Dihedral angles are predicted from vicinal coupling constant 3JHH[Hz] with 3-sphere approach, sphere or torus trigonometric equations of circle 1, 2 and circle inversion 3-5 for all cis-, trans-ee, trans-aa stereochemistry. The existence of circle inversion was demonstrated with conformational analysis on five and six membered rings. The sign and the stereochemistry result from vicinal coupling constant under trigonometric equations confirmed by algebraic equations, Hopf and Lie algebra theory. 3-Sphere, a hypersphere in 4D enable for all stereochemistry calculation dihedral angles under magnetic wave (NMR data -3JHH[Hz]), and all isomers from only one vicinal coupling constant.

The conformation of 9-O-(10,11-di-O-benzyl-12,14-O-benzylidene-α-D-galactopyranosyl)-1-butyl-2,3-O-isopropylidene-1,4-dideoxy-1,4-imino-1-N-dehydro-L-ribitol 1, phase angles of the pseudorotation of five (C3) and six (C14) membered rings, was analyzed with dihedral angles θHnHn+1[deg] calculated only from vicinal coupling constants 3JHH[Hz] with 3-Sphere approach and VISION molecular models. The dimension space around the six and five membered ring are established based on hypersphere equations results from calculation of the dihedral angles from carbon chemical shift. Higher biological activity was observed to date at iminocyclitols having dihedral or vicinal angles calculated in 2D. Tetrahedral angles in close relationships with dihedral angles are calculated from carbon and / or proton chemical shift with manifold equations, conic and rectangle geometries. Equations for calculation of the tetrahedral angles φCn[deg] only from vicinal coupling constant 3JHnHn+1[deg] or from chemical shift δCn[ppm] are analyzed for five and established for six membered ring, resulting general rules for calculation of tetrahedral angles. Conic as manifold in case of six membered ring enable calculation of dihedral angle θHnHn+1[deg] from tetrahedral angle φCn[deg] starting with tetrahedral angle on unit, and in case of five membered ring based on opposite relationship between dihedral and tetrahedral (sin versus tan function), unit start with dihedral angles. Rectangle as manifold enable calculation for both the tetrahedral angle from dihedral angle starting with dihedral angle on unit, for six membered ring using two or three units with three sets angles and in case of five membered ring only one unit with seven set angles. The bond distances lCnCn+1 [A0] of five and six membered ring are calculated from 3-Sphere-dihedral angles θHnHn+1[deg].

Java Script programs for calculation dihedral angles from NMR data with manifold equations of 3-Sphere approach: rectangle, Villarceau circles of cyclide (Torus – Dupin Cyclide), polar equations, Euler-Conic. Manifolds are curves or surface in higher dimension used for calculation of dihedral angles under wave character of NMR data, carbon and/or proton chemical shift δXn[ppm] and vicinal coupling constant 3JHnHn+1[Hz]. 3-Sphere approach for calculation of the dihedral angles from NMR data in four steps: 1. Prediction, or more exactly calculation of the dihedral angles from vicinal coupling constant with trigonometric equations, 2. Calculation of the dihedral angles from manifold equations; 3. Building units from angle calculated with one of the manifold equations; 4. Calculation the vicinal coupling constant of the manifold dihedral angle. In this paper are presented Java Script programs of step 2 and from step 3 only the Java Script program for calculation of seven sets angles. The bond distances lCnCn+1[A0] between two atoms of carbon are under different polar equations (i.e. limaçons or cardioid, rose or lemniscale), our expectation was to find different manifold equations for calculation the best angle, differences are smaller but can be find sometimes a preferred one for a vicinal coupling constant. 3-Sphere approach has the advantages of calculation from vicinal angle or/and chemical shift the dihedral angle, tetrahedral angle and the bond distance lCnCn+1[A0], with application on conformational and configurational analysis.

N-Alkyl-C1-dialkyl chains iminocyclitols with D or L-ribitol stereochemistry are synthesized with high diastereoselectivity after Grignard reagents addition to N-quaternary pyrrolines salts, and tested for antiviral activity in bovine viral diarrhea virus (BVDV), surrogate for hepatitis C virus (HCV). Dihedral angles are calculated from carbon chemical shift (δCn[ppm]) with 3-sphere method without building units. 3-Sphere, a hypersphere in 4D, under Hopf fibration and Lie algebra mathematics theories enable calculation of the dihedral angles from the NMR data (vicinal coupling constant 3JHnHn+1[Hz], chemical shift δCn[ppm]). Instead of 3D manifold equations on seven sets unit or six sets units are proposed equations between 4D – 2D, in function of the curvature. The relationship between the antiviral activity and the iminocyclitol structure reveals that monoalkyl chain, N-n-C1-dodecyl β-L-ribitol trifloroacetate salt 30 (IC50 1.5 uM) has higher antiviral activity in tangential space, relative to three alkyl chain, N-Methyl-C1-butil, nonyl-L-ribitol. HCl 26 (IC50 < 2 uM) with torus and Dupin cyclide coordinate, both with coordinates in 2D. Three alkyl chain isopropylidene protected pyrrolidine 25 has in 4D with all equations for calculation of the dihedral angles, and in protected pyrroline 19b double bond moves the coordinates in 2D.

