Fractal Patterns in Social, Institutional, Personal Change

Interpersonal coordination represents a very common phenomenon in daily-life activities. Three theoretical frameworks have been proposed to account for synchronization processes in such situations: the information processing approach, the coordination dynamics perspective, and the complexity matching effect. On the basis of a theoretical analysis of these frameworks, we propose three statistical tests that could allow to distinguish between these theoretical hypotheses: the first one is based on multifractal analyses, the second and the third ones on cross-correlation analyses. We applied these tests on series collected in an experiment where participants were instructed to walk in synchrony. We contrasted three conditions: independent walking, side-by-side walking, and arm-in-arm walking. The results are consistent with the complexity matching hypothesis.

The complexity matching effect supposes that synchronization between complex systems could emerge from multiple interactions across multiple scales, and has been hypothesized to underlie a number of daily-life situations. Complexity matching suggests that coupled systems tend to share similar scaling properties, and this phenomenon is revealed by a statistical matching between the scaling exponents that characterize the respective behaviors of both systems. However, some recent papers suggested that this statistical matching could originate from local adjustments or corrections, rather than from a genuine complexity matching between systems. In the present paper we propose an analysis method based on correlation between multifractal spectra, considering different ranges of time scales. We analyze several datasets collected in various situations (bimanual coordination, interpersonal coordination, walking in synchrony with a fractal metronome). Our results show that this method is able to distinguish between situations underlain by genuine statistical matching, and situations where statistical matching results from local adjustments.

Resumen El documento describe algunos errores y dificultades en relación con el objeto matemático dimensión resultado de una experiencia investigativa del grupo fractales DMA! UPN. Se pretende mostrar la complejidad del concepto de dimensión debido, entre otras ...

Los fractales han sido y son una herramienta muy útil para describir la distribución de galaxias en el Universo. En este artículo se hace un breve resumen de la teoría de fractales y su aplicación a la distribución de galaxias observada.Proyecto Alhambra. Estudio Sistemático de la Evolución Cósmica. (C-Consolider, AYA2006-14056

Research regarding interpersonal coordination can be traced back to the early 1960s when video recording began to be utilized in communication studies. Since then, technological advances have extended the range of techniques that can be used to accurately study interactional phenomena. Although such a diversity of methods contributes to the improvement of knowledge concerning interpersonal coordination, it has become increasingly difficult to maintain a comprehensive view of the field. In the present article, we review the main capture methods by describing their major findings, levels of description and limitations. We group them into three categories: video analysis, motion tracking, and psychophysiological and neurophysiological techniques. Revised evidence suggests that interpersonal coordination encompasses a family of morphological and temporal synchronies at different levels and that it is closely related to the construction and maintenance of a common social and affective sp...

Resumen El documento describe algunos errores y dificultades en relación con el objeto matemático dimensión resultado de una experiencia investigativa del grupo fractales DMA! UPN. Se pretende mostrar la complejidad del concepto de dimensión debido, entre otras ...

URL del artículo en la web de la Revista: https://www.upo.es/revistas/index.php/ripp/article/view/1878En el siguiente texto intentaré usar las imágenes que se derivan de las diferentes concepciones de la geometría, para ver espacios de acción social. Mientras en el espacio euclídeo se trabaja desde formas suaves, perfectas, planas, en la geometría fractal las formas son rugosas, cambiantes. Intentaremos traducir estas dos versiones a los hechos sociales, para ver como el espacio fractal nos puede servir para adquirir perspectivas más abiertas. Para ello explicaremos porque hemos considerado el imaginario matemático importante, resaltaremos los caracteres más importantes de la geometría fractal y euclidiana, para concluir con la crítica que nos surge desde el imaginario fractal.In the following text we try to use images that comes from different geometry conceptions to see social action spaces. Meanwhile,in the Euclidean space are working from smooth forms, perfects, plains; in Fract...

