1. The Conway–Radin pinwheel tiling 1, a variant of which is shown in figure 1, is a substitution tiling with tiles occurring in infinitely many orientations. Consequently, it is not of finite loca...

De Broglie gravitational waves form an axially symmetric class of vacuum solutions of the linearized Einstein equations. Unlike the well-known massless transverse perturbations in the weak-field regime, they possess an effective mass and exhibit longitudinal degrees of freedom, allowing them to act as guiding fields for elementary particles. In this paper, we show that the dynamics of the de Broglie gravitational wave, described in terms of the real part of a classical field $$\psi$$, is compatible with single particle Quantum Mechanics, (...) and that the complex field $$\psi$$ can be interpreted as the single particle wave function. In particular, we find that $$\psi$$ satisfies the Schrödinger equation for a free particle and, using the principles of gravitational lensing, the same holds in the presence of a weak central potential. We show that when our gravitational wave passes through two closely separated slits, it diffracts and interferes, creating the characteristic pattern of bright and dark bands observed in quantum experiments. We propose two different interpretations: a wave-particle picture, in which wave and particle are two distinct entities, and an effective background field perspective, in the spirit of soliton theory. A variant of the double-slit experiment that could serve, in principle, as an experimental test of the proposed framework is also discussed. (shrink)

Position measurement-induced collapse states are shown to provide a unified quantum description of diffraction of particles passing through a single slit. These states, which we here call ‘quantum location states’, are represented by the conventional rectangular wave function at the initial time of position measurement. We expand this state in terms of the position eigenstates, which in turn can be represented as a linear combination of energy eigenfunctions of the problem, using the closure property. The time-evolution of the location (...) states in the case of free particles is shown to have position probability density patterns closely resembling diffraction patterns in the Fresnel region for small times and the same in Fraunhofer region for large times. Using the quantum trajectory representations in the de Broglie–Bohm, modified de Broglie–Bohm and Floyd–Faraggi–Matone formalisms, we show that Fresnel and Fraunhofer diffractions can be described using a single expression. We also discuss how to obtain the probability density of location states for the case of particles moving in a general potential, detected at some arbitrary point. In the case of the harmonic oscillator potential, we find that they have oscillatory properties similar to that of coherent states. (shrink)

The first electron back-scattered diffraction Kikuchi patterns and grain orientation maps were captured for pure n -phase Ce. The sample preparation technique used for electron back-scattered diffraction orientation mapping of this surface-reactive metal included ion sputtering the surface using a scanning Auger microprobe followed by vacuum transfer of the sample from the scanning Auger microprobe to the scanning electron microscope. The effect of ion sputtering on the microstructure as well as preliminary electron back-scattered diffraction microstructural characterization is (...) presented. Based on the sputtering data, the room-temperature diffusivity of O in n -Ce was estimated. (shrink)

According to the Rayleigh criterion of classical optics, the finite resolving power of a microscope is due to the width of the central peak of the Fraunhofer diffraction pattern produced by the microscope's finite lens aperture. During the last few decades, theories and techniques for superresolution beyond the Rayleigh criterion have been developed in classical optics. Thus, Heisenberg's microscope could also in principle be made to give superresolution and thereby appear to violate the uncertainty relation. We believe that (...) this paradox is due to the inappropriate use of a definition, based purely on experimental convenience, to support a quantum mechanical theorem. (shrink)

The basic concepts appropriate for anS matrix theory for classical nonlinear physics are formulated here. These concepts are illustrated by a discussion of shock wave diffraction patterns. Other information concerning solutions of non-linear conservation laws is surveyed, so that a coherent picture of this theory can be seen. Within thisS matrix framework, a number of open problems as well as a few solved ones will be discussed.

