Research Article
3-Spherical Conformational Insights into Iminocyclitols with 1-α-D-Ribose and 1-β-L-Ribose Stereochemistry Under Homotopic Behaviour of Nuclear Magnetic Resonance Data
Issue:
Volume 14, Issue 1, February 2026
Pages:
1-11
Received:
22 December 2025
Accepted:
6 January 2026
Published:
2 February 2026
Abstract: 3-Sphere, a hypersphere in 4 dimensions approach, applied for calculating stereochemical parameters of iminocyclitol 1 – 5 with Hopf fibration and Lie algebra is described. Three angles have been considered, i.e. dihedral θHnHn+1 [deg] – tetrahedral φCn [deg] – phase angle of the pseudorotation P [deg] calculated from NMR data, vicinal coupling constant 3JHH [Hz] and carbon chemical shift δC [ppm]. This approach gave for 1-α-methyl-1,4-imino-1,4-dideoxy-D-ribitol 2 two conformers E3 and 3T2 having different dihedral θHnHn+1 [deg] and tetrahedral φCn [deg] angles with same vicinal angles ϕ [deg]. Notably, phase angle of pseudorotation P [deg] placed the conformations on the south side for D-ribitols 1 - 3 and on the north side for L-ribitol 4, excepting trifluoroacetate salt of L-ribitol 5. The wave character of NMR data introduced few homotopic switches, the transformation from torus to inverse of torus, the relationship between angles of set A and set B, the transformation from Planck constants h to h-bar, along with the transformation from Joule to Calorie (J 4.1868 ⇆ J-1 0.238). Two methods for calculation of tetrahedral angles φCn [deg], energy-graph and Euler conic with two ways for representing the angles, polyhedron and unit models are analyzed. The conformational parameters, phase angle of the pseudorotation P [deg] established with VISION molecular model and exocyclic 3-Sphere dihedral angles θHnHn+1 [deg] relative to endocyclic torsional angles θn,n+1 [deg] from Altona-Sundaralingan model have been evaluated. In addition, the corresponding angle of deviation from planarity θm [deg] has been determined.
Abstract: 3-Sphere, a hypersphere in 4 dimensions approach, applied for calculating stereochemical parameters of iminocyclitol 1 – 5 with Hopf fibration and Lie algebra is described. Three angles have been considered, i.e. dihedral θHnHn+1 [deg] – tetrahedral φCn [deg] – phase angle of the pseudorotation P [deg] calculated from NMR data, vicinal coupling con...
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Research Article
Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy
Issue:
Volume 14, Issue 1, February 2026
Pages:
12-24
Received:
20 January 2026
Accepted:
30 January 2026
Published:
26 February 2026
DOI:
10.11648/j.sjc.20261401.12
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Abstract: Conic projection as manifold enable calculation dihedral θHnHn+1[deg] angles from differences between two atoms of carbon ΔδCnCn+1[ppm] in three steps or from only one atom of carbon δCn[ppm] in close relationships with tetrahedral φCn[deg] angles under 3-Sphere approach. Hopf fibration and Lie algebra ensuring calculation dihedral θHnHn+1[deg] angles from vicinal ϕ[deg] angle, angle results from vicinal coupling constant 3JHH[Hz]. Real Hopf fibration for calculation dihedral θHnHn+1[deg] angle in real space, and R16 octonionic Hopf fibration, double of quaternionic R7, for all cis, trans-ee, trans-aa stereochemistry, unreal space relative to calculated dihedral θHnHn+1[deg] angle. Continue “deformation”, homotopic behaviour h ⇆ h-1 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-1, the tan function of h is equal with sin function of h-1. Dihedral θHnHn+1[deg] and tetrahedral φCn[deg] angles are from the trigonometric point of view under sin and tan function, or viceversa, homotopic behavior of NMR data under conic projection demonstrating that. Because the dihedral θHnHn+1[deg] angles are not found in first unit, for few vicinal coupling constants 3JHH[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 θHnHn+1[deg] angles in close relationships with tetrahedral φCn[deg] angles under seven sets unit. Building units through sets U or S calculated from sin or tan functions until calculated angles are almost equals with angles of unit U1 or S1, required long time for calculation.
Abstract: Conic projection as manifold enable calculation dihedral θHnHn+1[deg] angles from differences between two atoms of carbon ΔδCnCn+1[ppm] in three steps or from only one atom of carbon δCn[ppm] in close relationships with tetrahedral φCn[deg] angles under 3-Sphere approach. Hopf fibration and Lie algebra ensuring calculation dihedral θHnHn+1[deg] ang...
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,
Emerich Bartha,
Petru Filip