Fractals provide an innovative method for generating three-dimensional (3D) images of real-world objects by using computational modelling algorithms based on the imperatives of self-similarity, scale invariance, and dimensionality. Images such as coastlines, terrains, cloud mountains, and most interestingly, random shapes composed of curves, sets of curves, etc. present a multi-varied spectrum of fractals usage in domains ranging from multicoloured , multi-patterned fractal landscapes of natural geographic entities, image compression to even modelling of molecular ecosystems. Fractal geometry provides a basis for modelling the infinite detail found in nature. Fractals contain their scale down, rotate and skew replicas embedded in them. Of the many different types of fractals that have come into limelight since their origin, the fractals like the Koch curve, the Cantor Set, the Fern leaf, the Sierpinski gasket have eluded both mathematicians and computer scientists alike. And the two-dimensional (2D), 3D, etc. versions of the same have been realized based on the starting axioms/generators. This paper explains the generation of 3D versions of the above mentioned fractals that gives a real-world look and feel in the world of fractal images.

We report on a novel use of parallel coordinates as a pedagogical tool for illustrating the non-intuitive properties of high dimensional spaces with special emphasis on the phenomenon of Curse of Dimensionality. Also, we have collated what we believe to be a representative sample of diverse approaches that exist in literature to conceptualize the Curse of Dimensionality. We envisage that the paper will have pedagogical value in structuring the way Curse of Dimensionality is presented in classrooms and associated lab sessions.

Resumen El documento describe algunos errores y dificultades en relación con el objeto matemático dimensión resultado de una experiencia investigativa del grupo fractales DMA! UPN. Se pretende mostrar la complejidad del concepto de dimensión debido, entre otras ...

The essay develops a case study about a young boy playing with a toy train to address neoliberalism's problematic discourse that depicts learning as instrumental, as something that can be caused by teaching. This paper's perspective is enactive, taking the view that central to understanding learning is not the mind or brain working in isolation but involves the interrelationships between mind/brain, body, and world. The analysis revolves around the standing gap between teaching and learning, where navigating the gap involves a dynamic called 'the moment of study.' Three of Tyson Lewis's ideas about study-body gesture, time, and play-are used to explain the moment of study. The paper argues that a learner traverses the gap between teaching and learning through a body gesture of hesitation, during which there is a temporal turning away from the familiar and towards new possibilities. Traversing the gap occurs through the risk of improvisational play, propelling a forward movement in the face of not knowing how to go on. This depiction of such traversing, shown to be central to the activity of learning, undermines the causal account of learning associated with neoliberalism.

The estimation of intrinsic dimensionality of high-dimensional data still remains a challenging issue. Various approaches to interpret and estimate the intrinsic dimensionality are developed. Referring to the following two classifications of estimators of the intrinsic dimensionalitylocal/global estimators and projection techniques/geometric approaches-we focus on the fractalbased methods that are assigned to the global estimators and geometric approaches. The computational aspects of estimating the intrinsic dimensionality of high-dimensional data are the core issue in this paper. The advantages and disadvantages of the fractal-based methods are disclosed and applications of these methods are presented briefly.

In his first Epilogue to Naven, Bateson commented that “the writing of this book has been an experiment, or rather a series of experiments, in methods of thinking about anthropological materials . .  .  The present essay, in a very minor key, is a second-order experiment on that classic.  Here I employ fictional writing and the incorporation of ideas, notably those of William Burroughs, in a critical inquiry into the foundations of a canon in American anthropology.

Starting with Edmund Husserl’s seminal text[7] we would like to explore in this paper how musical perception is based on the conscious reduction of acoustic and spatiotemporal diversity into unity. In his lectures Husserl exemplifies his views with an analysis about the perception of a single sound and a sequence of sounds within a melody. They deliver an informative basis for us in order to rethink the perception of musical time and the perception of rhythms in particular. We demonstrate the impact of his phenomenology on music analysis and composition.

En el siguiente texto intentare usar las imagenes que se derivan de las diferentes concepciones de la geometria, para ver espacios de accion social. Mientras en el espacio euclideo se trabaja desde formas suaves, perfectas, planas, en la geometria fractal las formas son rugosas, cambiantes. Intentaremos traducir estas dos versiones a los hechos sociales, para ver como el espacio fractal nos puede servir para adquirir perspectivas mas abiertas. Para ello explicaremos porque hemos considerado el imaginario matematico importante, resaltaremos los caracteres mas importantes de la geometria fractal y euclidiana, para concluir con la critica que nos surge desde el imaginario fractal.