The classical wave-particle problem is resolved in accord with Newton's concept of the particle nature of light by associating particle density and flux with the classical wave energy density and flux. Point particles flowing along discrete trajectories yield interference and diffraction patterns, as illustrated by Young's double pinhole interference. Bound particle motion is prescribed by standing waves. Particle motion as a function of time is presented for the case of a “particle in a box.” Initial conditions uniquely determine the (...) subsequent motion. Some discussion regarding quantum theory is preseted. (shrink)

In this study, we solve analytically the Schrödinger equation for a macroscopic quantum oscillator as a central system coupled to two environmental micro-oscillating particles. Then, the double-slit interference patterns are investigated in two limiting cases, considering the limits of uncertainty in the position probability distribution. Moreover, we analyze the interference patterns based on a recent proposal called stochastic electrodynamics with spin. Our results show that when the quantum character of the macro-system is decreased, the diffraction pattern becomes more (...) similar to a classical one. We also show that, depending on the size of the slits, the predictions of quantum approach could be apparently different with those of the aforementioned stochastic description. (shrink)

ABSTRACT Rosalind Franklin is best known for her informative X-ray diffraction patterns of DNA that provided vital clues for James Watson and Francis Crick's double-stranded helical model. Her scientific career did not end when she left the DNA work at King's College, however. In 1953 Franklin moved to J. D. Bernal's crystallography laboratory at Birkbeck College, where she shifted her focus to the three-dimensional structure of viruses, obtaining diffraction patterns of Tobacco mosaic virus (TMV) of unprecedented detail and (...) clarity. During the next five years, while making significant headway on the structural determination of TMV, Franklin maintained an active correspondence with both Watson and Crick, who were also studying aspects of virus structure. Developments in TMV research during the 1950s illustrate the connections in the emerging field of molecular biology between structural studies of nucleic acids and of proteins and viruses. They also reveal how the protagonists of the “race for the double helix” continued to interact personally and professionally during the years when Watson and Crick's model for the double-helical structure of DNA was debated and confirmed. (shrink)

It is shown that the fact that an electron diffraction or interference pattern is not dependent on observer movement may be explained by de Broglie's phase waves, but cannot be explained by Schrödinger waves.

Wave-particle dualism of two equivalent pictures is untenable, methodically as well as physically. (1) Its origin, the alleged impossibility to explain electron diffraction patterns without wave interference is repudiated by the unitary particle mechanics under quantum rules. (2) The wave part of dualism of matter is unacceptable because the dual relations, γ = E/h and λ = h/p would hold only in one preferred reference system. And relativistically it would lead to further impossible empirical consequences, examplified by the « (...) oscillator dilemma ». (3) The main fault of dualism is that it coordinates frequencies and absolute values of the energy, when quantum theory and experiment show connection between frequency and energy difference, free of arbitrary additional constants. The dualistic philosophy attached to the (impossible) idea of two equivalent subjective pictures in place of one physical reality has led to abstruse epistemological consequences, of which several examples are quoted. The interpretation of the quantum world loses its enigmatic and controversial character when one holds on to the original meaning of the symbols used in the deduction of the quantum formalism from three non-quantal ground postulates, symmetry, generality of the probability law, and independence of arbitrary reference systems for the probability of transitions between physical states of mechanical systems. (shrink)

In this paper, we review the history of quasicrystals from their sensational discovery in 1982, initially “forbidden” by the rules of classical crystallography, to 2011 when Dan Shechtman was awarded the Nobel Prize in Chemistry. We then discuss the discovery of quasicrystals in philosophical terms of anomalies behavior that led to a paradigm shift as offered by philosopher and historian of science Thomas Kuhn in ‘The Structure of Scientific Revolutions’. This discovery, which found expression in the redefinition of the concept (...) crystal from being periodically arranged to producing sharp peaks in the Bragg diffraction pattern, is analyzed according to the Kuhn Cycle. We relate the quasicrystal revolution to the non-Euclidean geometry revolution and argues that since “great minds think alike” there is a diffusion of ideas between scientific revolutions, or a resonance between different disciplines at different times. The story behind quasicrystals is an excellent example of a paradigm shift, demonstrating the nature of scientific discoveries and breakthroughs. (shrink)

A correlation between the change in magnetic susceptibility (Δχexp) upon crystallisation of Cu–Zr and Hf metallic glasses (MG) with their glass forming ability (GFA) observed recently, is found to apply to Cu–Ti and Zr–Ni alloys, too. In particular, small Δχexp, which reflects similar electronic structures, ES, of glassy and corresponding crystalline alloys, corresponds to high GFA. Here, we studied Δχexp for five Cu–Ti and four Cu–Zr and Ni–Zr MGs. The fully crystalline final state of all alloys was verified from X-ray (...) diffraction patterns. The variation of GFA with composition in Cu–Ti, Cu–Zr and Cu–Hf MGs was established from the variation of the corresponding critical casting thickness, dc. Due to the absence of data for dc in Ni–Zr MGs their GFA was described using empirical criteria, such as the reduced glass transition temperature. A very good correlation between Δχexp and dc (and/or other criteria for GFA) was observed for all alloys studied. The correlation between the ES and GFA showed up best for Cu–Zr and NiZr2 alloys where direct data for the change in ES (ΔES) upon crystallisation are available. The applicability of the Δχexp (ΔES) criterion for high GFA (which provides a simple way to select the compositions with high GFA) to other metal-metal MGs (including ternary and multicomponent bulk MGs) is briefly discussed. (shrink)

