Tetrahedral angles of five membered ring are disclosed from the polyhedron geometry point of view, dodecahedron and icosahedron geometry under Fibonacci number. The number of carbons on carbasugar follows the Fibonacci sequence, each carbon is placed at 0.618034 per turn (222.5[deg]). The Fibonacci golden angle 137.5[deg] with its 0.382 turns is approximately equals with Sunflower (Ferment’s spiral – golden angle 136.18[deg]), icosahedron (138.16[deg]), or icositetrahedron (136.18[deg]). In case of five membered ring iminocyclitols are established four equations 18-21 for calculation the isomers of tetrahedral angles φCn[deg] under Fibonacci approach. As demonstrate by Aston et al in 1941 and confirmed by Pitzer in 1945, the cyclopentene conformation is puckered and the deformation is not static is dynamic, with the puckering displacements progressing pseudorotation. In case of five membered ring iminocyclitols with α-D ribitol (1-5) and β-L ribitol (6-8) stereochemistry the wave character of the NMR data on 3-sphere approach – Hopf fibration versus Lie algebra – point out the existence of the pseudorotation through the values of tetrahedral angles φCn[deg] around the five membered ring in close relationships with dihedral angles θHnHn+1[deg] and vicinal angles ϕ[deg], angles result from vicinal coupling constant 3JHnHn+1[Hz]. Hűckel theories under Hopf fibration and polyhedron geometry confirming the existence of tetrahedral angles able to fluctuate around the ring. The Fibonacci approach highlighting the existence of the pseudorotation through three characteristics number: 1.6, 1.9, 1.5 calculated from carbon chemical shift δCn[ppm] with equations 18-21. The main question, is that only a problem of geometry? or with some exception three indubitable isomers must be considered responsible for puckered five membered ring structure on its pseudo rotational itinerary. If you are exciting about this question follow the demonstration about the implication of topology on nature.
Published in | Science Journal of Chemistry (Volume 11, Issue 4) |
DOI | 10.11648/j.sjc.20231104.12 |
Page(s) | 146-154 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2023. Published by Science Publishing Group |
Tetrahedral Angle, Isomerization, Pseudorotation, Fibonacci Number, Golden Ratio, Golden Triangle, Polyhedron Geometry, 3-Sphere
[1] | Wikipedia: https://en.wikipedia/wiki/Fibonacci numbers, Fibonacci golden angle. |
[2] | Bridget L. Stocker, Emma M. Dangerfield, Anna L. Win-Mason, Gregory W. Haslett & Mattie S, M. Timmer, Recent Development in the Synthesis of Pyrrolidine-Containing Iminosugars, Eur. J. Org. Chem. 2010, 1615; DOI: 10.1002/ejoc.200901320. |
[3] | Wikipedia: https://en.wikipedia/wiki/Platonic solid, rhombic polyhedron, golden triangle. |
[4] | C.-I. Mitan, R. M. Moriarty, P. Filip, E. Bartha, C. Draghici, M. T. Caproiu, Golden ratio and tetrahedral angles. Relationships between the 13C-chemical shift and tetrahedral angles, 257th ACS National Meeting in Orlando, Florida, March 31- April 4, 2019, ANYL 93. Publisher: American Chemical Society, Washington, D. C. |
[5] | C. -I. Mitan, R. M. Moriarty, F. Petru, E. Bartha, C. Draghici, M. T. Ca proiu, Geometry and energy. Relationships between the 13C-NMR chemical shift and tetrahedral angles, 257th ACS National Meeting in Orlando, Florida, March 31- April 4, 2019, Sci-Mix ANYL 291. Publisher: American Chemical Society, Washington, D. C. |
[6] | R. M. Moriarty, C.-I. Mitan, N. Branza-Nichita, K. R. Phares, D. Parrish, exo-Imino to endo-Iminocyclitol Rearrangement. A General Route to Five-Membered Antiviral Azasugars, Org. Lett. 2006, 8, 3465; doi.org/10.1021/o1061071r. |
