The change in curvature from planar to spherical geometry, or from 3-dimensional space to 4-dimensional space, for example the Poincaré conjecture, the decreases of the third sphere during the translation to torus, as a fourth’s dynamic object passed through the three-dimensional space, are study in dimension 2 and 3. The four-dimensional Einstein field equation of general relativity, is a vectorially approach for the translation from the 2 to 4 dimension space, along the space and time axis. Hyperbolic manifold of genus 2, or Schwertz triangle, are two ways for representation of the pentagon ring in spherical surface; Riemann surface of genus one has hyperelliptic surface, and of genus two hyperbolic surface. The tetrahedral angles (φ) and internal angles (γ), of the pentagon surface of iminociclitol1 6a’, are determined using the Graph, Vectorially and Hückel approaches, from the differences in chemical shifts between two atoms of carbons (ΔδCnCn+1). The phase angle of pseudorotation (P), calculated with Altona model2 from the tetrahedral angles, or from the constant couplings varies between -0.2117 and 8.4281deg. Twist conformation 23T, or conformation with two atoms out of the planarity3, C4-exo and NH-exo, between 34T – 4E – 04T (36 – 72deg) occurs from Hückel's approach. Relationships between chemical shifts, dihedral angles and constant couplings are establishes with Conic section and Hückel Molecular Orbital theory. Six possible values of the dihedral angles, represented into the 180-three-concentric cones approach, fits well with the Hückel theory with six energy level. A collection of fibers over circles in S2 is a torus (S1xS1). Quaterionic Hopf fibration, or octonionic Hopf fibration are propose for prediction of the constant couplings. Relationship between the vicinal spin-spin constant coupling and dihedral angle of furanose and iminocyclitol rings, are study with polar equations, sinusoidal spirale, lemniscate, or cissoids of Diocles. The invers of limaçons is the equation of the conic section.

Conformational analysis on five membered ring by nuclear magnetic resonance spectroscopy. Relationships between constant couplings, chemical shift and dihedral angles

256th National Meeting (2018)

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