Conic Projection as Manifold and 3-Sphere Dihedral Angles θ<sub>HnHn+1</sub>[Deg] Under Homotopy

Carmen-Irena Mitan; Emerich Bartha; Petru Filip; Miron-Teodor Caproiu; Constantin Draghici; Calin Deleanu; Valeriu Dragutan; Robert Michael Moriarty

doi:doi:10.11648/j.sjc.20261401.12

Research Article |

| Peer-Reviewed

Conic Projection as Manifold and 3-Sphere Dihedral Angles θ_HnHn+1[Deg] Under Homotopy

Carmen-Irena Mitan^*

, Emerich Bartha, Petru Filip

, Miron-Teodor Caproiu, Constantin Draghici, Calin Deleanu

, Valeriu Dragutan, Robert Michael Moriarty

Published in Science Journal of Chemistry (Volume 14, Issue 1)

Received: 20 January 2026 Accepted: 30 January 2026 Published: 26 February 2026

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Abstract

Conic projection as manifold enable calculation dihedral θ_HnHn+1[deg] angles from differences between two atoms of carbon Δδ_CnCn+1[ppm] in three steps or from only one atom of carbon δ_Cn[ppm] in close relationships with tetrahedral φ_Cn[deg] angles under 3-Sphere approach. Hopf fibration and Lie algebra ensuring calculation dihedral θ_HnHn+1[deg] angles from vicinal ϕ[deg] angle, angle results from vicinal coupling constant ³J_HH[Hz]. Real Hopf fibration for calculation dihedral θ_HnHn+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 θ_HnHn+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 θ_HnHn+1[deg] and tetrahedral φ_Cn[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 θ_HnHn+1[deg] angles are not found in first unit, for few vicinal coupling constants ³J_HH[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 θ_HnHn+1[deg] angles in close relationships with tetrahedral φ_Cn[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.

Published in	Science Journal of Chemistry (Volume 14, Issue 1)
DOI	10.11648/j.sjc.20261401.12
Page(s)	12-24
Creative Commons	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.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Conic Projection, Homotopy, 3-Sphere Dihedral Angles, Tetrahedral Angles, Vicinal Coupling Constant

1. Introduction

The invers of any closed curve on S² will be an immersed torus in S³, and any compact Rieman surface of genus 1 can be embedded in the unit sphere S³CR⁴ as a flat torus. The embedding can be chosen as the intersecting of S³ with a quadric hypersurface in R⁴

[1]

. As claimed by Willmore Conjecture surface in R³ rather as S³.

[2]

Surface in 3D Euclidean space R³ is study with Möbius differential geometry,

[2]

measurement of angles instead of lengths, i.e. Plank-Einstein versus De Broglie

[3, 4]

Conic projection used on spherical map calculation (i.e. earth map), convert the latitude and longitude coordinates into two-dimensional coordinate, cartesian coordinate. Point of sphere or spheroid are transformed in tangent or secant of cone that is wrapped around the sphere (Figure 1, eq. (1)-(3))

[5]

θ = sec^-1h(1)

r= (h²- 1)^1/2/h(2)

z = cosθ = 1/h(3)

Where: h – height [radieni], r – radius of cone inscribes on sphere at tangent or secant point of the cone outside of sphere [radieni].

Six dihedral angles with cis, trans-aa, trans-ee stereochemistry can be represented on three concentric cons in 3D and on two intersecting circles in 2D. Point of intersection in 2D between two circles are half ϕ₁^A and ϕ₁^B, first angles of sets A and B or half angles ϕ₁^{A or B}/2 between trans-ee^3,2and trans-ee^4,1. Inside of the 3-Sphere, a hypersphere in 4D, are easy to drawn three sets angles, third set intersecting sets A and B at half first C angle, angle used for increased number of units

[6]

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Figure 1. Conic section representation.

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Figure 2. Dihedral and vicinal angles on 3-Sphere representation.

2. Results

Dihedral angles are calculated with conic projection (Figure 1) from differences between two atoms of carbon

[7]

or one atom of carbon in 3 steps as presented in Tables 1, 2 for iminocyclitols 1 - 8 (Figure 3)

[8, 9]

. The wave character of NMR data preserved homotopy, and angles calculated are under height h or invers of height h^-1 since the calculated vicinal angle must be almost equal with the recorded one or with very smaller differences. The tan function of h^-1 is equal with sin function of h ensuring the trigonometric rule between dihedral θ_HnHn+1[deg] and tetrahedral φ_Cn[deg] angles of five membered ring

[10]

(Table 2).

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Figure 3. Iminocyclitols with 1-α-D (1 – 3, 6 - 8) and 1-β-L-stereochemistry (4, 5, 9, 10) bearing at C₁ one alkyl chain or two alkyl chain.

2.1. Method for Calculation Dihedral Angles in Three Steps

Step 1: Height of cone outside of sphere (eq. (5)) will be calculated from differences between two atoms of carbon Δδ_CnCn+1[ppm] under Euler conic manifold (eq. (4)). The angle θ^AIn [deg] calculated with cos function from differences between two consecutive atoms of carbon chemical shift is transformed on first angle of the set A and calculate height of cone.

cos^-1R_m= θ^Ain(4)

Where θ^AIn → θ^AI1, n = 1 – 6.

cosθ^AI1= 1/h(5)

Table 1. Dihedral angles θ_HnHn+1[deg] calculated from differences between two atoms of carbons chemical shift Δδ_CnCn+1[ppm] with conic projection for iminocyclitols 1-9.

