Delambre-Gauss formulas for augmented, right-angled hexagons in hyperbolic 4-space

Abstract

We study the geometry of right-angled hexagons in the hyperbolic 4-space H 4 via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon and arbitrarily orient each line and plane involved, so that for the non-augmented sides, we can define quaternion half side-lengths whose angular parts are obtained from half the Euler angles associated to a certain orientation-preserving isometry of the Euclidean 3-space. We also define appropriate complex half side-lengths for the augmented sides of the augmented hexagon. We further explain how to geometrically read off the quaternion half side-lengths for a given oriented, augmented, right-angled hexagon in H 4. Our main result is a set of generalized Delambre-Gauss formulas for oriented, augmented, right-angled hexagons in H 4, involving the quaternion half side-lengths and the complex half side-lengths. We also show in the appendix how the same method gives Delambre-Gauss formulas for oriented right-angled hexagons in H 3, from which the well-known laws of sines and of cosines can be deduced. These formulas generalize the classical Delambre-Gauss formulas for spherical/hyperbolic triangles. © 2012 Elsevier Ltd.