Icosahedron's Secondary Great Circle Data


Now for the "secondary" great circle arcs of the Icosahedron's LCD spherical triangle.

Recall the primary Icosahedron's LCD spherical triangle.

We now use the "new" points D, E, and F generated by the primary rotation axes as new axes of rotation for the icosahedron.

All the axes passing through all the new D points generate the following new arcs.

All the axes passing through all the new E points generate the following new arcs.

All the axes passing through all the new F points generate the following new arcs.

If you get the impression that the arcs get "bounced" back into the LCD triangle as they "hit" the outer walls of the LCD spherical triangle you are correct. The arcs get "reflected" back into the LCD triangle. This is what Fuller meant by the energy holding patterns of the LCD spherical triangles. (Actually, because we are really spinning a polyhedron to generate a great circle equator, the arcs pass through the LCD edges and into the next LCD triangle. Because the adjacent LCD triangles are mirror images of each other and because we are mapping all the arc data into one LCD triangle, it looks like the arcs bounce back into the LCD triangle. There is validity in both ways of looking at it.)

All these new arcs, together with the original arcs, produce the secondary arcs for the icosahedron LCD triangle. This is shown next.

Now for the secondary arc angular data tables. If you know of any application for this data please let me know.

Icosahedron's Surface LCD Angular Data
Label Central Angle Edge Factor
A.1 7.434 809 922 8 0.098 623 200 0
A.2 7.473 084 037 9 0.099 136 697 6
A.3 7.761 243 907 0 0.103 005 664 7
A.5 8.227 129 662 3 0.109 272 179 4
A.13 11.211 433 552 5 0.149 801 503 5
B.E 13.282 525 588 5 0.239 363 534 5
B.F 10.812 316 963 5 0.190 983 005 6
B.34 11.354 628 092 2 0.206 011 329 5
C.13 9.693 723 895 3 0.138 873 631 0
C.20 7.201 242 121 1 0.103 850 861 7
C.25 6.645 371 907 7 0.095 491 502 8
C.30 6.349 948 932 1 0.090 535 082 2
C.35 6.405 037 162 6 0.090 588 747 8
C.39 6.716 268 278 8 0.094 632 483 3
C.40 8.300 774 799 5 0.118 033 988 7
D.1 3.377 507 040 6 0.045 714 367 2
D.6 4.172 280 248 1 0.057 950 324 3
D.8 4.386 177 559 1 0.059 016 994 3
E.11 4.499 294 987 5 0.068 938 410 7
E.22 4.675 036 786 5 0.068 476 708 2
E.31 4.269 282 630 9 0.061 803 398 8
E.34 4.273 238 778 0 0.065 146 502 5
F.34 3.488 005 128 9 0.055 641 755 9
F.42 5.454 720 361 5 0.085 410 196 6
1.2 2.867 486 668 5 0.038 142 083 6
1.4 3.321 598 905 2 0.044 306 525 2
2.3 1.067 453 366 8 0.014 214 715 3
2.4 1.190 705 307 7 0.016 022 260 0
3.5 1.174 468 353 4 0.015 667 991 3
3.7 1.801 498 225 1 0.024 316 338 9
4.7 1.520 969 542 4 0.020 410 037 3
4.8 2.994 501 554 3 0.040 778 013 3
5.10 2.658 120 790 0 0.036 062 751 3
5.13 4.743 940 904 8 0.063 858 197 1
6.8 6.050 821 841 2 0.082 711 823 2
6.11 4.610 950 352 8 0.066 760 432 2