Comp	³J_HH^a[Hz]	Φ [deg]	Δδ_CnCn+1[gaussx10]	θ^Ain[deg]	θ^AI1[deg]	1/h h [π]	r [π]	θ^AIIn[deg]	ϕ [deg]	³J_HnHn+1[deg]	θ_HnHn+1[deg]
1	4.1	67.24	0.755	39.11	20.88^b	h^-1: 0.934	0.381	22.43	67.56	4.109	22.43
	4.1	16.81				h: 1.0703	0.356	20.88^b	69.11^b	4.15	20.88
	5.4	116.64	0.022	88.71	28.71	h^-1: 0.877	0.547	33.219	116.78	5.40	-26.78
		29.1									-41.756
											30.312
									119.363	5.46	-41.073^c
									28.715	5.35	-33.219
						1.1402	0.4804	28.715	118.715	5.44	-28.715
									118.420	5.44	-40.428^c
	0		0.515	58.97	1.024	0.999	0.0178	1.0247	1.02471	0.5060.	-88.97, -91.024
	0					1.0001	0.0178	1.0246	0246	506	-88.97, -91.024
2	3.1	38.44	0.3949	66.736	6.7366	0.993	0.1181	6.7837	38.69	3.11	-53.216
		9.61							40.24	3.17	-32.861^S3
									9.648	3.106	-9.514^{d, US}
									9.648	3.106	-9.788^{d, US}
									39.648	3.14	50.351^US
						1.0069	0.1173	6.7366	38.691	3.11	-53.263
									9.790	3.12	-9.936
									9.70	3.12	-9.936
									39.790	3.15	50.209
	3.9	60.84	0.0056	89.679	29.679	0.8688	0.5699	34.743	60.320	3.88	-34.743
		15.21							62.530	3.95	-41.581
						1.1509	0.4951	29.679	60.320	3.88	29.679
									62.617	3.95	-40.106
									15.160	3.89	-14.655^e
	8.8	77.44	0.1372	82.110	22.110	0.9264	0.4062	23.971	78.014	8.82	-167.74, -135.63
	8.8	309.76				1.07938	0.3764	22.110	78.944	8.88	-168.73, -135.53
3	4.8	92.16	0.24649	75.729	15.729	0.9625	0.2816	16.358	92.716	4.814	-/+2.716^S2
	4.8	23.04				1.0389	0.2710	15.729	92.918	4.8519	-/+2.918^S3
	5.2	108.16	0.0420	87.591	27.591	0.886	0.5226	31.507	108.457	5.207	19.497
		27.04							27.591	5.25	-31.507
						1.1283	0.4631	27.591	28.188	5.30	-32.408
	0		0.13165	82.434	22.434	0.9243	0.4128	24.386	2.8069	0.83	-87.193
									1.4034	0.601	-88.596
						1.08188	0.3816	22.434	1.8913	0.687	-88.108
4	4.8	92.16	0.07563	85.662	25.662	0.9013	0.4804	28.715	92.568	4.81	-/+2.568
	4.8	23.04				1.10943	0.4330	25.662	92.168	4.8	-/+2.168
	5.2	108.16	0.04481	87.431	27.431	0.8875	0.5190	31.268	27.431	5.23	-31.268
		27.04							108.524	5.20	-19.577
						1.12667	0.4606	27.431	27.442	5.23	-31.284
									108.520	5.20	-19.571
	0		0.05042	87.109	27.109	0.8901	0.5119	30.793	0.793	0.44	-89.206
	0					1.12342	0.4556	27.109	1.445	0.60	-88.554
5	2.8	31.36	0.24649	75.729	15.729	0.9625	0.2816	16.358	31.048	2.78	-27.283
		7.84							8.09709	2.84	-8.1791
						1.03890	0.2710	15.729	31.459	2.80	58.545
										2.8	-37.135
										2.8	-27.556
									7.7917	2.79	-7.8647
	3.6	51.84	0.03641	87.913	27.913	0.8836	0.5297	31.989	51.3264	3.582	38.6736
		12.96									-37.9798
											-53.1678
						1.13165	0.4681	27.913	51.3912	3.584	38.6087
											-38.0049
											-52.9908
	8.8	77.44	0.26610	74.567	14.567	0.9678	0.2598	15.062	322.46	8.97	142.468
		309.76							79.958	8.94	169.958
						1.03321	0.2515	14.567	310.417	8.8	142.716
									79.711	8.92	168.711
6	4.9	96.04	0.19607	78.692	18.692	0.94725	0.3383	19.775	24.887	4.98	-22.823, -27.640
		24.01				1.05568	0.3294	18.692	24.618	4.96	-22.615
									24.346	4.93	-22.403, -26.903
	0^a		0.21848	77.379	17.379	0.95434	0.3129	18.239	2.64	0.812	-87.35
	0^a					1.04783	0.2987	17.379	3.93	0.99	-86.06
7	7.6	231.04	0.10364	84.051	24.051	0.9131	0.4462	26.506	230.54	7.59	37.67
		57.76				1.0950	0.4075	24.051	58.253	7.63	31.746
									58.253	7.63	-40.377
									58.253	7.63	-38.224
	3.8	57.76	0.29971	72.559	12.559	0.9760	0.2120	12.242	58.061	3.809	-38.563
									58.031	3.808	-38.616
									57.872	3.80	32.128
											-40.259
											-38.901
		14.44				1.0245	0.2174	12.559	58.565	3.79	-37.677
									57.559	3.79	32.440
											-40.162
											-39.465
8	6.6	174.24	0.10364	84.051	24.051	0.9131	0.4462	26.506	43.253	6.57	46.746
	6.6	43.56				1.09507	0.4075	24.051	174.018	6.57	-/+5.948
	3.5	49	0.29971	72.559	12.559	0.9760	0.2227	12.864	12.5517	3.54	-12.864
	3.5	12.25				1.02451	0.2174	12.559	12.2679	3.50	-12.559
9	7.3	213.16	0.112	83.56	23.56	0.916	0.436	25.86	214.13	7.31	-55.87, -42.67, 29.29
		53.29							212.29	7.28	-57.93, -38.78, 27.96
						1.090	0.399	23.566	53.566	7.31	36.433, -38.81, -47.57
									213.217	7.30	-56.78, 28.71, -40.90
	4.5	20.25	0.3417	70.01	10.01	0.984	0.176	10.17	20.34	4.51	69.65
		81							80.34	4.49	170.34
						1.0158	0.173	10.01	20.008	4.47	69.99
									80.008	4.47	170.01

