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.

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.

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.

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.

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.