[7] | R. M. Moriarty, C.-I. Mitan, B. Gu, T. Block, Hypersphere and Antiviral Activity of three Alkyl Chain Imnocyclitols with D and L Ribitol Stereochemistry, American Journal of Heterocyclic Chemistry 2023, 9 (1), 9-24. SciencePG: doi: 10.116481/jajhc20230901.12, ISSN: 2575-7059 (Print), ISSN: 2575-5722 (online). |
[8] | G. D. Zeiss, M. A. Whilehead, Hetero-atomic molecules: semi-empirical (π+σ) molecular orbital calculations and prediction of physical properties, J. Chem. Soc.(A), 1971, 1727; doi.org/10.1039/J19710001727. |
[9] | A. G. Evdokimov, A. Joseph, Kalb (Gilboa), Thomas F. Koetzle, Wim T. Klooster, Jan M. L. Martin, Structures of Furanosides: Density Functional calculations and High-Resolution X-ray and neutron diffraction crystal structures, J. Phys. Chem. A 1999, 103, 744; doi.org/10.1021/jp9837840. |
[10] | C.-I. Mitan, E. Bartha, C. Draghici, M. T. Caproiu, P. Filip, R. M. Moriarty, Tetrahedral angles of five membered ring iminocyclitols with ribitol stereochemistry beyond the dihedral angles, Rev. Roum. Chim. 2022, 67 (3), 161 – 166, DOI: 10.33224/rrch.2022.67.3.04. |
[11] | E. Bartha, C.-I. Mitan, C. Draghici, M. T. Caproiu, P. Filip, R. Moriarty, Program for prediction dihedral angle from vicinal coupling constant with 3-sphere approach, Rev. Roum. Chim. 2021, 66, 178-183; DOI: 10.33224/rrch.2021.66.2.08. |
[12] | C.-I. Mitan, E. Bartha, F. Petru, C. Draghici, M.-T. Caproiu, R. M. Moriarty, Manifold inversion on prediction dihedral angle from vicinal coupling constant with 3-sphere approach, Rev. Roum. Chim. 2023, accepted 20 Jun 2023. |
[13] | C.-I. Mitan, E. Bartha, C. Draghici, M. T. Caproiu, P. Filip, R. M. Moriarty, Hopf fibration on relationship between dihedral angle θHnHn+1[deg] and vicinal angle ϕ[deg], angles calculated from NMR data with 3-sphere approach and Java Script, Science Journal of Chemistry 2022, 10, 21. SciencePG: DOI: 10.11648/j.sjc.20221001.13. |
[14] | C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M. T. Caproiu, R. M. Moriarty, Two isomers with trans-aa5, 2 stereochemistry are calculated with 3-sphere trigonometric equations approach at circle inversion motion from NMR data. Sustainability in a changing world” ACS National Meeting in Chicago, IL, August 21- 25, 2022, CARB 3717658, 22 august 2022, Sci-Mix 23 august 2022; doi.org/10.1021/scimmetings.2c00523. |
[15] | C.-I. Mitan, R. Moriarty, P. Filip, E. Bartha, M. T. Caproiu, C. Draghici, Conformational analysis on five membered ring by Nuclear Magnetic Resonance Spectroscopy. Relationships between constant couplings 3JHH, chemical shifts and dihedral angles. 256th ACS National Meeting in Boston, MA, August 19-23, 2018, CARB 84, ID: 2972261, 51 pag. Publisher: American Chemical Society, Washington, D. C. |
[16] | J. E. Kilpatrick, K. S. Pitzer, R. Spitzer, The thermodynamics and molecular structure of cyclopentane, J. Am. Chem. Soc., 1947, 69, 2483; doi.org/10.1021/ja01202a069. |
[17] | W. J. Adams, H. J. Geise, L. S. Bartell, Structure, equilibrium conformation, and pseudorotation in cyclopentane. An electron diffraction study, J. Am. Chem. Soc. 1970, 92, 5013; doi.org/10.1021/ja00720a001. |
[18] | P. Herzyk, A. Rabczenko, A new description of equilateral five-membered rings during pseudorotation, J. Chem. Perkin Trans. II 1983, 213; doi.org/10.1039/P29830000213. |
[19] | P. Herzyk, A. Rabczenko, A general geometrical model for pseudorotation simulation in five-membered rings, J. Chem. Perkin Trans. II 1985, 1925; doi.org/10.1039/P29850001925. |
[20] | J. G. Aston, S. C. Schumann, H. L. Fink, P. M. Doty, The structure of alicyclic compounds, J. Am. Chem. Soc. 1941, 63, 2029; doi.org/10.1021/ja01852a508. |
[21] | K. S. Pitzer, Strain energies of cyclic hydrocarbons, Science 1945, 101, 672; doi: 10.1126/science.101.2635.672. |
[22] | J. R. Durig, D. W. Wertz, Vibrational spectra and structure of small-ring compounds. X. Spectroscopic evidence for pseudorotation in cyclopentane, J. Chem. Phys. 1968, 49, 2118; doi.org/10.1063/1.1670373. |