6.14 5.684 213 110 7 0.077 554 332 2
6.16 4.476 071 059 6 0.063 186 210 4
6.21 6.777 786 990 0 0.093 794 673 7
7.9 1.595 408 730 6 0.021 749 194 7
7.10 1.906 868 791 5 0.025 758 233 8
8.9 1.613 156 731 3 0.021 749 194 7
8.12 1.795 236 704 1 0.024 316 338 9
8.14 3.137 695 856 0 0.043 687 150 1
9.10 1.507 502 626 6 0.020 270 482 9
9.12 1.278 129 830 7 0.017 595 468 1
10.13 4.248 120 106 9 0.057 086 680 8
10.15 2.162 899 834 5 0.029 516 309 5
10.17 2.695 860 468 3 0.037 239 255 0
11.16 3.586 955 388 4 0.050 383 224 9
11.19 4.367 084 042 4 0.061 128 073 4
11.22 4.216 785 519 4 0.060 497 446 3
11.26 5.513 331 157 1 0.077 788 070 5
12.17 2.086 284 288 9 0.028 470 065 5
12.18 2.811 400 889 0 0.039 344 662 9
13.15 3.459 802 563 4 0.046 791 675 2
13.20 3.916 638 157 2 0.053 891 081 5
14.18 2.060 087 057 5 0.028 182 467 3
14.21 2.920 002 312 9 0.041 802 486 2
15.17 1.825 294 083 0 0.024 922 359 4
15.20 2.190 236 862 5 0.030 303 273 0
16.19 1.511 494 713 6 0.021 896 357 3
16.21 3.723 797 243 9 0.051 726 541 9
17.18 2.548 068 549 6 0.035 190 936 3
17.23 2.433 978 945 3 0.034 352 309 1
17.25 3.086 929 539 9 0.042 705 098 3
18.21 3.302 711 904 9 0.046 534 663 9
18.23 2.225 637 547 8 0.030 625 176 0
18.27 2.452 873 254 4 0.035 190 936 3
19.24 1.535 510 716 6 0.021 538 744 2
19.26 1.575 434 826 2 0.023 188 614 5
20.25 1.414 201 620 5 0.019 833 749 7
21.24 2.566 599 780 0 0.037 089 934 3
21.27 2.359 154 603 2 0.032 664 055 8
21.29 2.433 978 945 3 0.035 890 367 8
21.33 3.137 695 856 0 0.044 616 593 8
22.26 2.041 231 492 3 0.029 243 045 5
22.28 1.250 993 914 0 0.018 310 047 8
23.30 2.141 728 182 2 0.029 731 545 4
23.32 1.978 721 520 5 0.028 544 995 7
24.26 1.314 921 213 0 0.019 381 066 5
24.29 1.634 821 345 0 0.023 013 734 8
25.30 1.528 205 547 9 0.021 706 594 9
26.28 1.795 236 704 1 0.026 936 308 5
26.29 2.471 740 552 8 0.035 039 389 0
26.36 2.688 270 075 9 0.038 584 393 8
26.42 5.553 615 682 9 0.085 410 196 6
27.32 2.394 957 105 8 0.033 211 312 4
27.33 1.936 967 977 6 0.028 470 065 5
28.31 2.264 732 860 1 0.034 867 090 2
28.42 5.013 976 139 6 0.075 629 564 2
29.33 2.688 270 075 9 0.037 796 614 9
29.36 1.851 745 660 0 0.028 056 710 3
29.38 2.873 538 561 4 0.040 770 805 1
30.35 1.473 159 285 1 0.021 228 061 6
31.34 2.585 402 343 4 0.041 202 265 9
31.42 5.207 455 268 0 0.076 393 202 2
32.35 1.494 727 239 6 0.020 805 380 4
32.37 1.477 409 247 8 0.021 730 484 3
33.37 2.601 233 278 1 0.036 418 002 0
33.38 1.414 201 620 5 0.021 220 333 9
33.40 3.934 435 560 7 0.057 635 540 4
35.39 1.271 873 944 9 0.018 594 344 1
36.41 3.511 547 321 8 0.051 445 858 4
36.42 3.785 938 109 3 0.059 774 685 8
37.39 1.044 975 628 1 0.014 621 522 7
37.40 2.952 856 445 3 0.044 671 967 1
38.40 3.182 319 528 2 0.045 851 328 4
38.41 2.748 458 166 8 0.042 440 667 9
39.40 3.132 150 742 0 0.047 003 974 0
40.41 3.339 948 337 2 0.048 632 677 9
41.42 3.809 713 949 5 0.056 940 131 0