a.¹H, ¹³C-NMR data ind. 8, 9; b. θ(r(h))^sin = θ(r(h^-1)^tan; c. six sets; d. U to S giving U; e. from set D ; d polyhedron; e.U to S giving U.

Table 2. Dihedral angles calculated θ_HnHn+1[deg] from carbon chemical shift δ_Cn[ppm] with conic projection for iminocyclitols 1-10.

Comp	³J_HH[Hz]	Φ [deg]	δ_Cn [ppm]	δ_Cn [gaussx10]	θ^AIn [deg]	θ^AI1 [deg]	1/h h [π]	r [π]	θ^AIIn [deg]	Φ [deg]	³J_HH [deg]	θ_HnHn+1 [deg]	φ_Cn [deg]
1	4.1	67.24	C₁: 55.8	1.56302	50.224	9.775	0.9854	0.1722	9.9921	69.921	4.18	20.0787	109.88
		16.81					1.0147	0.1697	9.7757	69.775	4.17	20.2242	109.96
			C₂: 83.5	2.33893	64.688	4.688	0.9966	0.0820	4.7039	117.52	5.42	-41.567	99.218, 101.563^c
										115.29	5.36	-25.296	102.655, 101.56^b
										29.654	5.44	-34.703	102.655, 101.56^b
							1.0033	0.0817	4.6895	117.53	5.42	-41.563	99.216, 101.567^c
										115.31	5.369	-25.310	102.663, 101.565^b
										29.645	5.44	-34.689	102.663, 101.565^b
	5.4	116.6	C₃: 84.3	2.36694	65.008	5.008	0.9961	0.0876	5.0276	117.10	5.41	-41.675	99.162, 101.669^c
		29.16								114.97	5.36	-24.972	102.495, 101.669^b
										29.854	5.46	-35.027	102.495, 101.669^b
							1.0038	0.0873	5.0084	117.12	5.41	-41.669	99.16, 101.675^c
										114.99	5.36	-24.991	102.48, 101.67^b
										29.842	5.46	-35.008	102.48, 101.67^b
	0		C₄: 65.9	1.84593	57.198	2.801	0.9988	0.0489	2.8048	2.8048	0.837	-87.195	106.39
	0						1.0011	0.0488	2.8014	2.8014	0.836	-87.198	106.40
2	3.1	38.44	C₁: 57.4	1.60784	51.541	8.458	0.9891	0.1487	8.5524	38.552	3.104	51.447	109.229, 106.917^b
		9.61					1.0109	0.1470	8.4587	38.458	3.1	51.541	109.276, 107.104^b
			C₂: 71.5	2.00280	60.046	0.046	0.9999	0.0008	0.0462	60.046	3.874	29.953	104.97, 100.015^b
							1.0000	0.0008	0.0462	60.046	3.874	29.953	104.976, 100.015^b
	3.9	60.84	C₃: 71.7	2.00840	60.138	0.138	0.9999	0.0024	0.1383	60.138	3.877	29.861	104.93, 100.061^b
	3.9	15.21					1.0000	0.0024	0.1383	60.132	3.877	29.861	104.93, 100.046^b
	8.8	77.44	C₄: 66.8	1.87114	57.694	2.305	0.9991	0.0402	2.3081	76.154	8.72	-165.73	105.729^c
		309.7								79.230	8.90	-169.03	109.036
										76.923	8.77	-166.56	106.571^U2,c
										76.923	8.77	-135.75	105.752^d
							1.0008	0.0402	2.3053	76.152	8.72	-165.72	105.731^c
										79.231	8.9	-169.03	109.035
										76.926	8.77	-166.57	106.568^U2,c
										76.926	8.77	-135.75	105.752
3	4.8	92.16	C₁: 63.7	1.78431	55.913	4.086	0.9974	0.0714	4.0966	23.598	4.85	-25.903	107.043
		23.04								92.048	4.79	+/-2.04^e	106.02
							1.0025	0.0712	4.0862	23.606	4.85	-25.913	107.048
										92.043	4.796	+/-2.04^e	106.021
			C₂: 72.5	2.03081	60.500	0.500	0.9999	0.0087	0.5006	26.909	5.187	-30.500	104.749, 100.166^b
										109.83	5.21	19.833	99.916
							1.0000	0.0087	0.5006	26.909	5.18	-30.500	104.749, 100.166^b
										108.74	5.21	19.833	99.916
	5.2	108.1	C₃: 74.0	2.06511	61.037	1.037	0.9998	0.0181	1.0378	27.275	5.22	-31.037	104.481, 100.345^b
		27.04								108.58	5.21	19.654	99.827
							1.0001	0.0181	1.0376	27.275	5.22	-31.037	104.481, 100.345^b
										108.58	5.21	19.654	99.827
	0		C₄: 69.3	1.94117	58.992	1.007	0.9998	0.0175	1.0077	1.00771	0.501	-88.992	105.503, 109.664^b105.502, 109.664^b
	0		C₄: 69.3	1.94117	58.992	1.007	41.000	80.017	41.007	.0075	0.501	-88.992	105.503, 109.664^b105.502, 109.664^b
4	4.8	92.16	C₁: 68.4	1.91596	58.538	1.461	0.9996	0.0255	1.4622	92.924	4.81	-/+2.92	106.461
		23.04								22.541	4.74	-20.97	109.512
							1.0003	0.0255	1.4617	92.923	4.81	-/+2.92	106.462
										22.541	4.74	-20.974	109.512
			C₂: 71.1	1.99159	59.860	0.139	0.9999	0.0024	0.1396	26.661	5.163	-30.139	104.930
										108.84	5.216	19.953	110.023
							1.0000	0.0024	0.1396	26.661	5.163	-30.139	104.930
										108.84	5.213	19.953	110.023
	5.2	108.1	C₃: 72.7	2.03641	60.589	0.589	0.9999	0.0102	0.5898	26.971	5.193	-30.589	104.705
		27.04								108.71	5.217	19.8033	110.098
							1.0000	0.0102	0.5897	26.970	5.193	-30.589	104.705
										108.71	5.213	19.803	110.098
	0		C₄: 70.9	1.98599	59.766	0.233	0.9999	0.0020	0.1167	0.1167	0.178	89.883	105.116
	0						1.0000	0.0040	0.2335	0.233	0.241	89.766	105.058
5	2.8	31.36	C₁: 63.3	1.7731	55.668	4.331	0.9971	0.0757	4.3439	31.101	2.78	-37.104	109.33
