Exploring Non-Euclidean Geometry Through Virtual and Real Experiences to Unlock Non-Experts' Intuition

Three-dimensional non-Euclidean geometries are highly abstract mathematical constructs, typically introduced only at advanced levels of mathematical education due to their conceptual and technical complexity. This paper presents a study exploring whether intuitive understanding of one such geometry – S2× E, the product of a 2-sphere and a line – can be facilitated through an interactive virtual reality (VR) experience without relying on formal mathematical instruction. We designed an immersive VR environment that allows users to explore S2× E as inhabitants of this space. Participants, who were neither mathematics experts nor students specializing in mathematics, engaged with the system after a short introduction using a learn-by-doing approach. Results show that participants were able to infer and articulate key properties of S2× E without knowledge prior to the experiment, and that their performance was independent of their educational background or attitude towards mathematics. Our findings demonstrate the potential of VR as a medium for conveying abstract geometric concepts through embodied learning, making them accessible to non-expert audiences.