Icosahedron's LCD Central Angular Data
Label Spherical Surface AnglePlane Angle
AC20.905 157 44845+acos(1,2)
1.A.2 22.238 756 092 9 22.238 756 092 9
2.A.3 7.761 243 907 0 7.761 243 907 0
3.A.5 7.761 243 907 0 7.761 243 907 0
5.A.13 22.238 756 092 9 22.238 756 092 9
E.B.34 18.000 000 000 0 14.477 512 185 9
F.B.34 18.000 000 000 0 15.522 487 814 0
13.C.20 20.905 157 447 8 19.637 522 814 8
20.C.25 10.812 316 963 5 10.362 477 185 1
25.C.30 13.282 525 588 5 13.050 747 261 1
30.C.35 13.282 525 588 5 13.461 247 609 8
35.C.39 10.812 316 963 5 11.249 249 036 0
39.C.40 20.905 157 447 8 22.238 756 092 9
1.D.8 90.000 000 000 0 90.000 000 000 0
6.D.8 90.000 000 000 0 90.000 000 000 0
B.E.34 54.735 610 317 2 52.238 756 092 9
11.E.22 54.735 610 317 2 52.238 756 092 9
22.E.31 35.264 389 682 7 37.761 243 907 0
31.E.34 35.264 389 682 7 37.761 243 907 0
B.F.34 90.000 000 000 0 97.761 243 907 0
34.F.42 90.000 000 000 0 82.238 756 092 9
A.1.2 79.722 778 671 6 79.637 522 814 8
D.1.4 79.722 778 671 6 79.637 522 814 8
2.1.4 20.554 442 656 7 20.724 954 370 3
A.2.1 78.221 767 845 4 78.123 721 092 2
A.2.3 101.778 232 154 5 101.876 278 907 7
1.2.4 101.778 232 154 5 101.876 278 907 7
A.3.2 70.528 779 365 5 70.362 477 185 1
A.3.5 109.471 220 634 4 109.637 522 814 8
2.3.7 109.471 220 634 4 109.637 522 814 8
5.3.7 70.528 779 365 5 70.362 477 185 1
1.4.2 57.696 502 193 4 57.398 766 721 8
1.4.8 122.303 497 806 5 122.601 233 278 1
2.4.7 122.303 497 806 5 122.601 233 278 1
7.4.8 57.696 502 193 4 57.398 766 721 8
A.5.3 62.842 667 356 3 62.601 233 278 1
A.5.13 117.157 332 643 6 117.398 766 721 8
3.5.10 117.157 332 643 6 117.398 766 721 8
10.5.13 62.842 667 356 3 62.601 233 278 1
D.6.8 46.512 922 254 4 45.522 487 814 0
8.6.14 30.865 661 264 7 31.426 764 924 7
11.6.16 46.512 922 254 4 45.522 487 814 0
14.6.21 25.242 832 961 5 26.101 494 522 3
16.6.21 30.865 661 264 7 31.426 764 924 7
3.7.4 50.032 698 384 4 49.637 522 814 8
3.7.10 129.967 301 615 5 130.362 477 185 1
4.7.9 129.967 301 615 5 130.362 477 185 1
9.7.10 50.032 698 384 4 49.637 522 814 8
D.8.4 68.176 536 445 6 67.761 243 907 0
D.8.6 43.646 927 108 7 44.477 512 185 9
4.8.9 68.176 536 445 6 67.761 243 907 0
6.8.14 68.176 536 445 6 67.761 243 907 0
9.8.12 43.646 927 108 8 44.477 512 185 9
12.8.14 68.176 536 445 6 67.761 243 907 0
7.9.8 104.215 033 035 6 104.477 512 185 9
7.9.10 75.784 966 964 3 75.522 487 814 0
8.9.12 75.784 966 964 3 75.522 487 814 0
10.9.12 104.215 033 035 6 104.477 512 185 9
5.10.7 42.393 817 630 8 41.876 278 907 7
5.10.13 83.403 500 479 6 83.283 731 721 1
7.10.9 54.202 681 889 5 54.839 989 371 1
9.10.17 83.403 500 479 6 83.283 731 721 1
13.10.15 54.202 681 889 5 54.839 989 371 1
15.10.17 42.393 817 630 8 41.876 278 907 7
E.11.22 64.828 737 828 0 63.488 005 128 9
6.11.16 64.828 737 828 0 63.488 005 128 9
16.11.19 18.840 722 769 7 19.795 726 592 1
19.11.26 12.661 078 804 4 13.432 536 557 7
22.11.26 18.840 722 769 7 19.795 726 592 1
8.12.9 60.585 549 013 5 60.000 000 000 0
8.12.18 119.414 450 986 4 120.000 000 000 0
9.12.17 119.414 450 986 4 120.000 000 000 0