Recently, there has been increasing theoretical and experimental interest in Popper's gedanken experiment. We calculate in this paper, using the path integral approach, the diffraction patterns predicted by quantum mechanics for this arrangement. The calculations confirm the narrowing of the width of the pattern in absence of the slit obtained experimentally by Kim and Shih (Y. Kim and Y. Shih, Found. Phys. 29, 1849 (1999)).

We report the results of an optical analogue of the fullerene molecule diffraction experiment. Our results, and an analysis of the fullerene experiment, suggest that the patterns observed in the latter can be explained using a localized particle model. There is no evidence that the grating period contributed to the published fullerene diffraction pattern. De Broglie waves, if they exist, are unlikely to have played a significant part in the fullerene diffraction experiment. The observed patterns are (...) not consistent with those expected according to wave theory for the experimental geometry corresponding to the slit-detector system and the de Broglie wavelength. The measurements were performed in the near field, making the demonstration of wave properties difficult. We outline a new classical approach to the electron and neutron interference experiments. The magnetic moment is crucial to this model, which emphasizes a mechanism for generating narrow-band continuum X-radiation. Some experiments are proposed which can decide between the suggested model and quantum mechanics, and which can also rule out an alternative stochastic model. (shrink)

_This article reconstructs from Pāli texts a care (karu__ṇ__ā)-based philosophy for building an isonomic, complex society. The Greek term isonomia (lit. equality), in Karatani Kōjin’s sense of no-rule, means a categorical rejection of any ruler–ruled social hierarchy. I extend this use of isonomia to include practices of spiritual cultivation that relinquish habitual tendencies of the ruler–ruled mentality such as domination, coercion, and submission. To better appreciate this kind of care-based philosophy of isonomia, I argue, one needs to sidestep rights-based conversations, (...) adopt a processual paradigm, and rethink what counts as sociopolitical philosophy. _ _ __As the analysis in my article reveals, a core feature of relation-centered, care-based sociopolitical philosophies is that they diffract seemingly stable categories such as the inner psyche, personhood, group, community, institution, and nation-state into recurring patterns of motivated actions, coactions, and consequences of actions and then guide the reassembling of non-coercive patterns of thoughts, actions, habits, and social relations so to build a friendlier, more pluralistic future._ _ __This study is as much an effort to recover a marginalized voice from the past as it is a future-oriented project of testing out new ways of thinking about key issues concerning humanity and the planet. A baked-in flaw of established social contract theories is that they presume a rational, autonomous agent and in doing so relegate the young, the old, the sick, and the disabled to the realm of the unthought or the afterthought. In contrast, a care-based processual philosophy of isonomia has its theoretical origins in the existential__ reality of interconditionality and precarity. Thus, it offers vital insights into how to reimagine justice, equity, freedom, and pluralism in processual terms of care and equal support for life and liberation._ _ _. (shrink)

In this chapter, we trouble the traditional separation of axiology, ontology, and epistemology in mathematics education research by drawing together three researchers who posit an entanglement of these worlds. We use a diffractive methodology to look for patterns within the insights of Luis Radford on the ethics of mathematics teaching and learning, the work of Rochelle Gutiérrez on Indigenous ways of being-and-acting in the world and the associated mathematics (living mathematx), and the quantum physics derived ethico-onto-epistemology proposed by Karen Barad. (...) Exploring the reinforcements that emerge suggests that an ethics in mathematics education might include not only the teachers, the students, and the world, but also the performance and existence of mathematics itself—the joyful, the frustrating, the inhuman, and the unknowable. (shrink)

