In each experiment, participants moved freely within the arena for three 40-s intervals interspersed with two phases in which they were asked to navigate to a designated point. These distinct movement phases are clearly identifiable by analysing the temporal evolution of the average speed of the group (inset of Fig.\u00a02<\/a>, upper panel). Free movement periods (highlighted in colour) display a notably higher speed than the beginning of the experiment or the directed movement phases (in grey). To quantify the directionality of rotation, we employed the polarization parameter M defined at each time step as the average of the individual polarizations mi(t). The latter are computed as \\({m}_{i}(t)={\\widehat{v}}_{i}(t)\\cdot {\\widehat{e}}_{i}^{\\varphi }(t),\\) where \\({\\widehat{v}}_{i}(t)\\) is the normalized velocity vector of pedestrian i and \\({\\widehat{e}}_{i}^{\\varphi }(t)\\) is the azimuthal unit vector relative to a central point18<\/a>. A complete description of this measure and its variants is provided in the \u201cMethods\u201d. An example of the temporal evolution of M(t) is depicted in the inset of Fig.\u00a02<\/a> (lower panel). To quantify the system\u2019s net rotational tendency, we computed the time-averaged polarization \\(\\overline{M}\\) during each interval of free motion. \\(\\overline{M} > 0\\) corresponds to CCW motion, whereas \\(\\overline{M} < 0\\) indicates clockwise (CW) motion.<\/p>\n Fig. 2: Collective rotational behaviour on confined random motion in Spain.<\/b> The panels show the probability density functions (PDFs) of the collective polarization values (M), for groups with different numbers of participants: a<\/b> 16, b<\/b> 24, and c<\/b> 32. Different colours (see legend) are used for crowds with different percentages of right-turners (%RT) and for the case with only Left-Dominant (LD) pedestrians. The black line represents the aggregated distribution obtained by combining data from all experimental conditions. Inset: time series of the average speed of all participants (top) and the collective polarization (bottom). The values used to generate the PDFs are the ones marked on red. Intervals covering the initial stage of the experiment and periods of directed motion towards the walls (grey in the inset), were identified by analysing the average speed, and excluded from the analysis.<\/p>\n As explained, it was expected that increasing the proportion of right-turners in the experiment would favour CW rotation. But the results revealed that neither the number of participants nor the proportion of right-turners significantly influenced \\(\\overline{M}\\). Instead, across all experimental conditions, \\(\\overline{M}\\) consistently exhibited a positive value around \\(\\overline{M} \\sim 0.2\\) (Fig.\u00a01<\/a>d, orange), indicating a robust and persistent CCW bias. In this sense, it is noteworthy that even experiments A1 and A11 (in which 100% of pedestrians were right-turners and left-handed, respectively) revealed a similar, positive value of \\(\\overline{M}\\).<\/p>\n To further understand this observation, we analysed the probability density functions of M (Fig.\u00a02<\/a>). Interestingly, regardless of the global density (increasing from a to c) and the proportion of right-turners, all distributions are shifted towards positive values and are unimodal, with the peaks centred at M\u00a0~\u00a00.25. The low proportion of M\u00a0<\u00a00 values implies that the system maintains a constant CCW rotation. At the same time, the absence of values at M\u00a0~\u00a01 indicates that the CCW motion is not a global effect involving all pedestrians. Interestingly, the distributions become narrower as the number of pedestrians increases, which may suggest the existence of a collective effect that boosts the stability and robustness of the CCW rotation. More importantly, the overlap of all distributions obtained with the same number of pedestrians but different proportions of right-turners (especially for large crowds) indicates that individual turning preferences have a negligible impact on the emergence of CCW behaviour.<\/p>\n Next, aiming to elucidate the actual role of boundaries in the development of CCW motion, we analysed the spatial distributions of density, velocity, and polarization within the arena (Fig.\u00a03<\/a>). The density fields (first row, Fig.\u00a03<\/a>) show a rather homogeneous spatial distribution, yet some faint circular patterns can be perceived. These suggest that the position of the boundaries (or pedestrians\u2019 perception of them) affects the motion within the arena. The velocity fields (second row, Fig.