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Face-to-face conversations are central to human communication and a fascinating example of joint action. Beyond verbal content, one of the primary ways in which information is conveyed in conversations is body language. Body motion in natural conversations has been difficult to study precisely due to the large number of coordinates at play. There is need for fresh approaches to analyze and understand the data, in order to ask whether dyads show basic building blocks of coupled motion. Here we present a method for analyzing body motion during joint action using depth-sensing cameras, and use it to analyze a sample of scientific conversations. Our method consists of three steps: defining modes of body motion of individual participants, defining dyadic modes made of combinations of these individual modes, and lastly defining motion motifs as dyadic modes that occur significantly more often than expected given the single-person motion statistics. As a proof-of-concept, we analyze the mo...

In behavioral neuroscience, the adaptability of humans facing different constraints has been addressed on one side at the brain level, where a variety of functional networks dynamically support the same performance, and on the other side at the behavioral level, where fractal properties in sensorimotor variables have been considered as a hallmark of adaptability. To bridge the gap between the two levels of observation, we have jointly investigated the changes of network connectivity in the sensorimotor cortex assessed by modularity analysis and the properties of motor variability assessed by multifractal analysis during a prolonged tapping task. Four groups of participants had to produce the same tapping performance while being deprived from 0, 1, 2, or 3 sensory feedbacks simultaneously (auditory and/or visual and/or tactile). Whereas tapping performance was not statistically different across groups, the number of brain networks involved and the degree of multifractality of the inter-tap interval series were significantly correlated, increasing as a function of feedback deprivation. Our findings provide first evidence that concomitant changes in brain modularity and multifractal properties characterize adaptations underlying unchanged performance. We discuss implications of our findings with respect to the degeneracy properties of complex systems, and the entanglement of adaptability and effective adaptation.

The study of interpersonal bodily coordination, both in laboratory and in semi-naturalistic conditions, can reveal subtle phenomena that take place during social interactions (Bernieri, 1988; Yale et al., 2003; Oullier et al., 2008; Paxton and Dale, 2017). The coordination of interpersonal variables spans a range of timescales and has been associated with longer-term cognitive and affective aspects of interpersonal interaction; e.g., conversation (Abney et al., 2014), teacher-student interaction (Bernieri et al., 1988), synchrony in psychotherapy (Koole and Tschacher, 2016), and interpersonal influences on physiology (Palumbo et al., 2017). A general question of interest concerns the kinds of causal and constitutive links between interactive and unconscious coordination and interpersonal affect/cognition (De Jaegher et al., 2010). This question is particularly relevant for studies of embodied social interaction during psychotherapy.

Para la antropología, y la arqueología como parte de ésta,
el carácter y el alcance de las interrogantes que se plantean
determinan la profundidad, la escala y la duración de las investigaciones: preguntas complejas implican programas de
largo aliento, así como la formulación de diferentes escalas
y horizontes que suponen respuestas en el corto y el
mediano plazos. El Proyecto Valle del Mezquital ha buscado
profundizar el conocimiento de los hñahñu desde una perspectiva
diacrónica e interdisciplinaria, con un diálogo permanente
entre la arqueología, la etnografía, la etnohistoria,
la historia, la arqueología experimental y las teorías de la
complejidad.
Entramados en el Mezquital expresa los resultados,
conceptos y perspectivas de ese proyecto de larga data y
ambiciosos alcances, cuyo objetivo ha sido indagar en las
interacciones dinámicas que han tenido lugar en los tres
últimos milenios en el territorio del Valle del Mezquital.
Refleja también la evolución del pensamiento arqueológico,
así como los cambios en las ideas, los procedimientos
y las hipótesis, resultado del diálogo entre diferentes enfoques
y disciplinas. (Diego Prieto Hernández)

