The Banach-Tarski paradox for the hyperbolic plane
The Banach-Tarski paradox for the hyperbolic plane (II)
The second author found a gap in the proof of the main theorem in [J. Mycielski, Fund. Math. 132 (1989), 143-149]. Here we fill that gap and add some remarks about the geometry of the hyperbolic plane ℍ².
The Boundary at Infinity of a Rough CAT(0) Space
We develop the boundary theory of rough CAT(0) spaces, a class of length spaces that contains both Gromov hyperbolic length spaces and CAT(0) spaces. The resulting theory generalizes the common features of the Gromov boundary of a Gromov hyperbolic length space and the ideal boundary of a complete CAT(0) space. It is not assumed that the spaces are geodesic or proper
The classification of compact hyperbolic Coxeter d-polytopes with d + 2 facets.
The combinatorial structure of trigonometry.
The Gromov Invariant of Links.
The hyperbolic Carnot theorem in the Poincaré disc model of hyperbolic geometry.
The hyperbolic triangle centroid
Some gyrocommutative gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the center of momentum of three massive objects with equal masses located at the triangle vertices. Employing gyrovector space techniques we find...
The inner-product of iso-taxicab geometry.
The Regge symmetry is a scissors congruence in hyperbolic space.
The smallest arithmetic hyperbolic three-orbi-fold.
The theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry
In [Comput. Math. Appl. 41 (2001), 135--147], A. A. Ungar employs the Möbius gyrovector spaces for the introduction of the hyperbolic trigonometry. This Ungar's work plays a major role in translating some theorems from Euclidean geometry to corresponding theorems in hyperbolic geometry. In this paper we explore the theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry.
The theorems of Urquhart and Steiner-Lehmus in the Poincaré ball model of hyperbolic geometry
Traces, lengths, axes and commensurability
The focus of this paper are questions related to how various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. The paper is for the large part a survey of recent work.
Transformation of Hyperbolic Escher Patterns
Typisierung der elliptischen Dreiecke nach der Qualität ihrer Winkel und Seiten.
Typisierung und Angabe eines Baukastens für Tetraeder in Räumen konstanter Krümmung