Cube-Related Corner Coalesced Nets
Abstract
:1. Introduction
2. Results
2.1. Cube-Shape Corner Coalesced Nets
2.2. Spongy Corner Coalesced Nets
2.3. Hyper-Diamondoid Nets
3. Discussion
3.1. Cube-Shape Containing Structures
3.2. Spongy Corner Coalesced Nets
3.3. Spongy Diamond Nets
4. Methods
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix
Cn Cls | Network Letter and Vertex Classes | LM |
---|---|---|
1 | A (ortho) | T1: (v = 88; e = 168; r4 = 84; r6 = 48; r10 = 96; r12 = 998) |
1.1 | deg = 3; {40}; 4^3 | |
1.1.1 | (4.4); {8}; deg = 3; 4^3 (4.6); {8}; deg = 3; 4^3 | LC: {8}; 3.3.10.30.45.54.90.145.169.196. LR(4.4): {8}; 3.9.18.57.117.171.261.405.552.696.921. LR(4.6); {8}; 3.15.42.135.255.357.561.885.1170.1452.1983. |
1.1.2 | (4.4); {24}; deg = 3; 4^3 (4.6); {24}; deg = 3; 4^3.6^2 | LC: {24}; 3.9.23.35.47.82.127.153.194.277. LR(4.4): {24}; 3.15.45.90.141.225.354.495.657.885.1137. LR(4.6); {24}; 5.31.103.198.297.487.772.1047.1385.1907.2427. |
1.1.3 | (4.4); {8}; deg = 3; 4^3 (4.6); {8}; deg = 3; 4^3.6^6 | LC: {8}; 3.12.31.45.54.90.145.169.196.286. LR(4.4): {8}; 3.18.63.120.171.261.405.552.696.921.1218. LR(4.6): {8}; 9.42.141.258.357.561.885.1170.1452.1983.2628. |
1.2 | deg = 6; {48}; 4^6 | |
1.2.1 | {4.4}; {48}; deg = 6; 4^6 {4.6}; {48}; deg = 6; 4^6.6^8 | LC: {48}; 6.15.24.39.68.98.126.177.242.283. LR(4.4): {48}; 6.27.60.105.177.279.402.558.762.975.1161. LR(4.6); {48}; 14.61.130.225.385.601.852.1192.1636.2063.2437. |
2 | B (inclined) | T1: (v = 88; e = 168; r4 = 84; r6 = 48; r10 = 96; r12 = 998) |
2.1 | deg = 3; {32}; 4^3 | |
2.1.1 | (4.4); {32}; deg = 3; 4^3 (4.6); {32}; deg = 3; 4^3.6^6 | LC: {32}; 3.12.31.45.60.109.166.181.205.306. LR(4.4): {32}; 3.18.63.123.189.324.519.660.789.1056.1413. LR(4.6): {32}; 9.54.189.369.567.972.1557.1980.2367.3168.4239. |
2.2 | deg = 6; {56}; 4^6 | |
2.2.1 | (4.4); {56}; deg = 6; 4^6 (4.6); {56}; deg = 6; 4^6.6^12 | LC: {56}; 6.15.26.48.84.112.135.190.262.303. LR(4.4): {56}; 6.27.63.126.234.360.480.648.894.1131.1323. LR(4.6): {56}; 18.81.189.378.702.1080.1440.1944.2682.3393.3965. |
3 | C (ortho) | T1: 14(12.1.14); (v = 172; e = 336; r4 = 168; r8 = 300); T2: 12(12.1.14); (v = 152; e = 288; r4 = 144; r8 = 232). |
3.1 | deg = 3; {40}; 4^3 | |
3.1.1 | (4.4); {16}; deg = 3; 4^3 (4.8); {8}; deg = 3; 4^3.8^9 (4.8); {8}; deg = 3; 4^3.8^9 | LC: {16}; 3.6.12.19.60.93.105.111.264.374. LR(4.4): {16}; 3.12.27.48.111.240.369.420.594.1056.1446. LR(4.8): {8}; 12.48.132.252.588.1140.1716.2040.3072.4992.6648. LR(4.8): {8}; 12.60.132.240.588.1140.1716.2040.3072.4992.6648. |