Arquitectura en crecimiento modular. Ejemplos. Algunas similitudes y diferencias con el crecimiento fractal. Humanización complementaria por los usuarios en la configuración de espacios construidos y aparición de espacios residuales con anclaje emocional en el inconsciente particular y colectivo. Con esta comunicación se pretende, aún someramente, indagar y relacionar aspectos tales como Qué se entiende por crecimiento modular ? Metodología Ejemplos de aplicación de crecimiento modular Puntos de similitud y diferencias con el crecimiento fractal Interés de la arquitectura inacabada y participación Qué es un espacio construido residual y su interés emocional El crecimiento modular se basa en una combinatoria de células habitacionales que por agrupación y siguiendo unas determinadas directrices de crecimiento dan una arquitectura volumétricamente no prevista "a priori" sino como resultado a su sujeción a estas "leyes" de desarrollo. El arquitecto Rafael Leoz también investigó esta forma metodológica de proyectación a partir de volúmenes sencillos. Esta manera de proyectar ,tiene la ventaja añadida de-si se desea-permitir la industrialización de la construcción. En la "comunicación" se expone un ejemplo de proyectar con esta metodología Según puede leerse en Wikipedia :"Un fractal es un objeto geométrico cuya estructura básica, fragmentada o irregular, se repite a diferentes escalas"El resultado final es una forma similar a sí misma. Deriva del latín fractus, que significa quebrado o fracturado. Muchas estructuras naturales son de tipo fractal. La propiedad matemática clave de un objeto genuinamente fractal es que su dimensión métrica fractal es un número no entero." El mejor ejemplo es un árbol. Un árbol es claramente similar a sí mismo en el hecho de que sus ramas usualmente se dividen de modo regular a medida que se contraen (cambian de escala) con la distancia desde la raíz o del tronco principal. Cualquier parte del árbol imita al árbol en su totalidad, desde las raíces hasta las ramas e incluso en la estructura de sus hojas." (Batty, Michael; Longley, Paul (1997) "The Fractal City". En: Jencks, Charles; Kropf, Karl (2006) Theories and Manifestoes of Contemporary Architecture. London: Wiley Academy. pp 322, 324.). Volviendo al método de proyectar modularmente, donde la estructura y composición en combinatoria es fundamental, puede jugarse a ,una vez asentado en el terreno siguiendo el modelo de crecimiento, además de tener en cuenta la orientación adecuada, intervenir en un diseño mucho más pormenorizado o bien dejar la arquitectura "inacabada en lo accesorio" y que los futuros, y desconocidos, usuarios sean quienes, a pequeñas dosis, participen en la configuración de sus hábitats y entonces irán apareciendo espacios colectivos, además de los particulares, que podríamos denominar "residuales" es decir no controlados desde la mesa del proyectista sino como resultado, nunca final, del juego de implicación conjunta. Obviamente un resultado así está mucho más conectado emocionalmente-"apropiación"-con los beneficiarios del lugar. Espacios urbanos y particulares que hoy nos embelesan, que han crecido, apoyándose en unas estructuras quizás un tanto rígidas originales, pero con el valor añadido de esa creatividad colectiva los hay en todas partes y merecen ser conservados como valor Patrimonial.

This paper presents a brief overview of fractals and some possible applications of fractal geometry in architecture and civil engineering. The interest in this mathematical discipline has been steadily growing since the end of the 20 th century, due to the fascinating beauty of fractals as well as the generally accepted perception that many shapes in nature, if not all, are irregular and rugged, or chaotic (nature is fractal). Fractal geometry, in contrast to Euclidean geometry, offers considerably better methods for describing natural objects, and thereby for achieving harmony with nature, or harmony between clear precision and chaotic imperfection.