Jacques Lacan comments repeatedly on anamorphic art as it exemplifies for him how the mind from a certain angle perceives through law-like patterns the world that would otherwise be nothing but a chaos of arbitrary multiplicities. The angle, though, has a certain effect on what is perceived; an effect that, as such, cannot be perceived within the realm of experience. The article tries to make the link between diffraction laws of perception more explicit in the subject-object dichotomy and refers (...) for that purpose to the work of both Hegel and Lacan. A reference to Hegel is necessary, as Hegel was not only one of Lacan’s own most important sources of insights, but the author who first focused on justified true belief through a theory of a missed encounter between truth and knowledge. (shrink)

In this compendium of essays, some of the world's leading thinkers discuss their conceptions of space and time, as viewed through the lens of their own discipline. With an epilogue on the limits of human understanding, this volume hosts contributions from six or more diverse fields. It presumes only rudimentary background knowledge on the part of the reader. Time and again, through the prism of intellect, humans have tried to diffract reality into various distinct, yet seamless, atomic, yet holistic, independent, (...) yet interrelated disciplines and have attempted to study it contextually. Philosophers debate the paradoxes, or engage in meditations, dialogues and reflections on the content and nature of space and time. Physicists, too, have been trying to mold space and time to fit their notions concerning micro- and macro-worlds. Mathematicians focus on the abstract aspects of space, time and measurement. While cognitive scientists ponder over the perceptual and experiential facets of our consciousness of space and time, computer scientists theoretically and practically try to optimize the space-time complexities in storing and retrieving data/information. The list is never-ending. Linguists, logicians, artists, evolutionary biologists, geographers et cetera, all are trying to weave a web of understanding around the same duo. However, our endeavour into a world of such endless imagination is restrained by intellectual dilemmas such as: Can humans comprehend everything? Are there any limits? Can finite thought fathom infinity? We have sought far and wide among the best minds to furnish articles that provide an overview of the above topics. We hope that, through this journey, a symphony of patterns and tapestry of intuitions will emerge, providing the reader with insights into the questions: What is Space? What is Time? Chapter [15] of this book is available open access under a CC BY 4.0 license. (shrink)

This book re-energizes metafiction studies through the posthuman discourses of performativity, neocybernetics, and diffraction. My contention is that the human experience can be viewed as a metafictioning manifold, i.e., an active self-perpetuating entanglement and emergence of narrativizing structures. Metafictions, then, are living artifacts that model the metafictional processes of our experiences, while also actively re-organizing those experiences and acting as heuristics for engaging with the world with openness to the material ecology of the metafictioning manifold. Renewed attention should be (...) given to metafictionality, and to metafictional artifacts in particular, to better engage with our material reality as co-participant storytellers alongside the objects and systems around us. The introductory chapters set the critical and methodological stage. Chapter Three uses David Markson’s This is not a Novel (2001) to demonstrate the performativity of metafictions and objects. Chapter Four discusses The Third Policeman (1967) by Flann O’Brien and identifies metafictions as living systems. Chapter Five looks at Jerzy Kosinski’s Being There (1970) in order to theorize the agential natures of such object-systems, while Chapter Six investigates the heuristic ethos of a metafictioning manifold through Mark Z. Danielewski’s The Familiar (2015). The epilogue synthesizes these various threads to establish the metafictional novel as a particular type of apparatus for encountering the objects, systems, patterns, and practices of the metafictioning manifold. (shrink)

This short essay presents a critical cartography of the critical new materialist notion and methodology of diffraction.

Re-turning to our experiences of putting a diffractive methodology to work ourselves, as well as engaging with the writings of Donna Haraway and Karen Barad, we produce some propositions re...

This article explores the possibilities of considering how ‘matter and meaning are mutually constituted’ in the production of knowledge (Barad, 2007: 152) through presenting a diffractive analysis of a piece of interview data with a six-year-old boy in a preschool class. Inspired by Donna Haraway’s (1997) and Karen Barad’s (2007) theorising, I understand diffractive analysis as an embodied engagement with the materiality of research data: a becoming-with the data as researcher. Understanding the body as a space of transit, a series (...) of open-ended systems in interaction with the material-discursive ‘environment’, diffractive analyses constitute transcorporeal engagements with data. Stacy Alaimo’s (2010) theorisation of the transcorporeal is put to work diffractively with Gilles Deleuze and Felix Guattari’s (1987) thinking on the process of becoming minor or minoritarian. This implies a reconceptualisation of the very act of thinking as a transcorporeal process of engagement, going beyond the idea of reflexivity and interpretation as inner mental activities taking place in the mind of the researcher understood as separated from the data. Through my example, I argue that diffractive analysis can make visible new kinds of material-discursive realities that can have transformative and political consequences. (shrink)