		7.84								7.1593	2.65	-7.104	109.33
										32.008	2.82	-38.687	108.556
							1.0028	0.0755	4.3315	31.105	2.78	-37.112	108.552
										7.167	2.67	-7.112	108.552
										31.994	2.828	-38.663	109.343
			C₂: 72.1	2.01960	60.320	0.320	0.9999	0.0055	0.3206	50.213	3.54	39.789	100.106
												-37.539	101.229^d
												-56.383	103.191^d
							1.0000	0.0055	0.3206	50.213	3.54	39.786	100.106
												-37.540	101.230^d
												-56.383	103.191^d
	3.6	51.84	C₃: 73.4	2.0560	60.897	0.897	0.9998	0.0156	0.8974	50.598	3.55	39.401	100.2
		12.96										-37.693	101.153^d
												-55.276	102.615^d
							1.0001	0.0156	0.8974	50.598	3.55	39.401	100.299
												-37.693	101.153^d
												-55.231	102.638^d
	8.8	77.44	C4: 63.9	1.7899	56.035	3.964	0.9976	0.0693	3.9744	76.987	8.77	166.987	106.98^c
	8.8	309.7					1.0023	0.0691	3.9649	76.982	8.77	166.982	106.98^c
6	4.9	96.04	C₂: 78.4	2.19607	62.912	2.912	0.9987	0.0496	2.8471	24.530	4.95	-27.152	103.543
		24.01								95.694	4.89	-5.694^S2	102.687
										95.666	4.89	5.694^S2	102.687
										23.700	4.86	-21.89^U2	100.970
							1.0012	0.0508	2.9120	95.824	4.89	-5.824^S2	102.152
										95.794	4.89	5.824^S2	102.152
										23.755	4.87	-21.94^U2	100.949
			C₃: 85.4	2.39215	65.289	5.289	0.9957	0.0925	5.3122	95.312	4.88	-5.312	102.355
										95.289	4.88	5.3122	102.355
							1.0042	0.0921	5.2895	95.289	4.88	-5.289	102.343
										95.267	4.88	5.289	102.343
	0^a		C₄: 77.6	2.1736	62.609	2.609	0.9989	0.0455	2.6122	2.61222	0.808	-87.38	106.304106.306
	0^a		C₄: 77.6	2.1736	62.609	2.609	61.001	70.045	2.6095	.6095	0.807	-87.39	106.304106.306
7	7.6	231.0	C₂: 84.2	2.35854	64.913	4.913	0.9963	0.0859	4.9313	230.65	7.59	-55.068	102.543, 101.637^b
		57.76								57.342	7.57	-39.862	100.086
							1.0036	0.0856	4.9132	230.64	7.59	-55.086	102.534, 101.643^b
										57.361	7.57	-39.826	100.068
			C₃: 80.5	2.25490	63.673	3.673	0.9979	0.0642	3.6815	230.23	7.58	-56.318	103.163, 101.224^b
										229.77	7.57	37.363	101.326
							1.0020	0.0640	3.6739	230.23	7.58	-56.326	103.159, 101.227^b
										229.73	7.57	37.347	101.318
	3.8	57.76	C₄: 69.8	1.95518	58.238	0.761	0.9999	0.0138	0.7612	57.851	3.80	-79.747	110.126, 108.77^c
		14.44								14.165	3.76	-105.38^e	104.619^c, 97.309
							1.0000	0.0132	0.7612	57.851	3.80	-79.747	110.373, 108.77^c
										14.165	3.76	-105.38^e	104.619^c, 97.306
8	6.6	174.2	C₂: 84.2	2.35854	64.913	4.913	0.9963	0.0859	4.9313	44.300	6.65	-34.931	102.54, 101.63^b
		43.56					1.0036	0.0856	4.9132	44.262	6.65	-34.913	102.53, 101.643^b
			C₃: 80.5	2.25490	63.673	3.673	0.9979	0.0642	3.6815	43.159	6.56	46.840^e	103.163^c, 98.418
							1.0020	0.0640	3.6739	43.163	6.56	46.836^e	103.159^c, 98.420
	3.5	49	C₄: 69.8	1.95518	59.238	0.761	0.9999	0.0132	0.7612	49.746	3.52	-82.65	108.675^d
		12.25										-62.149	106.074^d
							1.0000	0.0132	0.7612	49.746	3.52	-82.65	108.675^d
												-62.149	106.074^d
9	7.3	213.1	C₂: 84.0	2.35294	64.849	4.849	0.9964	0.0848	4.8667	213.59	7.30	-41.622	101.616^c, 99.417
		53.29					1.0035	0.8453	4.8493	213.58	7.30	-41.616	101.167^c, 99.416
			C₃: 80.0	2.24089	63.496	3.496	0.9981	0.0611	3.5031	213.35	7.30	-41.167	101.165^c, 99.417
							1.0018	0.0609	3.4966	213.35	7.30	-41.165	101.167^c, 99.416
	4.5	81	C₄: 67.8	1.89915	58.227	1.772	0.9995	0.0309	1.7734	20.629	4.54	170.295	109.852^e, 110.295^c
	4.5	20.25					1.0004	0.0309	1.7725	20.630	4.54	170.295	109.852^e, 110.295^c
10	3.9	60.84	C₂: 75.0	2.10084	61.575	1.575	0.9996	0.0275	1.5761	61.576	3.92	28.423	104.212
		15.21								61.257	3.91	-40.525	100.525^c
							1.0003	0.0274	1.5755	61.575	3.92	28.424	104.211
										61.257	3.91	-40.525	100.525^c
			C₃: 68.8	1.92717	58.741	1.258	0.9997	0.0219	1.2584	61.258	3.91	28.741	104.370
										61.608	3.92	-40.419	100.419^c
							1.0002	0.0219	1.2581	61.258	3.91	28.741	104.370
										61.608	3.92	-40.419	100.419^c
	9.1	82.81	C₄: 74.9	2.09803	61.534	1.534	0.9996	0.0267	1.5346	331.38	9.10	146.931	106.53
		331.2								331.53	9.10	61.534	105.767
							1.0003	0.0267	1.5341	331.38	9.10	146.932	106.534
										331.53	9.10	61.534	105.767