Coordinated behavior promotes collaboration among humans. To shed light upon this relationship, we investigated whether and how interpersonal coordination is promoted by empathic perspective taking (EPT). In a joint music-making task, pairs of participants rotated electronic music-boxes, producing two streams of musical sounds that were meant to be played synchronously. Participants-who were not musically trained-were assigned to high and low EPT groups based on pre-experimental assessments using a standardized personality questionnaire. Results indicated that high EPT pairs were generally more accurate in synchronizing their actions. When instructed to lead the interaction, high and low EPT leaders were equally cooperative with followers, making their performance tempo more regular, presumably in order to increase their predictability and help followers to synchronize. Crucially, however, high EPT followers were better able to use this information to predict leaders' behavior and thus improve interpersonal synchronization. Thus, empathic perspective taking promotes interpersonal coordination by enhancing accuracy in predicting others' behavior while leaving the aptitude for cooperation unaltered. We argue that such predictive capacity relies on a sensorimotor mechanism responsible for simulating others' actions in an anticipatory manner, leading to behavioral advantages that may impact social cognition on a broad scale. Synchronous behavior is a universal means of communication and cooperation observable in diverse species 1-4. In humans, this kind of behavior can be witnessed during activities such as audiences clapping their hands in unison, military units marching, or groups of athletes or dancers coordinating their movements in space and time 5-9. Joint music making is a pervasive example of this phenomenon 10. Even a group of perfect strangers might end up singing together, for example during a competitive sporting game or a religious ritual-more likely so if the individuals support the same team, or believe in the same god, of course. When multiple individuals coordinate their movements in order to achieve interpersonal synchrony, the group benefits in terms of enhanced interpersonal bonding and social cohesion 11,12. Such pro-social effects of interpersonal synchronization have been reported in numerous laboratory studies using a variety of tasks 13-23. For instance, dyadic finger tapping in synchrony is more likely to increase feelings of affiliation 18 , trust, and likeability 16,24. These effects appear early in ontogenetic development, as evidenced by the finding that 14-month old infants, if moved in synchrony with another individual by an experimenter, are subsequently more likely to engage in cooperative helping behavior with that individual 19,25. Even the mere observation of synchronous group behavior can augment perceived social cohesion 26,27. These previous studies demonstrated that interpersonal synchrony is a sufficient condition to boost inter-personal pro-social processes. However, this is only one part of the story. Complementarily to these observations , some studies have shown that social personality traits can also affect interpersonal coordination skills. For instance, in one study, it was reported that individuals affected by Social Anxiety Disorder are impaired in leading a task requiring interpersonal coordination 28. Another study showed that individuals who have high internal Locus of Control (according to a questionnaire assessing to what extent individuals assume life events to be contingent on their personal behavior) are relatively good leaders in the sense that they prioritize stabilizing their own action timing over synchronization with an unreliable virtual partner 29. Taken alongside the above evidence, these results suggest that the link between synchrony and prosociality might be bidirectional: increased

The concept of social interaction is at the core of embodied and enactive approaches to social cognitive processes, yet scientifically it remains poorly understood. Traditionally, cognitive science had relegated all behavior to being the end result of internal neural activity. However, the role of feedback from the interactions between agent and their environment has become increasingly important to understanding behavior. We focus on the role that social interaction plays in the behavioral and neural activity of the individuals taking part in it. Is social interaction merely a source of complex inputs to the individual, or can social interaction increase the individuals' own complexity? Here we provide a proof of concept of the latter possibility by artificially evolving pairs of simulated mobile robots to increase their neural complexity, which consistently gave rise to strategies that take advantage of their capacity for interaction. We found that during social interaction, the neural controllers exhibited dynamics of higher-dimensionality than were possible in social isolation. Moreover, by testing evolved strategies against unresponsive ghost partners, we demonstrated that under some conditions this effect was dependent on mutually responsive co-regulation, rather than on the mere presence of another agent's behavior as such. Our findings provide an illustration of how social interaction can augment the internal degrees of freedom of individuals who are actively engaged in participation.

Arquitectura en crecimiento modular. Ejemplos. Algunas similitudes y diferencias con el crecimiento fractal. Humanización complementaria por los usuarios en la configuración de espacios construidos y aparición de espacios residuales con anclaje emocional en el inconsciente particular y colectivo.

In this paper I argue that there are gestalt principles underlying intersubjective interactions and that this means that intersubjective ‘units’, can be recognised as unified gestalt wholes. The nub of the claim is that interactions within a ‘plural subject’ can be perceived by others outside this plural subject. Framed from the first-person perspective: I am able to recognise intersubjective interactions between multiple others who are not me. I argue that the terminology of gestalt structures is helpful in framing and understanding the non-reducible make-up of these relational units. I consequently defend the legitimacy of the claim that we can attend to more than one other person at once, holding multiple others as a single focus of attention insofar as we can attend to multiple others as a gestalt whole. I argue that it is therefore legitimate to talk about attending to, perceiving and addressing multiple others at the same time, in the second-person plural. I argue that this can be identified in the phenomenology of such interactions and in an analysis of the core underlying structures of these interactions.

