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

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

2011-05-13

arxiv.org/abs/1105.2697v2

Advances in Mathematics 230 (2012), no. 3, 927-956

Mathematics

Geometric Topology

32 pages, 2 figures. Added new references, proof to Proposition 2.3, and Remark 5.11; and shifted Notation 5.7 to its new plac

Scientific paper

10.1016/j.aim.2012.03.009

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.

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