3.1.2 | (4.4); {24}; deg = 3; 4^3 (4.8); {24}; deg = 3; 4^3.8^9 | LC: {24}; 3.6.9.13.44.72.93.106.224.315. LR(4.4): {24}; 3.12.24.36.78.176.297.372.534.896.1263. LR(4.8): {24}; 12.52.112.184.416.844.1404.1800.2712.4252.5900. |
3.2 | deg = 4; {84}; 4^4 | |
3.2.1 | (4.4); {24}; deg = 4; 4^4 (4.8); {24}; deg = 4; 4^4.8^12 | LC: {24}; 4.10.16.45.72.93.106.224.315.337. LR(4.4): {24}; 4.15.28.54.112.210.312.459.688.984.1236. LR(4.8): {24}; 16.68.136.280.548.1024.1500.2260.3288.4728.5912. |
3.2.2 | (4.4); {24}; deg = 4; 4^4 (4.8); {24}; deg = 4; 4^4.8^16 | LC: {24}; 4.7.10.28.47.78.100.172.229.309. LR(4.4): {24}; 4.18.40.87.180.297.372.534.896.1263.1348. LR(4.8): {24}; 20.88.200.452.860.1404.1800.2712.4252.5900.6492. |
3.2.3 | (4.4); {24}; deg = 4; 4^4 (4.8); {24}; deg = 4; 4^4.8^16 | LC: {24}; 4.16.28.45.60.128.184.205.216.442. LR(4.4): {24}; 4.24.64.120.180.300.512.744.820.1080.1768. LR(4.8): {24}; 20.128.312.576.876.1520.2432.3488.3956.5472.8376. |
3.2.4 | (4.4); {24}; deg = 4; 4^4 (4.8); {12}; deg = 4; 4^4.8^20 | LC: {12}; 4.16.28.41.52.104.148.197.228.390. LR(4.4): {12}; 4.24.64.120.164.252.416.624.788.1068.1560. LR(4.8): {12}; 24.128.304.576.792.1264.1984.2976.3784.5296.7408. |
3.3 | deg = 6; {48}; 4^6 | |
3.3.1 | (4.4); {48}; deg = 6; 4^6 (4.8); {48}; deg = 6; 4^6.8^26 | LC: {48}; 6.12.30.48.78.100.172.229.309.352. LR(4.4): {48}; 6.24.60.120.213.312.459.688.984.1236.1578. LR(4.8): {48}; 32.120.304.580.1036.1500.2260.3288.4728.5912.7704. |
4 | D (inclined) | T1: 14(12.1.14); (v = 172; e = 336; r4 = 168; r8 = 300). |
4.1 | deg = 3; {32}; 4^3 | |
4.1.1 | (4.4); {32}; deg = 3; 4^3 (4.8); {32}; deg = 3; 4^3.8^9 | LC: {32}; 3.6.12.19.63.96.141.154.345.410. LR(4.4): {32}; 3.12.27.48.114.252.432.564.834.1380.1746. LR(4.8): {32};12.60.144.276.684.1308.2304.3156.4824.7116.9048. |
4.2 | deg = 4; {84}; 4^4 | |
4.2.1 | (4.4); {48}; deg = 4; 4^4 (4.8); {48}; deg = 4; 4^4.8^16 | LC: {48}; 4.10.16.48.76.126.146.304.360.420. LR(4.4): {48}; 4.18.40.90.192.354.504.762.1216.1578.1680. LR(4.8): {48}; 20.96.224.528.1008.1920.2784.4344.6336.8304.9280. |
4.2.2 | (4.4); {36}; deg = 4; 4^4 (4.8); {36}; deg = 4; 4^4.8^20 | LC: {36}; 4.16.28.53.72.168.216.273.272.542. LR(4.4): {36}; 4.24.64.132.212.384.672.960.1092.1416.2168. LR(4.8): {36}; 24.144.336.720.1176.2208.3504.5088.6040.8064.11248. |