a. ¹H, ¹³C-NMR data ind. 8, 9; b. Six sets, c. tetrahedral angles characteristics for 5 rings on same set with dihedral angles, d. tetrahedral angle established for dihedral angle calculated from sin^-1tan or tan^-1sin function on same set, e. unit was build from set E instead C.

Step 2: The radius r of the cone inscribe on sphere at corresponding tangent point can be calculated from h or h^-1 with eq. (6), then one angle of set AII with eq. (7) used for building units. Sometimes resulting directlly the vicinal angle or the dihedral angle, otherwise will be build units with seven sets angles.

r= [(a)²-1]^1/2/(a)(6)

Where a = h or 1/h.

sin^-1r = θ^AIIn(7)

Where θ^AIIn→ units.

Units are build from calculated angle θ^AIIn in seven sets angles with possibility to increase to more units with set C, as a general rule used until now dihedral angles

[6]

are considered in close relationships with tetrahedral angles between set A, B and D, E or F, G if dihedral angle is in set B. Alternatively, six sets angles, three sets angles of unit U and three of unit S, in first case with angles higher as 5[deg] and in second case smaller as 5[deg], totally 14 sets angles. In this paper some units are build from sets D, E or F, G, or sets F, G are transformed in A, B, as well as no more as three units C^nNi = U^nNi, S^nNi are used

[6]

Step 3: Characteristic equations for calculation dihedral angles – vicinal angles

[6]

and vicinal coupling constant

[11]

are used for choosing corresponding angles with calculated vicinal coupling constant values almost equals with recorded one.