17.12.18 60.585 549 013 5 60.000 000 000 0
A.13.5 40.908 063 754 0 40.362 477 185 1
C.13.20 40.908 063 754 0 40.362 477 185 1
5.13.10 33.851 773 963 2 34.115 035 000 7
10.13.15 30.480 324 565 3 31.044 975 628 1
15.13.20 33.851 773 963 2 34.115 035 000 7
6.14.8 81.111 708 177 0 80.811 991 168 2
6.14.21 98.888 291 822 9 99.188 008 831 7
8.14.18 98.888 291 822 9 99.188 008 831 7
18.14.21 81.111 708 177 0 80.811 991 168 2
10.15.13 95.382 039 903 5 94.115 035 000 7
10.15.17 84.617 960 096 4 85.884 964 999 2
13.15.20 84.617 960 096 4 85.884 964 999 2
17.15.20 95.382 039 903 5 94.115 035 000 7
6.16.11 68.789 009 702 0 70.989 507 056 9
6.16.21 111.210 990 297 9 109.010 492 943 0
11.16.19 111.210 990 297 9 109.010 492 943 0
19.16.21 68.789 009 702 0 70.989 507 056 9
C.17.23 53.022 528 567 6 52.238 756 092 9
10.17.12 53.022 528 567 6 52.238 756 092 9
10.17.15 53.022 528 567 6 52.238 756 092 9
12.17.18 73.954 942 864 6 75.522 487 814 0
13.17.25 73.954 942 864 6 75.522 487 814 0
18.17.23 53.022 528 567 6 52.238 756 092 9
C.18.27 73.614 860 764 3 73.050 747 261 1
12.18.14 73.614 860 764 3 73.050 747 261 1
12.18.17 45.504 099 046 0 44.477 512 185 9
14.18.21 60.881 040 189 5 62.471 740 552 8
17.18.23 60.881 040 189 5 62.471 740 552 8
21.18.27 45.504 099 046 0 44.477 512 185 9
11.19.16 49.992 418 197 5 51.193 780 464 8
11.19.26 130.007 581 802 4 128.806 219 535 1
16.19.24 130.007 581 802 4 128.806 219 535 1
24.19.26 49.992 418 197 5 51.193 780 464 8
C.20.13 118.403 872 466 1 120.000 000 000 0
C.20.25 61.596 127 533 8 60.000 000 000 0
13.20.15 61.596 127 533 8 60.000 000 000 0
15.20.25 118.403 872 466 1 120.000 000 000 0
6.21.14 56.012 156 418 5 54.710 496 645 8
6.21.16 38.059 129 171 4 39.562 742 132 1
14.21.18 38.059 129 171 4 36.716 268 278 8
16.21.24 47.869 585 238 6 49.010 492 943 0
18.21.27 47.869 585 238 6 49.010 492 943 0
24.21.29 38.059 129 171 4 36.716 268 278 8
27.21.33 38.059 129 171 4 39.562 742 132 1
29.21.33 56.012 156 418 5 54.710 496 645 8
E.22.11 60.585 549 013 5 64.273 238 778 0
E.22.28 119.414 450 986 4 115.726 761 221 9
11.22.26 119.414 450 986 4 115.726 761 221 9
26.22.28 60.585 549 013 5 64.273 238 778 0
17.23.18 66.139 669 285 2 65.289 503 354 1
17.23.30 113.860 330 714 7 114.710 496 645 8
18.23.32 113.860 330 714 7 114.710 496 645 8
30.23.32 66.139 669 285 2 65.289 503 354 1
19.24.21 113.411 197 083 8 111.193 780 464 8
19.24.26 66.588 802 916 2 68.806 219 535 1
21.24.29 66.588 802 916 2 68.806 219 535 1
26.24.29 113.411 197 083 8 111.193 780 464 8
C.25.20 107.669 807 107 0 109.637 522 814 8
C.25.30 72.330 192 892 9 70.362 477 185 1
17.25.20 72.330 192 892 9 70.362 477 185 1
17.25.30 107.669 807 107 0 109.637 522 814 8
11.26.19 37.377 368 140 6 37.761 243 907 0
11.26.22 41.810 314 895 7 44.477 512 185 9
19.26.24 63.434 948 822 9 60.000 000 000 0
22.26.28 37.377 368 140 6 37.761 243 907 0
24.26.29 37.377 368 140 6 37.761 243 907 0
28.26.42 63.434 948 822 9 60.000 000 000 0
29.26.36 41.810 314 895 7 44.477 512 185 9
36.26.42 37.377 368 140 6 37.761 243 907 0