Multivariate pattern analysis, or MVPA, has become one of the most popular analytic methods in cognitive neuroscience. Since its inception, MVPA has been heralded as offering much more than regular univariate analyses, for—we are told—it not only can tell us which brain regions are engaged while processing particular stimuli, but also which patterns of neural activity represent the categories the stimuli are selected from. We disagree, and in the current paper we offer four conceptual challenges to the use of (...) MVPA to make claims about neural representation. Our view is that the use of MVPA to make claims about neural representation is problematic. (shrink)

This essay focuses on the possibility of adopting a representational approach for technoscience, in which representation is considered as a situated process of dynamic “intra-action” (Barad 2007 ). Re-elaborating the recent critiques of representationalism (Thrift 2008 ), my analysis begins by analysing Hayles’s situated model of representation from an early essay where she explains her definition of constrained constructivism (Hayles [ 1991 ] 1997). The essay then discusses the notions of figuration and diffraction and the way they are employed (...) by Haraway in many of her writings for her critique of technoscience (Haraway 1991 , 1997 ). Finally, after considering diffraction through Barad’s reading of this practice in the context of her theory of agential realism ( 2007 ), it shows the links that relate constrained constructivism, situated knowledge and agential realism, and the way all of them work at “diffract[ing] the rays of technoscience” (Haraway 1997 : 16) through an alternative representational practice. (shrink)

Recent developments in critical policy analysis have occurred alongside the new materialisms in qualitative research. These lines of scholarship have unfolded along two separate, but related, tracks. In particular, the new materialist method of diffraction aligns with many elements of critical policy analysis. Both involve critical theory, complexity and multiple analyses. To examine diffraction as a potential method of critical policy analysis, this paper enacts Karen Barad’s method of diffraction through the theoretical writings of Jane Bennett and (...) Gloria Anzaldúa. Using the example of teacher leaders who are involved in educational policy, Bennett’s vibrant ecologies illustrate ecosystems of teacher leadership and Anzaldúa’s borderlands demonstrate how teacher leaders bridge their roles within multiple layers of community, governance and identity. Through the method of diffraction, critical policy analyses may produce broader perspectives regarding the ways in which differing stakeholders, groups, technologies and even theories collectively shape policy. Significantly, understanding educational policy differently may result in better educational policy-making. (shrink)

In this article, I assess the prospects for the limitarian thesis that someone has too much wealth if they exceed a specific wealth threshold. Limitarianism claims that there are good political and/or ethical reasons to prevent people from having such ‘surplus wealth’, for example, because it has no moral value for the holder or because allowing people to have surplus wealth has less moral value than redistributing it. Drawing on recent literature on distributive justice, I defend two types of limitarian (...) principles of justice. First, limitarian midlevel principles draw on the limitarian thesis to specify normative commitments for guiding institutional design and individual actions. Second, the limitarian presumption draws on that thesis to specify what a just allocation of wealth requires under epistemic constraints. Such a presumption says that without substantive reasons to the contrary, we should regard a distribution as unjust if some people’s wealth exceeds the limitarian threshold. Furthermore, I will argue that we must reject a possible but implausible interpretation of limitarianism as an ideal distributive pattern. Yet both as a midlevel principle and as a presumption, limitarianism can play an important role in theorizing about justice in the real world. (shrink)

Putting the New Materialist figure of diffraction to use in a set of readings – in which cultural texts are materially read against their contents and their themes, against their readers or against other texts – this volume proposes a criticalintervention into the practice of reading itself. In this book, reading and reading methodology are probed for their materiality and re-considered as being inevitably suspended between, or diffracted with, both matter and discourse. The history of literary and cultural reading, (...) including poststructuralism and critical theory, is revisited in a new light and opened-up for a future in which the world and reading are no longer regarded as conveniently separate spheres, but recognized as deeply entangled and intertwined. Diffractive Reading ultimately represents a new reading of reading itself: firstly by critiquing the distanced perspective of critical paradigms such as translation and intertextuality, in which texts encountered, processed or otherwise subdued; secondly, showing how all literary and cultural readings represent different ‘agential cuts’ in the world-text-reader constellation, which is always both discursive and material; and thirdly, the volume materializes, dynamizes and politicizes the activity of reading by drawing attention to reading’s intervention in, and (co)creation of, the world in which we live. (shrink)

