Conic Projection as Manifold and 3-Sphere Dihedral Angles θ<sub>HnHn+1</sub>[Deg] Under Homotopy

Conic projection as manifold enable calculation dihedral θ<sub>HnHn+1</sub>[deg] angles from differences between two atoms of carbon Δδ<sub>CnCn+1</sub>[ppm] in three steps or from only one atom of carbon δ<sub>Cn</sub>[ppm] in close relationships with tetrahedral φ<sub>Cn</sub>[deg] angles under 3-Sphere approach. Hopf fibration and Lie algebra ensuring calculation dihedral θ<sub>HnHn+1</sub>[deg] angles from vicinal ϕ[deg] angle, angle results from vicinal coupling constant <sup>3</sup><i>J</i><sub>HH</sub>[Hz]. Real Hopf fibration for calculation dihedral θ<sub>HnHn+1</sub>[deg] angle in real space, and R<sup>16</sup> octonionic Hopf fibration, double of quaternionic R<sup>7</sup>, for all <i>cis</i>, <i>trans-ee</i>, <i>trans-aa</i> stereochemistry, unreal space relative to calculated dihedral θ<sub>HnHn+1</sub>[deg] angle. Continue “deformation”, homotopic behaviour h ⇆ h<sup>-1</sup> characteristic for wave NMR data, probably a point of swich on Möbius band, in case of radius r of the cone inscribes on sphere at tangent point, calculated from height of cone h or inverse of height h<sup>-1</sup>, the tan function of h is equal with sin function of h<sup>-1</sup>. Dihedral θ<sub>HnHn+1</sub>[deg] and tetrahedral φ<sub>Cn</sub>[deg] angles are from the trigonometric point of view under sin and tan function, or <i>viceversa</i>, homotopic behavior of NMR data under conic projection demonstrating that. Because the dihedral θ<sub>HnHn+1</sub>[deg] angles are not found in first unit, for few vicinal coupling constants <sup>3</sup><i>J</i><sub>HH</sub>[Hz], the rule accepted until now are explored taking in consideration other sets for building unit along the set C, respectively D, E and F, G, or vicinal angle ϕ[deg] with its three possible dihedral θ<sub>HnHn+1</sub>[deg] angles in close relationships with tetrahedral φ<sub>Cn</sub>[deg] angles under seven sets unit. Building units through sets U or S calculated from <i>sin</i> or <i>tan</i> functions until calculated angles are almost equals with angles of unit U<sub>1</sub> or S<sub>1</sub>, required long time for calculation.