\u00a03<\/a>) reveal that the CCW motion extends over the whole arena but is slightly more pronounced near the boundaries. This is further confirmed by examining the polarization fields (third row, Fig.\u00a03<\/a>). On average, regions coded in blue are more abundant over the whole arena, but the colours are more intense near the boundaries; hence suggesting a possible role of those in the development of CCW rotation.<\/p>\n Fig. 3: Density, velocity and polarization fields for varying proportions of right-turners in the Spanish confined scenario.<\/b> Spatial distribution of temporally averaged density \\({\\overline{\\rho }}_{r}\\) (first row), velocity \\({\\overline{\\vec{v}}}_{r}\\) (second row), and polarization \\({\\overline{M}}_{r}\\) (third row) fields for a crowd of 16 pedestrians with different turning preferences as indicated at the top. The colour scales on the right (same for all cases) indicate (i) the average local density in persons\/m2(a\u2013d<\/b>); (ii) the average speed in m\/s (e\u2013h<\/b>); and (iii) the average local polarization value (i\u2013l<\/b>). In (e\u2013h<\/b>), the arrows indicate the average direction of the local velocity vector. The spatial units in both the vertical and horizontal directions are metres for all plots.<\/p>\n Boundary-free experiment in a schoolyard<\/p>\n From previous results, and in order to clarify whether the boundaries really trigger the CCW motion or just help to stabilise (and perhaps magnify) it, we designed a follow-up experiment in which pedestrians walked in an open and practically unconstrained setting (Fig.\u00a01<\/a>b). This consisted of a 50\u00a0\u00d7\u00a060\u2009m2 schoolyard in Spain, where over one hundred teenage students were gathered (see \u201cMethods\u201d for details). Surprisingly, despite the influence of boundaries being practically suppressed, the CCW rotation persisted, as reflected by the positive value of \\(\\overline{M}\\) depicted in Fig.\u00a01<\/a>d (red). In agreement with this, the analysis of the PDF(M) reveals again a unimodal distribution shifted towards positive values (Fig.\u00a04<\/a>a). Interestingly, the PDF(M) is even narrower than the ones presented in Fig.\u00a02<\/a>, suggesting that the variable controlling the width of the distribution is the total number of pedestrians, not the density\u2014which in this case is 6 times lower than in the sparser experiments of the first scenario.<\/p>\n Fig. 4: Probability density function (PDF) of collective polarization values (M) in different scenarios.<\/b> In (a<\/b>) the boundary-free motion of teenagers in Spain. In b<\/b>, c<\/b> the confined motion in Japan. In d<\/b> the children\u2019s motion in a Japanese nursery school. In each panel, colours are used to label different experimental conditions, as described in the legends. In a<\/b>, only one experimental condition is considered. In b<\/b>, each curve corresponds to a different crowd size. In c<\/b>, both the percentage of right-turners (%RT) and the group size (12, 24) varies. In d<\/b>, four realizations with different children of slightly different age are reported.<\/p>\n Confined random motion in Japan<\/p>\n After ruling out the pedestrian-boundary interactions as the origin of the CCW rotation, we focused on pedestrian-pedestrian interactions as a potential driving mechanism. Pedestrian-pedestrian interactions are key to various self-organising behaviours, with lane formation in bidirectional flows being a prime example4<\/a>,19<\/a>,20<\/a>. This process arises from local coordination, where individuals adjust their paths during head-on encounters to avoid collisions21<\/a>. In Spain (and most European countries), this avoidance manoeuvre is typically implemented by moving towards the right-hand side13<\/a>,22<\/a>, hence leading to the symmetry breaking in lane formation. After this fact, the hypothesis was that if pedestrians avoid collisions by stepping to the right side (thus leaving the incoming person to the left), in the circular arena they would end up moving CCW near the boundary walls.<\/p>\n To test this idea, we conducted experiments in Japan, a country where lanes in bidirectional flows conspicuously appear on the left side, as pedestrians generally avoid others by stepping to the left. First, we confirmed this left-side stepping tendency through a questionnaire where participants indicated their natural avoidance direction when viewing corridor walking images (see Supplementary Note\u00a01<\/a> for details). We then performed new experiments in an enclosure similar to the one used in Spain (Fig.