4.3 | deg = 6; {56}; 4^6 | |
4.3.1 | (4.4); {56}; deg = 6; 4^6 (4.8); {56}; deg = 6; 4^6.8^30 | LC: {56}: 6.12.33.53.105.132.240.277.399.407. LR(4.4): {56}; 6.24.63.132.261.420.648.960.1290.1596.1926. LR(4.8): {56}; 36.132.360.708.1452.2268.3600.5112.6996.8580.10524. |
Cn Cls | Network Letter and Vertex Classes | LM |
---|---|---|
1 | E (ortho) | T1: (rh12.1.14); (v = 14 ([6(4^4).8(4^3)]); e = 24; r4 = 12; r8 = 18). T2: Void = T1. |
1.1 | deg = 3; {8}; 4^3 | |
(4.4); {8}; deg = 3; 4^3 (4.8); {8}; deg = 3; 4^3.8^33 | LC: {8}; 3.18.15.67.45.166.90.305.150.478. LR(4.4): {8}; 3.24.54.120.201.360.498.720.915.1200.1434. LR(4.8): {8}; 36.288.648.1440.2412.4320.5976.8640.10980.14400.17208. | |
1.2 | deg = 8; {8}; 4^8 | |
(4.4); {6}; deg = 8; 4^8 (4.8); {6}; deg = 8; 4^8.8^88 | LC: {6}; 8.8.40.30.120.68.240.126.400.180. LR(4.4): {6}; 8.24.64.120.240.360.544.720.1008.1200.1440. LR(4.8): {6}; 96.288.768.1440.2880.4320.6528.8640.12096.14400.17280. | |
2 | F (ortho) | T1: (rh24.1.26); (v = 26 ([6(4^4).12(4^4).8(4^3)]); e = 48; r4 = 24; r8 = 15; r10 = 109). T2: Void = T1. |
2.1 | deg = 3; {8}; 4^3 | |
2.1.1 | (4.4); {8}; deg = 3; 4^3 (4.8); {8}; deg = 3; 4^3 | LC: {8}; 3.6.18.30.84.112.225.256.420.455. LR(4.4): {8}; 3.12.33.72.165.336.561.900.1218.1680.2115. LR(4.8): {8}; 3.30.105.180.525.840.1641.2250.3378.4200.5715. |
2.2 | deg = 4; {12}; 4^4 | |
2.2.1 | (4.4); {12}; deg = 4; 4^4 (4.8); {12}; deg = 4; 4^4.8^6 | LC: {12}; 4.14.24.68.96.206.230.372.426.670. LR(4.4): {12}; 4.22.56.132.272.488.824.1090.1488.2008.2680. LR(4.8): {12}; 10.70.140.420.680.1448.2060.3010.3720.5512.6700. |
2.3 | deg = 8; {6}; 4^8 | |
2.3.1 | (4.4); {6}; deg = 8; 4^8 (4.8); {6}; deg = 8; 4^8.8^24 | LC: {6}; 8.16.48.70.160.188.320.366.584.580. LR(4.4): {6}; 8.32.88.192.360.640.904.1280.1728.2336.2640. LR(4.8): {6}; 32.80.280.480.1080.1600.2536.3200.4752.5840.6960. |
3 | G (inclined) | T1: (rh24.1.26); (v = 26 ([6(4^4).12(4^4).8(4^3)]); e = 48; r4 = 24; r8 = 15; r10 = 109). T2: Void = T1. |
3.1 | (4.4); 6 | deg = 4 | 4^4 (4.8); 6 | deg = 4 | 4^4.8^10 | LC: {6}; 4.8.20.29.64.81.140.154.236.242. LR(4.4): {6}; 4.16.40.80.148.256.396.560.744.944.1168. LR(4.8): {6}; 14.44.104.220.374.704.1062.1540.2028.2596.3188. |
3.2 | (4.4); 12 | deg = 4 | 4^4 (4.8); 12 | deg = 4 | 4^4.8^7 | LC: {12}; 4.12.20.45.58.105.120.190.200.298LR(4.4): {12}; 4.20.48.100.180.284.420.580.760.964.1192. LR(4.8): {12}; 11.52.132.260.495.760.1155.1580.2090.2636.3278. |