Dihedral angles with right sign and stereochemistry are calculated with 3-sphere method in four steps having as main manifold equations the rectangle geometries, skew or middle lines transformed in circles. Non-coplanar Villarceau circles (eq. 4) gives better result for all stereochemistry of iminocyclitols 1-5, since middle (eq. 6) and antirectangle circles (eq. 7) are the best solution for vicinal coupling constants 3JHH[Hz] of 3.1 (2-cis-D-H1H2), 4.8 (3-cis-L-H1H2) and 5.4 (1-cis-D-H2H3), 5.2 (4-cis-L-H2H3). Results pointing out the influence of carbon and proton chemical shift delta[ppm] on calculation of the dihedral angle 0HnHn+1[deg] in close relationships with vicinal coupling constant 3JHH[Hz], through the vicinal angle [deg]. 0An = 2x[(delta delta HnHn+1x delta delta CnCn+1)/2)x90]1/2: n=2, cis-ea, ae, trans-ee 0An = [(delta delta HnHn+1 x delta delta CnCn+1)/2)x90]1/2: n=1, trans-aa6,1 0An = 2x[90x(& UDelta;delta HnHn+1 + delta delta CnCn+1)/2]1/2, n=2, cis-ea,-ae, trans-ee 0An = [90x(delta delta HnHn+1 -& U delta delta CnCn+1)/2]1/2, n = 1, trans-aa6,1 [deg].

3-Sphere theory, a hypersphere in four dimensions, is applied for calculation dihedral angles with the right stereochemistry and sign in D, -L ribitol series from proton and carbon chemical shift (∆δ XnXn+1 [ppm], X = H, C) and vicinal coupling constant ( 3 J HnHn+1 [Hz]) with Java Script.A method in three steps, easy to calculate by hand or with Java Script program: 1. prediction of the dihedral angle only from 3 J HnHn+1 [Hz], 2. calculation the angle of set A with manifold equation (conic section, Villarceau circles) from chemical shift, 3. building of the seven sets unit or six sets units, from which is chose an angle almost equal with the predicted one having its stereochemistry and sign.Angles of set A and set B, relationships between vicinal angle and dihedral angle (X 0 -X 15 ) are introduced instead of polar angle and azimuthal angle in spherical coordinates (eq. 1 versus eq.3).Hopf coordinates, trigonometric equations, confirmed by algebraic equations are disclosed for all cis-ae/ea, trans-ee, trans-aa stereochemistry.Octonionic fibration S 7 →S 15 →S 8 in R 16 , with real fibration S 0 →S 1 →S 1 as unit, reassembles all possible stereochemistry gives by the HCCH fragment on two congruent disks, each centered on the perimeter of the other with equilateral triangles as vertices.Complex Hopf fibration in R 4 ensuring the calculation of the dihedral angle from vicinal angle and vice versa, demonstrating the relationships between sets A, B, C.

Relationships between vicinal angles, angles result from vicinal coupling constant (3)J(HH)[Hz], and tetrahedral angles of five membered ring iminocyclitols with ribitol stereochemistry are demonstrate with polyhedron and 3-sphere methods. Tetrahedral angles phi[deg] and internal angles gamma[deg] are calculated from C-13-NMR, or H-1-NMR chemical shift delta[ppm] in case of heteroatom, with energy-graph theory approach. The vicinal coupling constant can be calculated from one atom of carbon chemical shift delta(Cn)[ppm], and also the corresponding dihedral angle under 3-sphere approach.

3-Sphere approach is applied on prediction dihedral angle θHnHn+1[deg] only from vicinal coupling constant 3JHnHn+1[Hz] with Java script, in comparation with angles calculated from the differences between two atoms of carbon chemical shift (ΔδCnCn+1[ppm]) and Karplus equations. The trigonometric equations 1, 2 ensuring the right sign along the D-, L series rule.