\u00a01<\/a>c), following the same methodology. Unexpectedly, the positive values of M reported in Fig.\u00a01<\/a>d reveal that the CCW motion persisted, hence refuting the idea that the stepping aside pedestrian manoeuvres were behind the collective development of CCW motion. Indeed, \\(\\overline{M} > 0\\) in all experimental trials but one (C9 in which \\(\\overline{M}\\approx 0\\)), an exception that we attribute to the intrinsic variability of human behaviour. Furthermore, as already observed in experiments 1 and 2, the distributions of M (Fig.\u00a04<\/a>b, c) remain skewed towards positive values, and the peak (at about M\u00a0~\u00a00.2) is more pronounced as the number of pedestrians in the arena increases; i.e., the fluctuations of M are smaller as the population size grows.<\/p>\n Random motion in a nursery school<\/p>\n We then addressed the question of whether social rules or learned behaviours\u2014potentially shaped by sporting events like athletics or other learned behavioural habits\u2014might be the cause of the CCW collective motion. To explore this option, we analysed experiments conducted in a nursery school, as previous research has demonstrated that young children differ from adults in their emergent movement patterns23<\/a>, and it can reasonably be assumed that they are less influenced by acquired adult conventions (e.g. signage norms or circulation habits). In these experiments, conducted by Ichikawa et al.17<\/a>, children (about 5 years old) participated in an eurhythmics activity involving free running (see \u201cMethods\u201d for more details). Interestingly, the CCW motion not only develops as in previous scenarios, but it becomes much more pronounced, as revealed by the higher values of \\(\\overline{M}\\), systematically above \\(\\overline{M}=0.7\\) (Fig.\u00a01<\/a>d, green). This behaviour is corroborated by the distributions of M (Fig.\u00a04<\/a>d), which show a noticeable peak near M\u00a0~\u00a01 that indicates a highly consistent and stable vortex-like motion, with all children moving in unison. This suggests that children, at least in this specific activity, tend to imitate their peers and end up walking in the same direction, which is, of course, the CCW one. This is consistent with prior research showing that young children are highly sensitive to peer consensus, prone to over-imitation, and tend to align more strongly with others during rhythmic or musical group activities24<\/a>,25<\/a>,26<\/a>,27<\/a>.<\/p>\n Social norm elicitation<\/p>\n In pedestrian dynamics, it is known that social influence is behind the observed behaviour in many different instances (for examples on the social influence or the emergence of social norms, see refs. 28<\/a>,29<\/a>,30<\/a> for non-emergencies and31<\/a> for emergency scenarios). Therefore, to investigate the possibility that unknown social norms were behind the emergence of CCW motion, we used the notion of social norm introduced by Bicchieri32<\/a>, which arises from the consideration of the expectations of people about a given situation. Two kinds of expectations are taken into account. Empirical expectations (often referred to as descriptive norms33<\/a>) correspond to what individuals think others in their reference group will do when faced with the situation of interest. Normative expectations (also called injunctive norms) refer to what individuals think the rest of their reference group expects them to do. Normative expectations are generally accompanied by the assumption that if individuals do not conform to the expected behaviour, they will be sanctioned or punished in a number of different ways. In this framework, we say that a social norm in a group exists if the majority of people in the group share common empirical and normative expectations, and the two types agree on the behaviour to be followed. Expectations are then elicited by means of a questionnaire34<\/a>. In our case, this test was composed of three different questions which allowed us to identify the personal beliefs (Q1), the empirical expectation (Q2), and the normative expectation (Q3) of participants (see Methods). This survey was performed in Spain with a group of 168 participants.<\/p>\n The results of this study are presented in Fig.