3.3 | (4.4); 8 | deg = 6 | 4^6 (4.8); 8 | deg = 6 | 4^6.8^6 | LC: {8}; 6.12.30.42.78.92.150.162.246.252. LR(4.4): {8}; 6.24.60.120.204.312.444.600.780.984.1212. LR(4.8): {8}; 12.66.156.330.552.858.1212.1650.2136.2706.3306. |
Cn Cls | Network Letter and Vertex Classes | LM |
---|---|---|
1 | pcu (ortho) | T1: (v = 8; e = 12; r4 = 6) |
1.1 | deg = 6; {8}; 4^12 | |
1.1.1 | (4.4); {8}; deg = 6; 4^12 | LC: {8}; 6.18.38.63.84.92.84.63.38–18 LR(4.4): {8}; 12.72.216.444.696.864.867.708.465.240.93 |
2 | flu (ortho) | T1: (12.4.14); (v = 14([6(4^4).8(4^3)]); e = 24; (r4 = 12; r8 = 18). |
deg = 4; {8}; 4^6 | ||
2.1 | (4.4); {8}; deg = 4; 4^6 (4.8); {8}; deg = 4; 4^6.8^36 | LC: {8}; 4.22.24.82.64.182.124.322.204.502.304. LR(4.4): {8}; 6.48.132.288.492.768.1092.1488.1932.2448.3012.3648. LR(4.8): {8}; 42.336.924.2016.3444.5376.7644.10416.13524.17136.21084. |
2.2 | (4.4); {6}; deg = 8; 4^12 (4.8); {6}; deg = 8; 4^12.8^72 | LC: {6}; 8.12.48.42.128.92.248.162.408.252.608. LR(4.4): {6}; 12.48.144.288.504.768.1104.1488.1944.2448.3024.3648. LR(4.8): {6}; 84.336.1008.2016.3528.5376.7728.10416.13608.17136.21168. |
3 | H (ortho) | T1H: (rh24.9.26). (v = 26 ([6(4^4).12(4^4).8(4^3)]); e = 48; r4 = 24; r8 = 15; r10 = 109). |
3.1 | deg = 4; {6}; 4^4 | |
(4.4); {6}; deg = 4; 4^4 (4.8); {6}; deg = 4; 4^4.8^20 | LC: {6}; 4.16.28.66.76.146.148.258.244.402. LR(4.4): {6}; 4.32.96.224.392.608.872.1184.1544.1952.2408. LR(4.8): {6}; 24.112.336.784.1392.2128.3072.4144.5424.6832.8448. | |
deg = 6; {8}; 4^12 | ||
3.2 | (4.4); {8}; deg = 6; 4^12 | LC: {8}; 6.18.30.66.78.146.150.258.246.402. |
(4.8); {8}; deg = 6; 4^12 | LR(4.4): {8}; 12.48.120.240.408.624.888.1200.1560.1968.2424. | |
LR(4.8): {8}; 12.168.360.840.1368.2184.3048.4200.5400.6888.8424. | ||
3.3 | deg = 6; {12}; 4^8 | |
(4.4); {12}; deg = 6; 4^8 | LC: {12}; 6.14.38.50.102.110.198.194.326.302. | |
(4.8); {12}; deg = 6; 4^8.8^20 | LR(4.4): {12}; 8.40.112.232.400.616.880.1192.1552.1960.2416. | |
LR(4.8): {12}; 28.120.392.792.1400.2136.3080.4152.5432.6840.8456. |
Cn Cls | Network Letter and Vertex Classes | LM |
---|---|---|
1 | Y (hyper-ada) | T1: (v = 528; e = 648; r6 = 68; r16 = 60). |
1.1 | (6.16); {120}; deg = 2; 16^2; (120 free) | LC: {120}; 2.4.3.5.9.12.18.22.26.22. LR: {120}; 2.6.8.11.18.28.36.44.58.57.69. |