The main purpose of my monograph is an attempt to explicate Alfred Gell’s notion of person. This objective, even though it had finally crystallized only during my research on the contributions of the British anthropologist, it determined the final profile of the monograph and its main interpretative thesis. The thesis concerns the understanding of person by the author of Art and Agency against the background of alternative conceptions offered especially by contemporary social and cultural anthropologists. For Gell, the problem of upholding distinctiveness between social and cultural anthropology is of great importance, and I follow his steps.  In this sense, Gell is an original and uncompromising researcher. I present this issue in the opening part of the book, discussing various paths of the evolvement of the anthropological studies along with different theoretical views.  In this part of the book I also demonstrate many Gell’s innovative ideas, and numerous paradoxes disclosed both in his research as well as in his life attitude.
The issue encapsulated in the title of this book crowns, in a sense, my previous scientific inquiries, which revolved around the understanding of human being, whether philosophical or artistic, and in particular theatrical.  With time these were augmented with the performing arts and performance studies. This trajectory brought me finally into the realm of „performance philosophy”, which in one of my papers I called ‘the knowledge of agent’. „Performance philosophy” as a new discipline has been developed since 2012 by Laura Cull and her colleagues from University of Surrey – England.

Desde una búsqueda diferente de las fuentes históricas de la Colonia y con la idea de que el altépetl se destruye en una trayectoria compleja después de la Conquista española, los autores realizan un estudio de caso sobre la vieja cabecera de Itzmiquilpan en la que demuestran que la segregación de los barrios indígenas sujetos respecto a su antigua cabecera, heredera del gobierno indígena prehispánico, se configuró de tal manera que el tiempo transcurrido entre los eventos es una función del número de pueblos independizados, el tamaño de las fronteras que construyeron y la superficie del territorio enajenado, de forma tal que, incluso, la dimensión fractal de las fronteras y de la superficie independizada es una función logarítmica del tiempo, donde las grandes extensiones territoriales tienden a reducirse hasta una dimensión fractal que converge en un valor cercano al 1.3 y finalmente exploran la idea de que existe una autosimilitud espacial en la acumulación de los procesos, también como una función del tiempo.

The paper describes an approach to extending the Koch curve into three dimensions. An intermediate stage is a two-dimensional Koch "leaf", a shape with zero area bounded by three Koch curves. The final delta-shaped fractal solid has an infinite number of planar facets, each of which is a Koch leaf.

La nocion de fractal, introducida en la decada de 1970 por Benoit Mandelbrot para designar conjuntos con ciertas propiedades geometricas —en particular, la autosimilitud y la invariancia por cambio de escala—, hizo su aparicion en antropologia en el decenio de 1990 y fue objeto de debates mas recientes entre los mesoamericanistas. El presente articulo revisa la literatura para contestar varias preguntas: ¿adaptaron los antropologos la definicion matematica? ¿Qué tipo de producciones humanas calificaron como fractales? ¿Es util esta nocion para el estudio de los amerindios? Tras definir de manera precisa la geometria fractal, se examinan dos grupos de estudios antropoló- gicos —los que tratan de las producciones geometricas culturales y el fractal por analogia— para concluir sobre las posibles aplicaciones de la nocion en el Mexico antiguo y contemporaneo.

The Fractal: A useful notion for the Americanist Anthropology?
The notion of fractal, introduced in the 1970s by Benoit Mandelbrot to designate sets possessing particular geometrical properties as self-similarity and invariance by change of scale, was introduced in anthropology from 1991. Since that date, it has been the object of debates on the part of the specialists of Mesoamerica. The present article settles for purpose to review the literature to answer several questions: did the anthropologists take back the mathematical definition? What kind of human productions did they qualify of fractal? Is this notion useful for the study of the American Indians? Having defined the notion of fractal geometry, the article distinguishes two uses of the notion: the fractal in the social geometrical productions and the fractal by analogy, to conclude on the possible applications of the notion to the ancient and contemporary Mexico.