18.27.21 86.676 742 876 3 86.511 994 871 0
18.27.32 93.323 257 123 6 93.488 005 128 9
21.27.33 93.323 257 123 6 93.488 005 128 9
32.27.33 86.676 742 876 3 86.511 994 871 0
22.28.26 82.056 495 259 9 77.965 517 314 9
22.28.31 97.943 504 740 0 102.034 482 685 0
26.28.42 97.943 504 740 0 102.034 482 685 0
31.28.42 82.056 495 259 9 77.965 517 314 9
21.29.24 75.385 673 586 0 74.477 512 185 9
21.29.33 75.385 673 586 0 74.477 512 185 9
24.29.26 29.228 652 827 9 31.044 975 628 1
26.29.36 75.385 673 586 0 74.477 512 185 9
33.29.38 29.228 652 827 9 31.044 975 628 1
36.29.38 75.385 673 586 0 74.477 512 185 9
C.30.25 94.471 803 731 1 96.586 775 553 6
C.30.35 85.528 196 268 8 83.413 224 446 3
23.30.25 85.528 196 268 8 83.413 224 446 3
23.30.35 94.471 803 731 1 96.586 775 553 6
E.31.28 107.502 684 108 4 104.477 512 185 9
E.31.34 72.497 315 891 5 75.522 487 814 0
28.31.42 72.497 315 891 5 75.522 487 814 0
34.31.42 107.502 684 108 4 104.477 512 185 9
C.32.37 79.287 906 235 8 78.750 750 963 9
23.32.27 79.287 906 235 8 78.750 750 963 9
23.32.35 100.712 093 764 1 101.249 249 036 0
27.32.37 100.712 093 764 1 101.249 249 036 0
21.33.27 48.657 434 208 8 46.949 252 738 8
21.33.29 48.657 434 208 9 50.811 991 168 2
27.33.37 82.685 131 582 2 82.238 756 092 9
29.33.38 82.685 131 582 2 82.238 756 092 9
37.33.40 48.657 434 208 9 50.811 991 168 2
38.33.40 48.657 434 208 9 46.949 252 738 8
B.34.E 107.669 807 107 0 113.283 731 721 1
B.34.F 72.330 192 892 9 66.716 268 278 8
E.34.31 72.330 192 892 9 66.716 268 278 8
F.34.31 107.669 807 107 0 113.283 731 721 1
C.35.30 81.270 746 278 3 83.125 527 943 7
C.35.39 98.729 253 721 6 96.874 472 056 2
30.35.32 98.729 253 721 6 96.874 472 056 2
32.35.39 81.270 746 278 3 83.125 527 943 7
26.36.29 62.842 667 356 3 61.044 975 628 1
26.36.42 117.157 332 643 6 118.955 024 371 8
29.36.41 117.157 332 643 6 118.955 024 371 8
41.36.42 62.842 667 356 3 61.044 975 628 1
32.37.33 90.000 000 000 0 90.000 000 000 0
32.37.39 90.000 000 000 0 90.000 000 000 0
33.37.40 90.000 000 000 0 90.000 000 000 0
39.37.40 90.000 000 000 0 90.000 000 000 0
29.38.33 68.119 128 653 6 66.716 268 278 8
29.38.41 111.880 871 346 3 113.283 731 721 1
33.38.40 111.880 871 346 3 113.283 731 721 1
40.38.41 68.119 128 653 6 66.716 268 278 8
C.39.35 70.528 779 365 5 71.876 278 907 7
C.39.40 109.471 220 634 4 108.123 721 092 2
35.39.37 109.471 220 634 4 108.123 721 092 2
37.39.40 70.528 779 365 5 71.876 278 907 7
C.40.39 49.797 034 113 4 49.637 522 814 8
33.40.37 41.409 622 109 2 39.188 008 831 7
33.40.38 19.498 154 831 9 19.767 015 540 0
37.40.39 19.498 154 831 9 18.123 721 092 2
38.40.41 49.797 034 113 4 53.283 731 721 1
36.41.38 55.690 639 534 4 53.283 731 721 1
36.41.42 62.154 680 232 7 66.716 268 278 8
38.41.40 62.154 680 232 7 60.000 000 000 0
F.42.31 55.105 900 902 9 60.000 000 000 0
26.42.28 18.699 407 085 1 17.965 517 314 9
26.42.36 25.544 395 554 4 23.283 731 721 1
28.42.31 25.544 395 554 4 26.511 994 871 0
36.42.41 55.105 900 902 9 52.238 756 092 9
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