. It was also found that the topology of the network of lines of sight In the age of cyber attacks, ethical education is being introduced also greatly affected the propensity to synchronise. Interestingly, to most computer science programmes. But given what is now the case of all-to-all coupling (Figure 3a) which you might intu- possible using automatically implemented mathematical models, itively expect to be more chaotic, was at least as effective as the is it also time that ethics should become a compulsory part of any case where the topology ensures a natural leader (the star graph mathematics degree? in d). Also of interest is the fact that the results of these tests could be replicated and explained, using a heterogeneous network R of Kuramoto oscillators of the form (1). 1 Strogatz, S.H. (2003) Sync: How Order Emerges from Chaos in The Another fascinating question addressed in the same EU Universe, Nature and Daily Life, Hyperion, New York. project was the extent to which synchronised motion can occur if one or more of the humans were replaced by a computer that 2 Strogatz, S.H., Abrams, D.M., McRobie, A., et al. (2005) Crowd syn- is attempting, through automatic control algorithms, to mimic chrony on the Millennium Bridge, Nature, vol. 438, pp. 43–44. the actions of a human [6]. Although standard feedback con- 3 Macdonald, J.H.G. (2009) Lateral excitation of bridges by balancing trol algorithms could be used, the aim was to design a controller pedestrians, Proc. R. Soc. Lond. A, vol. 465, pp. 1055–1073. that felt human-like, including the kind of perception-cognition 4 Kuramoto, Y. (1984) Chemical Oscillations, Waves, and Turbulence, pathway delays that a human would display. The solution was to Springer, Berlin. use model-reference control with a (more sophisticated form of) Kuramoto-like oscillator as the reference. It was found that en- 5 Alderisio, F., Fiore, G., Salesse, R.N., et al. (2017) Interaction pat- terns and individual dynamics shape the way we move in synchrony, sembles containing one or more such computer participants (im- Sci. Rep., vol. 7, p. 6846. plemented through dots moving on a screen) could be made to perform similarly to entirely human groups. 6 Alderisio, F., Lombardi, M., Fiore, G. and di Bernardo, M. (2017) A One possible application of this technology is in rehabilita- novel computer-based set-up to study movement coordination in hu- Front. Psychol. tion robotics. Often physiotherapy following injury involves a man ensembles, , vol. 8, art. no. 967. patient repeatedly copying the movements of a therapist who is 7 Cohen, L., Khoramshahi, M., Salesse, N.R., et al. (2017) Influence of subtly interacting with them, adjusting to their specific impair- facial feedback during a cooperative human-robot task in schizophre- ment. Could a robot be trained to perform the role of the thera- nia, Sci. Rep., vol. 7, p. 15023. pist, thus extending the time available for such treatment? A per- 8 O’Neil, C. (2016) Weapons of Math Destruction: How Big Data In- haps more surprising motivation for the EU project was in the creases Inequality and Threatens Democracy, Crown, New York.
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