Vicinal coupling constants are under two rules: Case 1: smaller vicinal coupling constant for trans-ee and cis stereochemistry and higher for trans-aa stereochemistry

[6]

, iminocyclitols 1-6, 10, Case 2: higher vicinal coupling constant for cis stereochemistry and smaller for trans-ee and trans-aa vicinal coupling constant

[11]

, iminocyclitols 7-9.

2.2. Calculated Dihedral θ_HnHn+1[deg] and Tetrahedral φ_Cn[deg] Angles on Seven and Six Sets Unit

In Table 1 are calculated dihedral θ_HnHn+1[deg] angles for imoncyclitols 1 - 10 from differences between two atoms of carbons Δδ_CnCn+1[deg]. Dihedral angles are chosen from sets A and B having in mind relationship with tetrahedral angles φ_Cn[deg]. The presence of vicinal angles on sets A and B gives other possibility to calculate dihedral angles without building units (Table 1, comp 1, ³J_H2H3 5.4[Hz]). In case of compound 2, vicinal angle ϕ[deg] under second rule

[11]

results from unit U^S calculated from unit U₁ with equation characteristic for transformation

[12]

U to S (ϕ = 9.64 or 9.79[deg]). Dihedral angles θ_H3H4 and θ_H2H3of compounds 2 and 10 are calculated from sets D or G, using a different way for building units, relative to increased units from set C. In this case was reduced the number of units, method used also in Table 2 for calculation dihedral angles of 3 in close relationships with tetrahedral angles from carbon C₁ chemical shift δ_C1[ppm], or for compounds 9 and 7. In Table 1, compound 6, for a vicinal coupling constant of 4.6[Hz] unit was switch from set G to set A and angles are calculated from vicinal angles ϕ[deg]. Tetrahedral angle φ_C4[deg] for a vicinal coupling constant of 8.8[Hz] usually can be considered an angle from same sets with dihedral and vicinal angle, but must be contemplate as possible angle also increasing the unit from sets D, E or F, G.

2.3. Conic of Revolution and Dihedral Angle θ_HnHn+1[deg]

Cone of revolution, results from any Riemann surface of genus one embedded in the unit sphere S³CR⁴ as a flat torus,

[1]

can be used for calculation the distances d_HnHn+1i[A⁰] and bond length l_CnCn+1[A⁰]. Distances between H_nH_n+1ⁱcan be calculated from rectangle geometry for cis stereochemistry and rhombus for trans stereochemistry

[17]

with conic approach or from equilateral triangles having as based d_HnHn+1ⁱ[A⁰]. In the light of distances demonstration on the conic approach, the bond lengths l_CnCn+1[A⁰] for all stereochemistry are calculated with eq. 11. In case of rhombus geometry

[17]

sin instead of cos give the required diagonal for trans-ee^3,2and trans-ee^4,1.

Rectangle:

d_HnHn+1ⁱ= 1.57xcos^1/2θ_HnHn+1[A⁰](88)

Triangle:

d_HnHn+1ⁱ= 0.5x[1.57xcos^1/2(θ_HnHn+1/2)][A⁰] (9)

d_HnHn+1ⁱ= 1.57xcos^1/2(θ/2)[A⁰](10)

where: d_HnHn+1ⁱ- distance between H₁ and H₂ⁱ in A⁰,

l_CnCn+1= {1.54x[1.57xcos^1/2(θ_HnHn+1/2)]}^1/2[A⁰](11)

where θ_HnHn+1 – dihedral angles in deg.

3. Discussion

Dihedral angles θ_HnHn+1[deg] calculated from chemical shift δ [ppm] and vicinal coupling constant ³J_HH[Hz] with 3-sphere approach are usefully for determination phase angle of pseudorotation P[deg] and angle of deviation from planarity θ_m[deg].

[6, 13]

Dihedral angles θ_HnHn+1[deg] calculated with conic projection eq. (1)-(3) ensures the relationship between dihedral θ_HnHn+1[deg] and tetrahedral φ[deg] angles under sin and tan functions in opposite,

[10]

a model for calculation used to date for five and six membered

[6]

ring carbasugar. Higher number of units for choosing dihedral angle θ_HnHn+1[deg] with values almost equal with dihedral angle calculated only from recorded vicinal coupling constant have in opposite with tetrahedral φ_Cn[deg] angles a complicatedly system for establishing the trigonometric equations. The main question, are dihedral and tetrahedral angles in opposite under strict sin and tan trigonometric functions for building units or simply relationships between sets A, B and D, E, and F, G ensure sin versus tan functions? Probably the conic projection manifold is the answer.

Conic projection as manifold proves that in case of radius of the cone inscribes on sphere at tangent point (r), 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(Eq. (12)).

[7]

60 - cos^-1R_m= θ^AN1= sin^-1r, θ^AI1= θ^AII1(12)

with N = I, II, n = 1 – 6.

A program for calculations dihedral θ_HnHn+1[deg] and tetrahedral φ_Cn[deg] angles in opposite

[10]

from sin and tan functions, can be simplified using first the trigonometric equations ensuring relationship between dihedral θ_HnHn+1[deg] and tetrahedral φ_Cn[deg] on seven sets angle and six angles, and second can be chose in opposite dihedral θ_HnHn+1[deg] with tetrahedral φ_Cn[deg] angles.

