Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model

ABSTRACT

This paper explores a deep transformation in mathematical epistemology and its consequences for teaching and learning. With the advent of non-Euclidean geometries, direct, iconic correspondences between physical space and the deductive structures of mathematical inquiry were broken. For non-Euclidean ideas even to become thinkable the mathematical community needed to accumulate over twenty centuries of reflection and effort: a precious instance of distributed intelligence at the cultural level. In geometry education after this crisis, relations between intuitions and geometrical reasoning must be established philosophically, rather than taken for granted. One approach seeks intuitive supports only for Euclidean explorations, viewing non-Euclidean inquiry as fundamentally non-intuitive in nature. We argue for moving beyond such an impoverished approach, using dynamic geometry environments to develop new intuitions even in the extremely challenging setting of hyperbolic geometry. Our efforts reverse the typical direction, using formal structures as a source for a new family of intuitions that emerge from exploring a digital model of hyperbolic geometry. This digital model is elaborated within a Euclidean dynamic geometry environment, enabling a conceptual dance that re-configures Euclidean knowledge as a support for building intuitions in hyperbolic space—intuitions based not directly on physical experience but on analogies extending Euclidean concepts.

KEYWORDS:

• Epistemology
• hyperbolic geometry
• dynamic geometry
• model
• structure
• representation
• intuition
• digital learning environment

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Unless the definition of ‘triangle’ is expanded to admit the degenerate case of three collinear points, pairwise connected by segments. In this case, the three perpendicular bisectors are also parallel. The hyperbolic case is of course striking in that the situation holds for non-degenerate triangles with positive area.