The Magic Numbers of Ih Quasicrystal Beyond 20,000 shells

The Magic Numbers of Ih Quasicrystal Beyond 20,000 shells ---From an n‑Shell Lattice Model of 3D Quasicrystals n=20,000, NV=12, NE=599,970, NF=3,999,400,020, Nn=4,000,000,002 N(QC)=26,668,666,740,001, N(QC)/13=2,051,435,903,077. ---Calculated with the n-shell Lattice Model for 3D Quasicrystals. Abstract When the number of shells n = 20,000, the corresponding quasicrystal contains N(QC) = 26,668,666,740,001 atoms—a value known as a magic number; dividing this number by 13 yields the super-magic number, namely N(QC)/13 = 2,051,435,903,077. Based on the physical picture of the Ih point group, utilizing the Rubik's Cube (Oh) [1] and the Megaminx (Ih) [1, 2] as physical models, and employing a 3D lattice model [3] grounded in the Atomic Environment Calculation (AEC) [4] framework of the 230 space groups, this study calculated the magic numbers and super-magic numbers for quasicrystals up to n=20,000 using a recursive method inspired by Fibonacci numbers. The structure of the quasicrystal 3D lattice model [3] is divided into three levels: shell, cell, and quasicrystal (QC). A "cell" (also known as a QC cell) is the fundamental symmetry unit constituting a quasicrystal; it contains the lattice points required to generate the complete set of orbitals possessing Ih point-group symmetry and a degeneracy of 1[2]. The "shell" is the growth front of the cell, as well as the growth front of the QC. If the QC cell is detached from the quasicrystal, it becomes a conventional crystal; its lattice parameters and calculation methods can be found in reference [3]. Therefore, the recurrence mechanism for quasicrystal magic numbers is analogous to the Fibonacci sequence, as the recurrence is based on the shell formula u+v+w=n [3]; this constitutes the growth mechanism of quasicrystals. In terms of the hierarchical relationship, the Shell belongs to the Cell, and the Cell belongs to the QC. The geometric shape of the shell is an equilateral triangle. There are three types of sites on the shell: vertices, edges, and faces. By construction, for the current shell (n), the shell's vertices are the vertices of the Cell and the QC; the shell's edges are the edges of the Cell and the QC; and the shell's faces are the faces of the Cell and the QC. It should be emphasized that, although icosahedral and dodecahedral quasicrystals share the same lattice (both possess Ih point group symmetry), the relationships between their vertices, edges, and faces differ. Table 1 Relationship between the number of shells and the number of atoms in QC n denotes the number of shells, NV, NE and NF denote the number of atoms at vertices, edges and faces respectively; Nn denotes the number of atoms in nth shell; NQC denotes the number of atoms of QC with n shells; the last column is the number of atoms divided by 13. n NV NE NF Nn (Shell) N (QC) N(QC)/13 ----------------------------------------------------------------------- 1 12 0 0 12 13 1 2 12 30 0 42 55 4.2307692308 3 12 60 20 92 147 11.3076923077 4 12 90 60 162 309 23.7692307692 5 12 120 120 252 561 43.1538461538 6 12 150 200 362 923 71 7 12 180 300 492 1415 108.8461538462 8 12 210 420 642 2057 158.2307692308 9 12 240 560 812 2869 220.6923076923 10 12 270 720 1002 3871 297.7692307692 11 12 300 900 1212 5083 391 12 12 330 1100 1442 6525 501.9230769231 13 12 360 1320 1692 8217 632.0769230769 14 12 390 1560 1962 10179 783 15 12 420 1820 2252 12431 956.2307692308 16 12 450 2100 2562 14993 1153.3076923077 17 12 480 2400 2892 