Delambre-Gauss Formulas for Augmented, Right-Angled Hexagons in Hyperbolic 4-Space

TL;DR

This paper extends classical Delambre-Gauss formulas to oriented, augmented right-angled hexagons in hyperbolic 4-space using quaternion and complex half side-lengths, providing new geometric insights and formulas.

Contribution

It introduces generalized Delambre-Gauss formulas for hexagons in H^4, utilizing Clifford numbers and quaternion half side-lengths, extending prior work from H^3.

Findings

Derived quaternion half side-lengths for hexagons in H^4.

Established generalized Delambre-Gauss formulas in H^4.

Connected formulas in H^4 to classical laws in H^3 and spherical/hyperbolic geometry.

Abstract

We study the geometry of oriented right-angled hexagons in H^4, the hyperbolic 4-space, via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon 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. This generalizes the complex half side-lengths of oriented right-angled hexagons in H^3. We also define appropriate complex half side-lengths for the augmented sides of the 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 sine and cosine laws can be deduced. These formulas generalize the classical Delambre-Gauss formulas for spherical/hyperbolic triangles.