Euclidean coordinates in (n+1) space (Figure 4): 0-Sphere: pair of points (c-r, c+r), boundary of a line segment (1-ball); 1-Sphere: Abelian Lie grup structure U(1), circle group; 2-Sphere 3-dimensional Euclidean space (3-ball); 3-Sphere: also namely Glome, 4-dimension Euclidean space. Cylindrical coordinates having conic coordinates as limited case, surface of revolution with zero Gaussian curvature, giving the possibility to analyzed their geodesics on the equivalent flat surface are characteristic for torus. Unroll cone cut along a meridian become straight lines in the plane, and represent a way to translate in two-dimension.

[14]

Conic projection versus conic section, as published already are coordinates used for hypersphere tessellation

[12, 15]

, a map projection that transform point from a sphere in tangent or secant of cone that is wrapped around the sphere

[5]

Download: Download full-size image

Figure 4. Euclidian coordinates in (n+1) space.

From the trigonometric point of view vicinal ϕ[deg] angle is in close relationship with dihedral θ_HnHn+1[deg] angle on two sets angles, two intersecting circles, real Hopf fibration, relative to R⁴-complex Hopf fibration where both angles can be vicinal ϕ[deg] or dihedral θ_HnHn+1[deg] angles resulting two vicinal coupling constant ³J_HH[Hz]. A hole inside of sphere gives two tori, topologically a ring torus is homeomorphic to the cartesian product of two circles T² = S¹xS¹, a compact 2-manifold of genus 1. The transformation from sphere to torus simulate better by building unit S from trans-ee^3,2of unit U, and represent a good explanation for R¹⁶ octonionic Hopf fibration, double of quaternionic R⁷, for all cis, trans-ee, trans-aa stereochemistry, observed on two units representation with six sets angles or four sets angles from the strict trigonometric point of view. Since trigonometric equations don’t differentiate between trans-ee^4,1and trans-ee^3,2, algebraic equations have different values and place angles on two units U and S with first angles higher or smaller as 5[deg].

[16]

Polyhedron equation

[13]

under Fibonacci numbers Fⁿ, golden ratio φ and golden triangle, can be used for transforming tetrahedral angles φ_Cn[deg] presented in Table 2 from calculated angles in close relationships with dihedral angle, in predicted angles in agreement with five membered ring trigonometry, i.e. compound 2, φ_C2104.94[deg] under polyhedron equation become 101.243[deg] and φ_C2105.729[deg] under polyhedron equation become 106.796[deg]. Based on other conformational analysis

[13]

on compound 2, in case of calculated phase angle of the pseudorotation E₃ and E₂ – ³T₂ in close relationship with dihedral angles on seven sets angles, E₃ conformation gives angles for C_1,4around 105 and for C_2,3around 104[deg] on seven sets angles. Calculated tetrahedral angles φ_Cn[deg] can be considered transition states (TS), or only values result from six sets angles must be considered, polyhedron transformation giving values comparable with six sets angles. Since between E₃ and E₂– ³T₂ are not expected higher differences on values of tetrahedron angles, probably one from two methods must be applied as well as at list the values of tetrahedron angles to be in line with five membered ring geometry

[13]

As observation on Table 2, dihedral angles 10-θ_H3H4[deg] with ³J_HH 9.1[Hz] and trans-aa 146.9[deg] and trans-ee 61.53[deg] stereochemistry are calculated with positive sign from vicinal angle ϕ 331.3 or 331.5[deg], angles characteristic for L-stereochemistry, relative to 82.81[deg] with only one possible positive dihedral angle, but inexistent on first units. Second vicinal angle giving negative sign for D-stereochemistry.

Conic of revolution on calculation of the distances d_HnHn+1i[A⁰] and bond length l_CnCn+1[A⁰] gives the relationship between dihedral angle and cis, trans- stereochemistry on rectangle and equilateral triangle.

4. Conclusions

Dihedral angles θ_HnHn+1[deg] are calculated with conic projection as manifold from differences between two atoms of carbon Δδ_CnCn+1[ppm] and vicinal coupling constant ³J_HH[Hz], or for carbon chemical shift δ_Cn[ppm] and vicinal coupling constant ³J_HH[Hz] in close relationships with tetrahedral angles φ_Cn[deg] on seven or six sets angles unit(s). In attempt to reduce the number of units calculated by hand, relative to units build through set C, in few cases units are constructed also from sets F and G. Probably a model for calculation all the possible dihedral angles with values almost equals with recorded that must be applied for calculation the number of units until the calculated angles are almost equals with angles on first unit.

[16]

The relationships between dihedral θ_HnHn+1[deg] and tetrahedral φ_Cn[deg] angles in opposite from the trigonometric point of view was demonstrated with conic projection.

Conic of revolution on calculation of the distances d_HnHn+1i[A⁰] and bond length l_CnCn+1[A⁰] with eq. 8-11 was point out.

Abbreviations

RMN data

Nuclear Magnetic Resonance Data

Conflicts of Interest

The authors have not conflicted of interest.