Highly idealized models, such as the Hawk-Dove game, are pervasive in biological theorizing. We argue that the process and motivation that leads to the introduction of various idealizations into these models is not adequately captured by Michael Weisberg’s taxonomy of three kinds of idealization. Consequently, a fourth kind of idealization is required, which we call hypothetical pattern idealization. This kind of idealization is used to construct models that aim to be explanatory but do not aim to be explanations.

This article introduces a 'spiritual self-pattern' into the Resonance-Inference Model (RIM) of psychotherapy (Leidig, 2025), proposing it as the supreme regulator of the balance between predictive confidence and corrective evidence (E-I balance) that underpins consciousness. The neurocognitive rationale is grounded in the work on criticality and affective E-I control by Tucker, Luu, and Friston (2025) and Gallagher's (2013) process-oriented Self-Pattern Theory. We argue that the spiritual self-pattern—the dynamic search for meaning, purpose, and connection—functions as the highest regulatory (...) level of this E-I balance. Spiritual needs are conceptualized as the most fundamental priors in the generative self-model, whose fulfillment depends on this optimal balance between confidence (E, elation) and corrective evidence (I, anxiety). Consequently, spiritual dissonance is defined as a chronic E-I imbalance that drives the system into rigid, maladaptive attractor states. In contrast, spiritual resonance describes the synergetic phase transition into a state of criticality, enabling maximum complexity, adaptability, and well-being. The spiritual self-pattern thus acts as the supreme control parameter governing the psyche's entire attractor landscape. This model provides a process-based explanation for the efficacy of spiritually integrated therapies, framing the spiritual dimension not as an add-on, but as a neurocognitive imperative for understanding sustainable change. (shrink)

In recent years, philosophers have developed accounts of cognitive and affective scaffolding to describe the contribution of environmental resources to the realization of mental abilities. However, an integrative account, which captures scaffolding relations in general terms and across domains, is currently lacking. To close this gap, this paper proposes a pattern theory of scaffolding. According to this theory, the functional and causal role of an environmental resource for an individual agent or a group of agents concerning a mental ability (...) in a situational context can be specified by multiple conditions that a scaffold can satisfy. On this view, scaffolds can be described in terms of patterns of relevance-weighted, gradually satisfied scaffolding conditions. The resulting theory will be applied to cases of mathematical cognition to demonstrate its plausibility and feasibility. The pattern theory of scaffolding, it will be shown, has important implications for research on embodied, embedded, extended, and enactive (4E) cognition. (shrink)

Patterns and pattern ontologies are a powerful way for pragmatists to address metaphysical issues by rejecting a false dichotomy between pluralism and realism. However, there is a common misconception about patterns that I call the philosophically perverse patterns (PPP) problem. Here, critics of patterns invent perverse examples that meet the metaphysical criteria to count as patterns. I defuse this concern by showing how PPP misunderstands what the pragmatist metaphysics of patterns is supposed to accomplish: the bare definition should not (...) rule out, or in, substantive examples of patterns that instead should involve methodological considerations. I use this response to the PPP problem to show how the metaphysical definition of 'pattern' allows the pragmatist to capture the rich intricacies of ontologies in the sciences and yields two illustrative norms by which methodology can be guided in developing or refining ontologies: cohesion and coherence. (shrink)

Central pattern generators (CPGs) are circuits of neurons – coupled by chemical synapses and often gap junctions (Marder et al., Dev Neurobiol 77:597–609, 2017) as well – that produce a limited repertoire of network outputs; the outputs typically are used to drive a motor behavior. The motor behaviors in invertebrates include (but are not limited to) driving a heartbeat (Stent et al., Physiol Rev 59:101–136, 1979), grinding up food, flying, and walking. In the case of vertebrates, they include (but (...) are not limited to) locomotion, breathing, chewing, and swallowing. The output of a CPG is often rhythmic but need not be – think of a sneeze or a yawn. Many of the functions of CPGs are essential to life. (shrink)