[23] | J. B. Houseknecht, C. Altona, C. H. Hadad, T. L. Lowary, Conformational analysis of furanose rings with PSEUROT: parametrization for rings possessing the arabino, lyxo, ribo and xylo stereochemistry and application to arabinofuranosides, J. Org. Chem. 2002, 67, 4647; doi.org/10.1021/jo025635q. |
[24] | K. N. Slessor, A. S. Tracey, Couplings into methylene groups: a new nuclear magnetic resonance approach to stereochemistry, Canadian Journal of Chemistry, 1971, 49 (1), 2875; doi.org/10.1139/v71-477. |
[25] | M. G. Constantino, G. V. Jose da Silva, Stereochemistry in substituted cyclopentenes: an approach to the analysis by proton NMR; Tetrahedron 1998, 54, 11363; doi.org/10.1016/S0040-4020(98)00630-9. |
[26] | E. Bartha, C.-I. Mitan, P. Filip 3-Sphere Torsional Angles and Six Membered Ring Conformation, American Journal of Quantum Chemistry and Molecular Spectroscopy 2023, 7 (1), SciencePG: doi: 10.116481j.ajqcms.20230701.12. |
[27] | C.-I. Mitan, E. Bartha, P. Filip, C. Draghici, M. T. Caproiu, R. M. Moriarty, Dihedral angles calculated with 3-sphere approach as integer in conformational analysis on D-, L- ribitol series, Rev. Roum. Chim. 2022, 66 (21), 941, DOI: 10.33224/rrch.2021.66.12.07. |
[28] | C.-I. Mitan, E. Bartha, A. Hîrtopeanu, C. Stavarache, C. Draghici, M. T. Caproiu, M. Maganu, P. Filip, American Journal of Quantum Chemistry and Molecular Spectroscopy 2023, 7 (1), 1-8. SciencePG: doi: 10.11648/j.ajqcms.20230701.11. |
[29] | E. Westhof, M. Sundaralingam, A method for the analysis of puckering disorder in five-membered rings: the relative mobilities of furanose and proline rings and their effects on polynucleotide and polypeptide backbone flexibility, J. Am. Chem. Soc. 1983, 105, 970; doi.org/10.1021/ja00342a054. |
[30] | M. Tomimito, N. Gõ, Analytic theory of pseudorotation in five-membered rings. Cyclopentane, tetrahydrofuran, ribose, and deoxyribose, J. Phys. Chem. 1995, 99, 563; doi.org/10.1021/j100002a019. |
APA Style
Carmen-Irena Mitan, Emerich Bartha, Petru Filip. (2023). Topology on Nature: Isomerization of Five Membered Ring Tetrahedral Angles. Science Journal of Chemistry, 11(4), 146-154. https://doi.org/10.11648/j.sjc.20231104.12
ACS Style
Carmen-Irena Mitan; Emerich Bartha; Petru Filip. Topology on Nature: Isomerization of Five Membered Ring Tetrahedral Angles. Sci. J. Chem. 2023, 11(4), 146-154. doi: 10.11648/j.sjc.20231104.12
AMA Style
Carmen-Irena Mitan, Emerich Bartha, Petru Filip. Topology on Nature: Isomerization of Five Membered Ring Tetrahedral Angles. Sci J Chem. 2023;11(4):146-154. doi: 10.11648/j.sjc.20231104.12
@article{10.11648/j.sjc.20231104.12, author = {Carmen-Irena Mitan and Emerich Bartha and Petru Filip}, title = {Topology on Nature: Isomerization of Five Membered Ring Tetrahedral Angles}, journal = {Science Journal of Chemistry}, volume = {11}, number = {4}, pages = {146-154}, doi = {10.11648/j.sjc.20231104.12}, url = {https://doi.org/10.11648/j.sjc.20231104.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20231104.12}, abstract = {Tetrahedral angles of five membered ring are disclosed from the polyhedron geometry point of view, dodecahedron and icosahedron geometry under Fibonacci number. The number of carbons on carbasugar follows the Fibonacci sequence, each carbon is placed at 0.618034 per turn (222.5[deg]). The Fibonacci golden angle 137.5[deg] with its 0.382 turns is approximately equals with Sunflower (Ferment’s spiral – golden angle 136.18[deg]), icosahedron (138.16[deg]), or icositetrahedron (136.18[deg]). In case of five membered ring iminocyclitols are established four equations 18-21 for calculation the isomers of tetrahedral angles φCn[deg] under Fibonacci approach. As demonstrate by Aston et al in 1941 and confirmed