1.2 | (6.16); {120}; deg = 2; 6.16; (168 free) | LC: {120}; 2.4.6.8.9.13.20.20.24.28. LR: {120}; 2.7.12.18.20.24.39.52.58.68.82. |
1.3 | (6.16); {120}; deg = 3; 6.16^2; (168 free) | LC: {120}; 3.3.5.6.9.15.19.24.23.29. LR: {120}; 3.6.10.14.20.30.40.51.54.66.77. |
1.4 | (6.16); {168}; deg = 3; 6.16^3; (72 free) | LC:{168}; 3.5.7.9.12.14.16.20.23.28. LR:{168}; 4.10.15.18.22.29.36.50.66.70.71. |
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# | Net Name | Letter | Shape (Symbol) | Tile (Topology) |
---|---|---|---|---|
1 | Corner coalesced pcu (ortho) | A | rh6.8 (cub [4^6]) | T1A: (14(rh6.1.8)); (v = 88; e = 168; r4 = 84; r6 = 48; r10 = 96; r12 = 998). T2A: (rh6.e.8@6(rh6.8)); (v = 32; e = 48; r4 = 6; r6 = 12). |
2 | Corner coalesced pcu (inclined) | B | rh6.8 (cub [4^6]) | T1A: Hyp[rh12.14](rh6.1.8).88. |
3 | Rhombellated-A; rbl(A (ortho)) | C | rh12.14 (rdo [4^12]) | T1C: (14(rh12.1.14)); (v = 172; e = 336; r4 = 168; r8 = 300). T2C: 12(rh12.1.14); (v = 152; e = 288; r4 =1 44; r8 = 232). |
4 | Rhombellated-B; rbl(B (inclined)) | D | rh12.14 (rdo [4^12]) | T1C: Hyp[rh12.14](rh12.1.14).172 |
5 | Corner coalesced flu (spongy; ortho) | E | rh12.14 (rdo [4^12]) | T1E: (rh12.1.14); (v = 14 ([6(4^4).8(4^3)]); e = 24; r4 = 12; r8 = 18). T2E: Void = T1E. |
6 | Rhombellated-E; rbl(E (spongy; ortho)) | F | rh24.26 (mtp [4^24]) | T1F: (rh24.1.26); (v = 26 ([6(4^4).12(4^4).8(4^3)]); e = 48; r4 = 24; r8 = 15; r10 = 109). T2F: Void = T1F. |
7 | Corner coalesced pcu (spongy; inclined) | G | rh24.26 (mtp [4^24]) | T1F; T2F |
8 | Deffect (mtp) pcu (spongy; ortho) | H | rh24.26 (mtp [4^24]) | T1H: (rh24.9.26). |
9 | Etheric hyper-diamond | Y | CC.60 | T1Y: (10CC.60); (v = 528; e = 648; r6 = 68; r16 = 60). CC.60 (v = 60; e = 72; r6 = 8; r16 = 6). T1Y: Hyp[ada.10](CC.60).528T1Ys: Hyp[ada.10](CC.156)).1270 |
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Diudea, M.V. Cube-Related Corner Coalesced Nets. Molecules 2019, 24, 1221. https://doi.org/10.3390/molecules24071221
Diudea MV. Cube-Related Corner Coalesced Nets. Molecules. 2019; 24(7):1221. https://doi.org/10.3390/molecules24071221
Chicago/Turabian StyleDiudea, Mircea V. 2019. "Cube-Related Corner Coalesced Nets" Molecules 24, no. 7: 1221. https://doi.org/10.3390/molecules24071221
APA StyleDiudea, M. V. (2019). Cube-Related Corner Coalesced Nets. Molecules, 24(7), 1221. https://doi.org/10.3390/molecules24071221