17885 1375.7692307692 18 12 510 2720 3242 21127 1625.1538461538 19 12 540 3060 3612 24739 1903 20 12 570 3420 4002 28741 2210.8461538462 21 12 600 3800 4412 33153 2550.2307692308 22 12 630 4200 4842 37995 2922.6923076923 23 12 660 4620 5292 43287 3329.7692307692 24 12 690 5060 5762 49049 3773 25 12 720 5520 6252 55301 4253.9230769231 26 12 750 6000 6762 62063 4774.0769230769 27 12 780 6500 7292 69355 5335 28 12 810 7020 7842 77197 5938.2307692308 29 12 840 7560 8412 85609 6585.3076923077 30 12 870 8120 9002 94611 7277.7692307692 31 12 900 8700 9612 104223 8017.1538461538 32 12 930 9300 10242 114465 8805 33 12 960 9920 10892 125357 9642.8461538462 34 12 990 10560 11562 136919 10532.2307692308 35 12 1020 11220 12252 149171 11474.6923076923 36 12 1050 11900 12962 162133 12471.7692307692 37 12 1080 12600 13692 175825 13525 38 12 1110 13320 14442 190267 14635.9230769231 39 12 1140 14060 15212 205479 15806.0769230769 40 12 1170 14820 16002 221481 17037 41 12 1200 15600 16812 238293 18330.2307692308 42 12 1230 16400 17642 255935 19687.3076923077 43 12 1260 17220 18492 274427 21109.7692307692 44 12 1290 18060 19362 293789 22599.1538461538 45 12 1320 18920 20252 314041 24157 46 12 1350 19800 21162 335203 25784.8461538462 47 12 1380 20700 22092 357295 27484.2307692308 48 12 1410 21620 23042 380337 29256.6923076923 49 12 1440 22560 24012 404349 31103.7692307692 50 12 1470 23520 25002 429351 33027 51 12 1500 24500 26012 455363 35027.9230769231 52 12 1530 25500 27042 482405 37108.0769230769 53 12 1560 26520 28092 510497 39269 54 12 1590 27560 29162 539659 41512.2307692308 55 12 1620 28620 30252 569911 43839.3076923077 56 12 1650 29700 31362 601273 46251.7692307692 57 12 1680 30800 32492 633765 48751.1538461538 58 12 1710 31920 33642 667407 51339 59 12 1740 33060 34812 702219 54016.8461538462 60 12 1770 34220 36002 738221 56786.2307692308 61 12 1800 35400 37212 775433 59648.6923076923 62 12 1830 36600 38442 813875 62605.7692307692 63 12 1860 37820 39692 853567 65659 64 12 1890 39060 40962 894529 68809.9230769231 65 12 1920 40320 42252 936781 72060.0769230769 66 12 1950 41600 43562 980343 75411 67 12 1980 42900 44892 1025235 78864.2307692308 68 12 2010 44220 46242 1071477 82421.3076923077 69 12 2040 45560 47612 1119089 86083.7692307692 70 12 2070 46920 49002 1168091 89853.1538461538 71 12 2100 48300 50412 1218503 93731 72 12 2130 49700 51842 1270345 97718.8461538462 73 12 2160 51120 53292 1323637 101818.2307692308 74 12 2190 52560 54762 1378399 106030.6923076923 75 12 2220 54020 56252 1434651 110357.7692307692 76 12 2250 55500 57762 1492413 114801 77 12 2280 57000 59292 1551705 119361.9230769231 78 12 2310 58520 60842 1612547 124042.0769230769 79 12 2340 60060 62412 1674959 128843 80 12 2370 61620 64002 1738961 133766.2307692308 81 12 2400 63200 65612 1804573 138813.3076923077 82 12 2430 64800 67242 1871815 143985.7692307692 83 12 2460 66420 68892 1940707 149285.1538461538 84 12 2490 68060 70562 2011269 154713 85 12 2520 69720 72252 2083521 160270.8461538462 86 12 2550 71400 73962 2157483 165960.2307692308 87 12 2580 73100 75692 2233175 171782.6923076923 88 12 2610 74820 77442 2310617 177739.7692307692 89 12 2640 76560 79212 2389829 183833 90 12 2670 78320 81002 2470831 190063.9230769231 91 12 2700 80100 82812 2553643 196434.0769230769 92 12 2730 81900 84642 2638285 202945 93 12 2760 83720 86492 2724777 209598.2307692308 94 12 2790 85560 88362 2813139 216395.3076923077 95 12 2820 87420 90252 2903391 223337.7692307692 96 12 2850 89300 92162 2995553 230427.1538461538 97 12 2880 91200 94092 3089645