by Pitzer in 1945, the cyclopentene conformation is puckered and the deformation is not static is dynamic, with the puckering displacements progressing pseudorotation. In case of five membered ring iminocyclitols with α-D ribitol (1-5) and β-L ribitol (6-8) stereochemistry the wave character of the NMR data on 3-sphere approach – Hopf fibration versus Lie algebra – point out the existence of the pseudorotation through the values of tetrahedral angles φCn[deg] around the five membered ring in close relationships with dihedral angles θHnHn+1[deg] and vicinal angles ϕ[deg], angles result from vicinal coupling constant 3JHnHn+1[Hz]. Hűckel theories under Hopf fibration and polyhedron geometry confirming the existence of tetrahedral angles able to fluctuate around the ring. The Fibonacci approach highlighting the existence of the pseudorotation through three characteristics number: 1.6, 1.9, 1.5 calculated from carbon chemical shift δCn[ppm] with equations 18-21. The main question, is that only a problem of geometry? or with some exception three indubitable isomers must be considered responsible for puckered five membered ring structure on its pseudo rotational itinerary. If you are exciting about this question follow the demonstration about the implication of topology on nature.}, year = {2023} }
TY - JOUR T1 - Topology on Nature: Isomerization of Five Membered Ring Tetrahedral Angles AU - Carmen-Irena Mitan AU - Emerich Bartha AU - Petru Filip Y1 - 2023/07/21 PY - 2023 N1 - https://doi.org/10.11648/j.sjc.20231104.12 DO - 10.11648/j.sjc.20231104.12 T2 - Science Journal of Chemistry JF - Science Journal of Chemistry JO - Science Journal of Chemistry SP - 146 EP - 154 PB - Science Publishing Group SN - 2330-099X UR - https://doi.org/10.11648/j.sjc.20231104.12 AB - Tetrahedral angles of five membered ring are disclosed from the polyhedron geometry point of view, dodecahedron and icosahedron geometry under Fibonacci number. The number of carbons on carbasugar follows the Fibonacci sequence, each carbon is placed at 0.618034 per turn (222.5[deg]). The Fibonacci golden angle 137.5[deg] with its 0.382 turns is approximately equals with Sunflower (Ferment’s spiral – golden angle 136.18[deg]), icosahedron (138.16[deg]), or icositetrahedron (136.18[deg]). In case of five membered ring iminocyclitols are established four equations 18-21 for calculation the isomers of tetrahedral angles φCn[deg] under Fibonacci approach. As demonstrate by Aston et al in 1941 and confirmed by Pitzer in 1945, the cyclopentene conformation is puckered and the deformation is not static is dynamic, with the puckering displacements progressing pseudorotation. In case of five membered ring iminocyclitols with α-D ribitol (1-5) and β-L ribitol (6-8) stereochemistry the wave character of the NMR data on 3-sphere approach – Hopf fibration versus Lie algebra – point out the existence of the pseudorotation through the values of tetrahedral angles φCn[deg] around the five membered ring in close relationships with dihedral angles θHnHn+1[deg] and vicinal angles ϕ[deg], angles result from vicinal coupling constant 3JHnHn+1[Hz]. Hűckel theories under Hopf fibration and polyhedron geometry confirming the existence of tetrahedral angles able to fluctuate around the ring. The Fibonacci approach highlighting the existence of the pseudorotation through three characteristics number: 1.6, 1.9, 1.5 calculated from carbon chemical shift δCn[ppm] with equations 18-21. The main question, is that only a problem of geometry? or with some exception three indubitable isomers must be considered responsible for puckered five membered ring structure on its pseudo rotational itinerary. If you are exciting about this question follow the demonstration about the implication of topology on nature. VL - 11 IS - 4 ER -