\u00a05<\/a>. Panel (a) illustrates the hypothetical scenario shown to the survey respondents, while panel (b) summarizes their responses. As can be seen, if a social norm is indeed present, it would surprisingly be to move CW: nearly 40% of respondents exhibited aligned empirical and normative expectations in the CW direction (i.e., Q2 and Q3 both indicated CW) and also reported a personal inclination to move CW (Q1), while another 15% of the participants also shared those expectations even if they would move in the CCW direction. This must be compared to roughly 20% of respondents who expect to move CCW, while approximately 25% provided conflicting answers. Therefore, we must conclude that a clear norm does not exist, but in case we would accept slightly more than the majority\u2019s expectations as a norm, it surprisingly would go against the observed behaviour.<\/p>\n Fig. 5: Influence of social norms.<\/b> a<\/b> Photograph used in the survey, where participants responded to three questions (Q1, Q2, Q3) about the direction of rotation they would choose. See Methods for the complete survey form. b<\/b> Proportion of responses to the survey questions. Answers, limited to CW or CCW, are grouped into three categories: (i) The same answer for all three questions (all CW or all CCW), indicating a strong influence of social norms, (ii) The same answers for Q2 and Q3 but different from Q1, suggesting a moderate influence of social norms, and (iii) Mixed answers, reflecting a weak influence of social norms.<\/p>\n The above result has interesting cognitive implications. In most social interaction contexts, people follow explicit and implicit norms that tell them how to behave in those situations, typically coming from moral, ethical, legal regulations, conventions or shared social norms. Explicit norms are a direct result of codified rules (e.g., laws or regulations), whereas implicit norms are situation-specific and act without conscious awareness. The norm in place emerges through interactions between people and the environment, potentially resulting in a situation where there is a contrast between the norm eventually adopted by people and what is expected through explicit rules. Our experiment shows that turning direction is likely not determined by explicit rules, as confirmed through the results from the questionnaire contrasting what is expected from codified motion rules. One could then conclude that the CCW turning norm (if there is any) is more likely implicit, consequently acting with participants not being aware of it. However, although the explicit vs. implicit framework allows to examine the observed CCW turning behaviour from a more systematic perspective, the mechanisms leading to this specific norm (or behavioural repertoire, to use an alternative expression) are still unclear and would deserve an in-depth investigation.<\/p>\n Individual behaviour<\/p>\n Thus far, the analysis of collective polarization M across different experiments has demonstrated the consistency and robustness of the CCW rotation effect. Moreover, the distributions of M revealed that this effect persists over time, with fluctuations around the average being dependent on the total number of pedestrians and not on the density of them. More importantly, the absence of values at M\u00a0~\u00a01 in all the systems but in the Japanese nursery school, indicates that the CCW motion is not a global effect involving all pedestrians. This seems reasonable, as pedestrian behaviour exhibits inherent variability and, although on average the collectivity is always rotating CCW, there might be individuals moving in the opposite direction.<\/p>\n To quantify this, we took advantage of our experimental capabilities, which enable precise tracking of each pedestrian and analysed individual behaviour using the individual polarization parameter mi. Unlike M, which captures collective motion, mi quantifies each pedestrian rotation pattern, providing insight into the individual behaviour. As an example, in Fig.\u00a06<\/a>a\u2013d we illustrate four typical trajectories from the Japan experiment together with their corresponding PDF(mi) (Fig.\u00a06<\/a>e\u2013h). Figure\u00a06<\/a>a displays a stable CCW trajectory characterized by a unimodal and sharply skewed distribution that peaks near mi\u00a0~\u00a01; in much the same way, Fig.\u00a06<\/a>b corresponds to a stable CW trajectory. Figure\u00a06<\/a>c exemplifies another type of pedestrian behaviour in which the rotating direction changes during the experiment (in this case, it changes twice). Accordingly, the PDF(mi) shows a bimodal distribution with two marked peaks at mi\u00a0~\u00a01 and mi\u00a0~\u00a0\u2212\u00a01. Finally, Fig.