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.

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.

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3-Sphere and Lie group theories (Fig. 1, 2) enable calculations of the dihedral angle with right sign and stereochemistry from NMR data[1, 2]: carbon, proton chemical shift ( 1 H, 13 C [ppm]), and vicinal constant coupling ( 3 J HH [Hz]), and manifold equations transform NMR data in angles. The dihedral angles of the mixture 1-α:1-β-L-lyxose are calculated with Lambert R value under 3-sphere approach rule from the ratio of α H1 -H 3 H 4 :β H1 -H 3 H 4 , or with manifold equation from carbon or proton chemical shift under conic section. Six sets angles on two units, U1 and its parallel unit S1, are build with manifold angles under equilateral triangle rule. The coordinates[3] on R 4 (X 0 = cosϕsinθ HnHn+1 , X 1 = sinϕsinξ*/2, X 2 = cosθ HnHn+1 sinϕ, X 3 = sinθ HnHn+1 sinξ/2), confirmed by algebraic equations (cis-, trans-ee: 3 J HH = ϕ 1/2 /2, trans-aa: 3 J HH = ϕ 1/2 ), gives greet relationships between the dihedral angle and the vicinal angle. Scoop of the paper: Our aim is to demonstrated once again the gift of NMR data for calculation dihedral angles with 3-sphere approach. In the absence of one of the NMR data the manifold equations are solutions for all cases.

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