The generation and classification of tile-k-transitive tilings of the Euclidean plane,the sphere and the hyperbolic plane

Algorithms based on the theory of Delaney-Dress symbols are discussed that can be used to recursively produce all possible equivariant types of tile-k-transitive tilings of the Euclidean plane, the sphere and the hyperbolic plane, for any (reasonable)k. A number of results can be obtained using computer implementations of the algorithms.     

The uses of homogeneous barycentric coordinates in plane Euclidean geometry

The notion of homogeneous barycentric coordinates provides a powerful tool for analysing problems in plane geometry. The paper explains the advantages over the traditional use of trilinear coordinates, and illustrates its power in leading to discoveries of new and interesting collinearity relations of points associated with a triangle.     

Applications of a distance formula in hyperbolic plane geometry

This note may be used in model-based courses on the classical geometries. Given two points P₁(x₁, y₁) and P₂(x₂, y₂) in the Poincaré upper half-plane model of hyperbolic plane geometry with x₁≠x₂, a Cartesian equation of thebowed geodesic passing through P₁ and P₂ and an integral expression for the hyperbolic distance between P₁ and P₂ are developed. These formulas depend only on the coordinates of P₁ and P₂, and it is easy to implement them with the numerical integrators of modern technology. The distance formula is used to find the coordinates of points of division along a bowed geodesic and, thus, leads to activities discovering that results such as Ceva's Theorem are valid in hyperbolic geometry. The distance formula is also used in verifying that the model satisfies the axioms of absolute geometry.     

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

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.     

Defining co-punctuality in terms of line-orthogonality in plane hyperbolic geometry

Summary We present a definition for the co-punctuality of three lines in the hyperbolic plane in terms of line-orthogonality, correcting an omission in [2] and [1].     

The hyperbolic geometry of the symmetrized bidisc

We solve the Caratheodory and Kobayashi extremal problems for the open symmetrized bidisc

Lambert or Saccheri quadrilaterals as single primitive notions for plane hyperbolic geometry

With the aim of revealing their purely geometric nature, we rephrase two theorems of S. Yang and A. Fang [S. Yang, A. Fang, A new characteristic of Möbius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2006) 660-664] characterizing Möbius transformations as definability results in elementary plane hyperbolic geometry. We show not only that elementary plane hyperbolic geometry can be axiomatized in terms of the quaternary predicates λ or σ, with λ(abcd) to be read as ‘abcd is a Lambert quadrilateral’ and σ(abcd) to be read as ‘abcd is a Saccheri quadrilateral’, but also that all elementary notions of hyperbolic geometry can be positively defined (i.e. by using only quantifiers (∀ and ∃) and the connectives ∨ and ∧ in the definiens) in terms of λ or σ.     

On the hyperbolic unitary geometry

Hans Cuypers (Preprint) describes a characterisation of the geometry on singular points and hyperbolic lines of a finite unitary space—the hyperbolic unitary geometry—using information about the planes. In the present article we describe an alternative local characterisation based on Cuypers’ work and on a local recognition of the graph of hyperbolic lines with perpendicularity as adjacency. This paper can be viewed as the unitary analogue of the second author’s article (J. Comb. Theory Ser. A 105:97–110, 2004) on the hyperbolic symplectic geometry.     

Higher dimensional Euclidean and hyperbolic matrix spaces

Journal d'Analyse Mathématique -