Holonomy is a virtual environment based on the mathematical concept of hyperbolic geometry. Unlike other environments, Holonomy allows users to seamlessly explore an infinite hyperbolic space by physically walking. They use their body as the controller, eliminating the need for teleportation or other artificial VR locomotion methods. This paper discusses the development of Holonomy, highlighting the technical challenges faced and overcome during its creation, including rendering complex hyperbolic environments, populating the space with objects, and implementing algorithms for finding shortest paths in the underlying non-Euclidean geometry. Furthermore, we present a proof-of-concept implementation in the form of a VR navigation game and some preliminary learning outcomes from this implementation.

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The need to understand the structure of hierarchical or high-dimensional data is present in a variety of fields. Hyperbolic spaces have proven to be an important tool for embedding computations and analysis tasks as their non-linear nature lends itself well to tree or graph data. Subsequently, they have also been used in the visualization of high-dimensional data, where they exhibit increased embedding performance. However, none of the existing dimensionality reduction methods for embedding into hyperbolic spaces scale well with the size of the input data. That is because the embeddings are computed via iterative optimization schemes and the computation cost of every iteration is quadratic in the size of the input. Furthermore, due to the non-linear nature of hyperbolic spaces, euclidean acceleration structures cannot directly be translated to the hyperbolic setting. This article introduces the first acceleration structure for hyperbolic embeddings, building upon a polar quadtree. We compare our approach with existing methods and demonstrate that it computes embeddings of similar quality in significantly less time. Implementation and scripts for the experiments can be found at https://graphics.tudelft.nl/accelerating-hyperbolic-tsne .@en

Chip-firing was first introduced as a probabilistic game. Subsequently, it was generalized to arbitrary graph configurations and investigated mostly with regard to two-dimensional quad-grid layouts. In this paper, we lift chip-firing to the third dimension. Aside from the arising three-dimensional shapes, we are interested in the internal, two-dimensional structures. Furthermore, we explore the different shapes obtained by chip firing processes on various neighborhoods, such as the face-centered and the cube-centered grid as well as on a neighborhood inspired by knight moves.@en

Recent years saw a rapid increase in conference formats that take place either fully online or in a hybrid fashion with some people on-site and others online. While these formats brought new challenges, they also opened up new opportunities. In the present article, we first outline advantages and disadvantages of different conference formats as discussed in the literature. We then share our own experiences based on two mathematics and art events that occurred during the respective annual meetings of the German Mathematical Society in 2020 and 2021. This is to illustrate the main benefits of online formats, in particular for the MathArt community. We conclude by highlighting two specific aspects–the facilitated presentation of large artworks and the availability of talk recordings–and give a brief outlook on hybrid events.

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During a surface acquisition process using 3D scanners, noise is inevitable and an important step in geometry processing is to remove these noise components from these surfaces (given as point set or triangulated mesh). The noise removal process (denoising) can be performed by filtering the surface normals first and by adjusting the vertex positions according to filtered normals afterward. Therefore, in many available denoising algorithms, the computation of noise-free normals is a key factor. A variety of filters have been introduced for noise removal from normals, with different focus points like robustness against outliers or large amplitude of noise. Although these filters are performing well in different aspects, a unified framework is missing to establish the relation between them and to provide a theoretical analysis beyond the performance of each method.

In this paper, we introduce such a framework to establish relations between a number of widely used nonlinear filters for face normals in mesh denoising and vertex normals in point set denoising. We cover robust statistical estimation with M-smoothers and their application to linear and nonlinear normal filtering. Although these methods originate in different mathematical theories—which include diffusion-, bilateral-, and directional curvature-based algorithms—we demonstrate that all of them can be cast into a unified framework of robust statistics using robust error norms and their corresponding influence functions. This unification contributes to a better understanding of the individual methods and their relations with each other. Furthermore, the presented framework provides a platform for new techniques to combine the advantages of known filters and to compare them with available methods.@en

A core assumption often heard in public health discourse is that increasing trust in national political leaders is essential for securing public health compliance during crises such as the COVID-19 pandemic (2019–ongoing). However, studies of national government trust are typically too coarse-grained to differentiate between trust in institutions versus more interpersonal trust in political leaders. Here, we present multiscale trust measurements for twelve countries and territories across the West, Oceania and East Asia. These trust results were used to identify which specific domains of government and social trust were most crucial for securing public health compliance (frequency of mask wearing and social distancing) and understanding the reasons for following health measures (belief in effectiveness of public health measures). Through the use of linear regression and structural equation modeling, our cross-cultural survey-based analysis (N = 3369 subjects) revealed that higher trust in national and local public health institutions was a universally consistent predictor of public health compliance, while trust in national political leaders was not predictive of compliance across cultures and geographical regions. Institutional trust was mediated by multiple types of transparency, including providing rationale, securing public feedback, and honestly expressing uncertainty. These results highlight the importance of distinguishing between components of government trust, to better understand which entities the public gives the most attention to during crises.

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Various computer simulations regarding, e.g. the weather or structural mechanics, solve complex problems on a two-dimensional domain. They mostly do so by splitting the input domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This process of splitting can be automatized by using subdivision schemes. Given the wide range of simulation problems to be tackled, an equally wide range of subdivision schemes is available. This paper illustrates a subdivision scheme that splits the input domain into pentagons. Repeated application gives rise to fractal-like structures. Furthermore, the resulting subdivided domain admits to certain weaving patterns. These patterns are subsequently generalized to several other subdivision schemes. As a final contribution, we provide paper models illustrating the weaving patterns induced by the pentagonal subdivision scheme. Furthermore, we present a jigsaw puzzle illustrating both the subdivision process and the induced weaving pattern. These transform the visual and abstract mathematical algorithms into tactile objects that offer exploration possibilities aside from the visual.

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In this paper, we define and evaluate a weighting scheme for neighborhoods in point sets. Our weighting takes the shape of the geometry, i.e., the normal information, into account. This causes the obtained neighborhoods to be more reliable in the sense that connectivity also depends on the orientation of the point set. We utilize a sigmoid to define the weights based on the normal variation. For an evaluation of the weighting scheme, we turn to a Shannon entropy model for feature classification that can be proven to be non-degenerate for our family of weights. Based on this model, we evaluate our weighting terms on a large scale of both clean and real-world models. This evaluation provides results regarding the choice of optimal parameters within our weighting scheme. Furthermore, the large-scale evaluation also reveals that neighborhood sizes should not be fixed globally when processing models. Finally, we highlight the applicability of our weighting scheme within the application context of denoising.

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In this paper, we are interested in classifying the different arising (topological) structures of three-dimensional Turing-like patterns. By providing examples for the different structures, we confirm a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate how these structures are distributed in the parameter space of the discrete model. We found twofold versions of so-called "zero-" and "one-dimensional" structures as well as "two-dimensional" structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases. @en

In this paper, we are interested in classifying the different arising (topological) structures of three-dimensional Turing-like patterns. By providing examples for the different structures, we confirm a conjecture regarding these structures within the setup of three-dimensional Turing-like pattern. Furthermore, we investigate how these structures are distributed in the parameter space of the discrete model. We found twofold versions of so-called "zero-" and "one-dimensional" structures as well as "two-dimensional" structures and use our experimental findings to formulate several conjectures for three-dimensional Turing-like patterns and higher-dimensional cases.@en

The approximation of a golden logarithmic spiral by quarter circles is well known. Starting from this, we show that any logarithmic spiral can be approximated by quarter circles in a similar way. Using our construction on a rectangle with aspect ratio √휙 and performing a coordinate reparametrization, we obtain an aesthetic partition of the plane as our main artwork.@en