\u00a06<\/a>d shows a scenario in which the pedestrian rotates, but also performs a number of straight paths that give rise to more values of mi different from \u00a0\u00b1\u00a01, and therefore to a broader distribution.<\/p>\n Fig. 6: Four representative cases of individual rotational behaviour during confined motion in Japan.<\/b> a\u2013d<\/b> Individual trajectories of four different pedestrians over 40\u2009s coloured according to the instantaneous individual polarization value (mi) (see colour bar on the right). Green arrows indicate the direction of motion at the start and end of each trajectory. Spatial units in both the vertical and horizontal directions are metres. e\u2013h<\/b> The corresponding probability density functions (PDFs) of mi for each trajectory.<\/p>\n Considering the particularities of these distributions and aiming to reflect the individual behaviour using a single parameter, we computed the time-averaged individual polarization (\\(\\overline{{m}_{i}}\\)) for each pedestrian. In this way, \\(\\overline{{m}_{i}} \\sim 1\\) corresponds to pedestrians walking always CCW, \\(\\overline{{m}_{i}} \\sim -1\\) corresponds to pedestrians walking always CW, while intermediate values (and particularly, those close to \\(\\overline{{m}_{i}} \\sim 0\\)) reflect both pedestrians that change rotating direction as in Fig.\u00a06<\/a>c and those performing straight trajectories as in Fig.\u00a06<\/a>d. In Fig.\u00a07<\/a> we represent the distributions of \\(\\overline{{m}_{i}}\\) for all pedestrians that participated in each experiment (note that, for each experiment, we combined the results obtained in different conditions). Remarkably, in all cases the distributions show a notable peak at \\(\\overline{{m}_{i}} \\sim 1\\), revealing the presence of a number of people determinedly walking CCW, no matter the specific conditions at which the experiment was implemented. Also, the distributions suggest the existence of an analogous peak at \\(\\overline{{m}_{i}} \\sim -1\\), but this is in general less prominent and altogether absent in the case of the nursery school experiments.<\/p>\n Fig. 7: Average polarization values for each pedestrian in the four studied scenarios.<\/b> Each panel includes a stripchart at the bottom, showing the time-averaged polarization values for each pedestrian (\\({\\overline{m}}_{i}\\)) and the corresponding probability density function (PDF) of the whole set of these \\({\\overline{m}}_{i}\\) values at the top. a<\/b> Confined motion in Spain, b<\/b> boundary-free motion of teenagers in Spain, c<\/b> confined motion in Japan, and d<\/b> kids motion in a Japanese nursery school.<\/p>\n Overall, Fig.\u00a07<\/a> indicates substantial individual variability in rotational behaviour. Despite this variability, in all experiments there is an important proportion of pedestrians exhibiting a determined preference for CCW rotation. Notably, this behaviour at the individual level helps to explain the main features reported for the collective polarization parameter M. In this way, the consistent positive values of M can be justified by the presence of a larger proportion of pedestrians moving CCW than CW. Similarly, the absence of a peak at M\u00a0~\u00a01 can be explained by the intrinsic variability of the pedestrian type of motion; with the exception of kids, in all cases there will be people walking CW or straight. Also, the findings reported in Fig.\u00a07<\/a> suggest that the correlation between the sharpness of the PDF(M) and the crowd size is merely a statistical effect. When the crowd is small, each value of M is computed using a small number of values of mi, and then the fluctuations increase just for statistical reasons.<\/p>\n Beyond this remarkable correlation between the macroscopic behaviour and the individual one, the results of Fig.\u00a07<\/a> suggest that the prevalent preference for CCW rotation is not a collective effect but an individual one. Interestingly, this hypothesis is supported by the fact that the distribution with the sharpest peak at m\u00a0~\u00a01 occurs for the scenario in which pedestrians move with more freedom; i.e. the teenagers walking in a space free of boundaries (Fig.\u00a07<\/a>b).<\/p>\n Individual motion<\/p>\n Aiming to confirm that the CCW motion symmetry breaking is not caused by a collective effect but a result of individual preferences of motion, we implemented a new set of experiments in which over 200 participants walked alone (one at a time) in an enclosed arena (Fig.