Este libro constituye un análisis pionero de planteamiento y análisis interdisciplinario para el estudio de las manifestaciones rupestres con la contribución de las ciencias exactas, la arqueología y la antropología cultural. La aplicación de los modelos fractales en este estudio representa una nueva forma de teoría sistemática para explicar la complejidad de los numerosos y diversos sitios rupestres hallados en el valle del Mezquital, región del estado de Hidalgo en México, empleando para ello una serie de técnicas de fotografía digital y software que se utilizan para llegar a un mejor entendimiento y una interpretación más precisa de quién y cómo se crearon las manifestaciones rupestres.

Over the years, a number of measures have been defined for the purpose of determining the number of independent dimensions contained in a space. The most common dimensionality measures are the topological dimensionality and various kinds of fractal dimensionalities. While each of these dimensionality measures is useful in its own right, none of them accurately quantifies the effective number of independent directions passing through locations contained in a local region of a space. This article introduces a new dimensionality measure, called the connectivity dimensionality field, which is the true measure for the effective number of independent directions passing through locations in a space. In contrast to the fractal dimensionality, the connectivity dimensionality field is a topological property because its value at each material location is invariant to deformations of the space preserving connectivity. The connectivity dimensionality field is a fundamental concept that applies to many different kinds of discrete spaces, continuous spaces, and discrete-continuous dual spaces. A discrete space is a space in which positions cannot be varied differentially, and a continuous space is a space in which positions can be varied differentially. A discrete-continuous dual space has complementary discrete and continuous representations, and a process called discrete-continuous dual matching relates the discrete and continuous representations to each other. This article formally defines two basic types of discrete-continuous duality: (a) asymptotic and (b) strict. A rigorous method is provided for computing the connectivity dimensionality field in edge-vertex graphs, continuous spaces, and discrete-continuous dual spaces. Many examples are given to illustrate the key concepts. Single points, unbranched lines, and periodic lattices are examples of discrete-continuous dual spaces in which the connectivity dimensionality field is a constant nonnegative integer. In other types of discrete-continuous dual spaces, the connectivity dimensionality field contains inherent uncertainty. For the first time, a comprehensive theory is derived that predicts the inherent uncertainty associated with the connectivity dimensionality field in discrete-continuous dual spaces. The study of discrete-continuous dual spaces with variable connectivity dimensionality fields transcends variable-based mathematics.

¿Qué tienen en común la costa marítima de Inglaterra, los patrones de asentamientos arqueológicos al oriente de Esparta, la disposición de las rutas en las modernas metrópolis, los vaivenes de las cotizaciones de la bolsa de valores, las "curvas monstruosas" de la matemática abstracta, los ornamentos de las máscaras de las comunidades Dan de Liberia, algunas cruces etíopes, la música browniana, los Kolam Tamil del sur de la India, la ornamentación detrás del Espejo celta de Desborough, la forma de las nubes y la de las montañas, el sistema circulatorio, o la disposición de los Kraal (corrales) Ba-ila de Zimbabwe?

In this essay, we explain time estimation on the basis of principles of self-organization. Timing behavior can be seen as an outcome of the coupling and coordination across physiological events, overt behavior, and task demands. Such coupling reveals itself in scaling relations known as fractal patterns. The self-organization hypothesis posits a coherent relation between frequency and amplitude of change, as a single coordinated unity, that possess fractal features. Empirical data lend support of this hypothesis, initiating a discussion on how fractal properties of time estimation can be altered by the interplay of voluntary and involuntary control of behavior.

Recent research on fractal scaling in simple human behaviors (e.g., reaction time tasks) has demonstrated that different aspects of the performance (e.g., key presses and key releases) all reveal pink noise signals but yet are uncorrelated with one another in time. These studies have suggested that the independence of these signals might be due to the functional independence of these different sub-actions, given the task constraints. The current experiments investigated whether under a different set of constraints (e.g., finger tapping with and without a metronome) nested sub-actions might show interrelated dynamics, and whether manipulations affecting the fractal scaling of one also might have consequences for the scaling of others. Experiment 1 revealed that the inter-tap intervals and key-press durations of participants' tapping behavior were dynamically related to one another and that the fractal scaling of both changed in the switch from self-paced to metronome-paced tappin...