References

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[2]	Udo Hertrich-Jeromin Bulletin (New Series) of the American Mathematical Society 2005, 42(4), 549, ISBN 0-521-53569-7.
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Mitan, C., Bartha, E., Filip, P., Caproiu, M., Draghici, C., et al. (2026). Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy. Science Journal of Chemistry, 14(1), 12-24. https://doi.org/10.11648/j.sjc.20261401.12

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Mitan, C.; Bartha, E.; Filip, P.; Caproiu, M.; Draghici, C., et al. Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy. Sci. J. Chem. 2026, 14(1), 12-24. doi: 10.11648/j.sjc.20261401.12

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Mitan C, Bartha E, Filip P, Caproiu M, Draghici C, et al. Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy. Sci J Chem. 2026;14(1):12-24. doi: 10.11648/j.sjc.20261401.12

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@article{10.11648/j.sjc.20261401.12,
  author = {Carmen-Irena Mitan and Emerich Bartha and Petru Filip and Miron-Teodor Caproiu and Constantin Draghici and Calin Deleanu and Valeriu Dragutan and Robert Michael Moriarty},
  title = {Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy},
  journal = {Science Journal of Chemistry},
  volume = {14},
  number = {1},
  pages = {12-24},
  doi = {10.11648/j.sjc.20261401.12},
  url = {https://doi.org/10.11648/j.sjc.20261401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20261401.12},
  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.},
 year = {2026}
}

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TY  - JOUR
T1  - Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy
AU  - Carmen-Irena Mitan
AU  - Emerich Bartha
AU  - Petru Filip
AU  - Miron-Teodor Caproiu
AU  - Constantin Draghici
AU  - Calin Deleanu
AU  - Valeriu Dragutan
AU  - Robert Michael Moriarty
Y1  - 2026/02/26
PY  - 2026
N1  - https://doi.org/10.11648/j.sjc.20261401.12
DO  - 10.11648/j.sjc.20261401.12
T2  - Science Journal of Chemistry
JF  - Science Journal of Chemistry
JO  - Science Journal of Chemistry
SP  - 12
EP  - 24
PB  - Science Publishing Group
SN  - 2330-099X
UR  - https://doi.org/10.11648/j.sjc.20261401.12
AB  - 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.
VL  - 14
IS  - 1
ER  -

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Carmen-Irena Mitan

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Romanian Academy, Bucharest, Romania

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http://orcid.org/0000-0002-4790-3122
Emerich Bartha

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Romanian Academy, Bucharest, Romania

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Petru Filip

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Romanian Academy, Bucharest, Romania

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http://orcid.org/0000-0003-2076-7244
Miron-Teodor Caproiu

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Romanian Academy, Bucharest, Romania

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Constantin Draghici

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Romanian Academy, Bucharest, Romania

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Calin Deleanu

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Romanian Academy, Bucharest, Romania

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http://orcid.org/0000-0001-8206-5227
Valeriu Dragutan

Department of Organic Chemistry, “C. D. Nenitescu” Institute of Organic and Supramolecular Chemistry of Romanian Academy, Bucharest, Romania

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Robert Michael Moriarty

Department of Chemistry, University of Illinois at Chicago, Chicago, US

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APA Style

Mitan, C., Bartha, E., Filip, P., Caproiu, M., Draghici, C., et al. (2026). Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy. Science Journal of Chemistry, 14(1), 12-24. https://doi.org/10.11648/j.sjc.20261401.12

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ACS Style

Mitan, C.; Bartha, E.; Filip, P.; Caproiu, M.; Draghici, C., et al. Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy. Sci. J. Chem. 2026, 14(1), 12-24. doi: 10.11648/j.sjc.20261401.12

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AMA Style

Mitan C, Bartha E, Filip P, Caproiu M, Draghici C, et al. Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy. Sci J Chem. 2026;14(1):12-24. doi: 10.11648/j.sjc.20261401.12

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@article{10.11648/j.sjc.20261401.12,
  author = {Carmen-Irena Mitan and Emerich Bartha and Petru Filip and Miron-Teodor Caproiu and Constantin Draghici and Calin Deleanu and Valeriu Dragutan and Robert Michael Moriarty},
  title = {Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy},
  journal = {Science Journal of Chemistry},
  volume = {14},
  number = {1},
  pages = {12-24},
  doi = {10.11648/j.sjc.20261401.12},
  url = {https://doi.org/10.11648/j.sjc.20261401.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjc.20261401.12},
  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.},
 year = {2026}
}

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TY  - JOUR
T1  - Conic Projection as Manifold and 3-Sphere Dihedral Angles θHnHn+1[Deg] Under Homotopy
AU  - Carmen-Irena Mitan
AU  - Emerich Bartha
AU  - Petru Filip
AU  - Miron-Teodor Caproiu
AU  - Constantin Draghici
AU  - Calin Deleanu
AU  - Valeriu Dragutan
AU  - Robert Michael Moriarty
Y1  - 2026/02/26
PY  - 2026
N1  - https://doi.org/10.11648/j.sjc.20261401.12
DO  - 10.11648/j.sjc.20261401.12
T2  - Science Journal of Chemistry
JF  - Science Journal of Chemistry
JO  - Science Journal of Chemistry
SP  - 12
EP  - 24
PB  - Science Publishing Group
SN  - 2330-099X
UR  - https://doi.org/10.11648/j.sjc.20261401.12
AB  - 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.
VL  - 14
IS  - 1
ER  -

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