\u00a08<\/a>a). In these new tests, we looked for a connection between this hypothetical CCW motion preference of the individuals and some biological features, such as handedness, footedness, or eye dominance. To this end, each participant was asked about their dominant hand, foot, and eye (left or right). If they were unsure, dominance was determined through a series of performance tests (see Methods for details). Participants who showed no clear dominance (i.e., were ambidextrous or had indeterminate eye preference) were excluded from the analysis. Furthermore, 49 participants were asked to walk with a patch covering the right eye, a strategy that was aimed at evaluating whether artificially constraining the right visual field could have a significant effect on the rotational bias.<\/p>\n Fig. 8: Rotational motion of individuals.<\/b> a<\/b> Snapshot of the last experimental setup, where individual pedestrians were instructed to walk alone and freely within an enclosure. The recent trajectory followed by the participant is shown in orange, with the current position marked in red. For image rights purposes, the pedestrian has been removed from this snapshot; only the arena background and the superimposed trajectory are displayed. b<\/b> The trajectories of two pedestrians over a 60-s period are depicted, with colours representing their instantaneous individual polarization mi, as indicated by the colour scale above. On the right, the probability density functions (PDFs) of mi for each trajectory are shown. c<\/b> Probability density function (PDF) of the individual time-averaged polarization values (\\({\\overline{m}}_{i}\\)). d<\/b> Synthetic PDF of the collective polarization \\(\\widetilde{M}\\), constructed by aggregating instantaneous individual polarization values mi randomly selected from different pedestrians at arbitrary times. Note that \\(\\widetilde{M}\\) is not a genuine collective measure, but rather a synthetic construct designed to emulate its statistical properties. e<\/b> Box plots of \\({\\overline{m}}_{i}\\) of the data grouped by individual features: handedness preference (Hand-Pref), footedness preference (Foot-Pref), eye dominance (Eye-Pref), sex, and whether the right eye is patched or not (Right-Eye Patch). Note that the `Yes’ group corresponds to all participants wearing a patch, including both left- and right-eye dominant individuals. Each box depicts the median together with the interquartile range (IQR), while the whiskers extend to the most extreme values lying within 1.5\u2009IQR. Outliers are not displayed; instead, each dot represents an individual data point associated with a pedestrian. The number in parentheses above each category indicates the sample size.<\/p>\n For each pedestrian, we extracted the complete trajectory within the arena (left panels of Fig.\u00a08<\/a>b) and obtained the instantaneous individual polarization mi(t). Then, we computed the probability density function of the polarization values of each individual. In the right panels of Fig.\u00a08<\/a>b, we show two examples corresponding to a pedestrian who is consistently walking CCW (top panels) and a pedestrian with several changes in the rotation direction (bottom panels). From these distributions, we calculated \\(\\overline{{m}_{i}}\\) for each pedestrian, and then built the distributions of PDF(\\(\\overline{{m}_{i}}\\)) (as in Fig.\u00a07<\/a>) by considering all participants, irrespective of their condition. Clearly, the distribution exhibits a pronounced peak near \\(\\overline{{m}_{i}} \\sim 1\\), much higher than the one at \\(\\overline{{m}_{i}} \\sim -1\\). This result provides evidence that the origin of the CCW motion is not at the crowd level, but at the individual one. Interestingly, the distribution also presents a peak for values of \\(\\overline{{m}_{i}}\\) slightly greater than 0 that is more prominent than for the individuals moving within a crowd (Fig.\u00a07<\/a>). We speculate that this might be related to psychological aspects as moving in an empty space with no other pedestrians might become unengaging, hence provoking the change in the rotation direction of pedestrians as in Fig.\u00a08<\/a>b bottom panels. Overall, the key result of Fig.