Stereochemistry of D, L-ribitol Iminocyclitols Between 4 and 2 Dimensions

The transformation from 4D to 3D and 2D exemplified recently with Lie algebra and Hopf fibration on cis , trans-ee , trans-aa stereochemistry, now is demonstrated with hypersphere trigonometric equations for calculation dihedral angles θ HnHn+1 [deg] from carbon chemical shift δ Cn [ppm], then the vicinal angle ϕ [deg] and the other five possible dihedral angles θ HnHn+1 [deg] for one known vicinal coupling constant 3 J HnHn+1 [Hz]. Cis , trans-ee and trans-aa stereochemistry placed on six sets angles on two units have trans -ee 4, 1 stereochemistry under 2D dimension on unit S and trans-ee 3, 2 stereochemistry particularized with algebraic equations on unit U under 4D. Both trans-ee stereochemistry representing a great example for continuum transformation for hypersphere to torus. B 2 root system θ = 150 [deg] as alternative model for building unit under Hopf fibration, along A 2 root system θ = 120 [deg] unit will be disclosed, the second one ensuring also relationships between dihedral θ HnHn+1 [deg] and vicinal angles ϕ[deg]. Three pair of sets angles representative for five membered rings simulated for a vicinal coupling constant, sets A, B - dihedral and vicinal angles, set C used to build unit U 2 with set A and B calculated with 3α + 2β, and pair of sets A and B of unit S 1 calculated with 3α + 3β.