\u00a08<\/a>c is that the CCW asymmetry exists at the individual level. To statistically assess this bias, we performed a two-tailed one-sample Wilcoxon signed-rank test against 0: z\u00a0=\u00a0\u2212\u00a05.63, n\u00a0=\u00a0156, P < 0.001, \u2223r\u2223 = 0.45. The median \\(\\overline{{m}_{i}}\\) was positive (bootstrap 95% CI for the median: 0.12\u20130.27), indicating that the median differs statistically from 0 and supporting the presence of a CCW bias at the individual level.<\/p>\n Next, we grouped the data according to pedestrian particularities such as handedness, footedness, eye dominance, and sex. Also, we discriminated the pedestrians who were asked to use a patch over their right eye. As shown in the box plots of \\(\\overline{{m}_{i}}\\) in Fig.\u00a08<\/a>e, the CCW bias remains consistent across all subgroups. Two-tailed Mann\u2013Whitney U-tests showed no statistically significant differences in \\(\\overline{{m}_{i}}\\) between right- and left-handed participants (U(142,\u00a014)\u00a0=\u00a0898, z\u00a0=\u00a0\u2212\u00a00.60, P\u00a0=\u00a00.554, r\u00a0=\u00a0\u2212\u00a00.05); the Hodges\u2013Lehmann (HL) estimate of the location shift (Right\u00a0\u2212\u00a0Left) was \u00a0\u2212\u00a00.05 with a bootstrap 95% CI of [\u00a0\u2212\u00a00.34,\u00a0\u20090.18]. Analogously, no statistically significant differences were found between right- and left-footed participants (U(138,\u00a018)\u00a0=\u00a01134.5, z\u00a0=\u00a0\u2212\u00a00.60, P\u00a0=\u00a00.553, r\u00a0=\u00a0\u2212\u00a00.05; HL Right\u00a0\u2212\u00a0Left: \u00a0\u2212\u00a00.04, 95% CI [\u00a0\u2212\u00a00.21,\u00a0\u20090.09]), between right- and left-eyed participants (U(96,\u00a060)\u00a0=\u00a02851, z\u00a0=\u00a0\u2212\u00a00.11, P\u00a0=\u00a00.917, r\u00a0=\u00a0\u2212\u00a00.01; HL Right\u00a0\u2212\u00a0Left: 0.00, 95% CI [\u00a0\u2212\u00a00.15,\u00a0\u20090.12]), or between male and female participants (U(62,\u00a094)\u00a0=\u00a02542, z\u00a0=\u00a0\u2212\u00a01.35, P\u00a0=\u00a00.178, r\u00a0=\u00a0\u2212\u00a00.11; HL Male\u00a0\u2212\u00a0Female: \u00a0\u2212\u00a00.08, 95% CI [\u00a0\u2212\u00a00.24,\u00a0\u20090.03]). Likewise, the comparison between participants with and without a right-eye patch showed no statistically significant difference in \\(\\overline{{m}_{i}}\\) (U(156,\u00a049)\u00a0=\u00a04385, z\u00a0=\u00a01.55, P\u00a0=\u00a00.120, r\u00a0=\u00a00.11; HL No Patch\u00a0\u2212\u00a0Patch: 0.10, 95% CI [\u00a0\u2212\u00a00.03,\u00a0\u20090.25]). While these results indicate a consistent CCW bias regardless of these factors, we note that the sample sizes of certain subgroups\u2013particularly left-handed (n\u2009=\u200914) and left-footed (n\u2009=\u200918) individuals\u2013were relatively small, limiting statistical power. Therefore, although the trend is robust, we cannot entirely exclude the possibility of weak or subtle effects. Together, these results support the hypothesis that the CCW motion bias arises from individual locomotion trends rather than group-level phenomena. We found no statistically significant evidence that this intrinsic breaking of symmetry depends on the laterality-related biological features considered in this study.<\/p>\n Going a step further, we try to connect the individual motion results with the values of collective polarization (M mostly around 0.2 in Fig.\u00a01<\/a>) as well as their distributions, where we observed a clear narrowing of the peak with the crowd size (Figs.\u00a02<\/a>, 4<\/a>). To this end, in Fig.\u00a08<\/a>d we synthetically build the PDFs of M by taking the actual values of m from the individual pedestrians moving alone (as shown in Fig.\u00a08<\/a>b) in a random manner. In particular, for each crowd size, we randomly select a subset of individuals and take one polarization measurement from each to calculate the hypothetical group-average polarization, \\(\\widetilde{M}\\). We repeat this process for 1000 subsets, obtaining a statistically robust distribution of \\(\\widetilde{M}\\) values as shown in Fig.\u00a08<\/a>d. Importantly, from data obtained for pedestrians walking alone, we obtain synthetic distributions of global polarization that peak at \\(\\widetilde{M} \\sim 0.25\\) and become systematically narrower as the crowd size increases; exactly as it happened with real experimental distributions of collective polarization (Figs.\u00a02<\/a>, 4<\/a>). This finding corroborates the idea that the individual preferences of motion are likely the most important features observed at the collective level.<\/p>\n","protected":false},"excerpt":{"rendered":"In this section, we present our results on the statistical properties of motion observed in each experiment. 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