# Combined VE & Icosahedron Great Circle Data

It is possible to fit an icosahedron inside a VE. The triangular faces of the icosahedron will then be skewed with respect to the triangular faces of the VE. The icosahedron face can be skewed (rotated) either clockwise or counter clockwise about the face-center-to-face-center axis of the icosahedron.

Consider a triangle and square face pair of the VE and two triangular faces of the icosahedron with their primary great circle arcs as shown below.

Next the icosahedron triangular face pair is rotated about an axis passing through the face center of the "top" icosahedron triangle. The rotation can be either clockwise or counter clockwise. These rotated pairs are then superimposed onto the VE triangle and square face pair matching up the triangular face center points. The icosahedron triangular faces are highlighted in a bold outline.

If you now look for the LCD spherical triangle of the VE you will see the new LCD spherical triangle for the combined VE and icosahedron system. This next illustration shows this LCD triangle with labels for the outer edge intersection points. Unfortunately, because of the size and quality of the image I don't think I can label all the inner intersection points. So you'll have to go to Fullr's book Synergetics 2 for the remainder of the labels.

So here are the tables containing the angular data for the combined VE and icosahedron systems.

Comb. VE & Icosahedron Surface Data
LABEL Central Angle Edge Factor
A.1 11.658 290 899 9 0.168 468 538 1
A.2 10.812 316 963 5 0.155 936 971 1
A.3 10.909 891 609 4 0.157 378 651 6
A.4 11.211 433 552 5 0.161 840 005 7
A.7 13.581 110 920 6 0.197 246 400 0
B.7 5.890 109 713 8 0.091 428 734 5
B.15 7.761 243 907 0 0.118 033 988 7
B.18 2.952 856 445 3 0.052 786 404 5
C.14 1.126 133 218 5 0.018 576 030 0
C.15 3.132 150 742 0 0.048 632 677 9
C.16 3.584 117 536 7 0.056 940 131 0
C.21 4.568 891 002 4 0.076 895 015 4
D.8 9.476 737 898 9 0.182 857 469 0
D.17 7.761 243 907 0 0.145 898 033 7
D.19 8.225 780 113 0 0.166 925 267 7
D.34 13.282 525 588 5 0.270 090 756 7
E.26 3.137 695 856 0 0.048 632 677 9
E.32 2.771 639 301 6 0.038 447 507 7
E.33 5.641 161 811 3 0.076 895 015 4
E.40 6.777 786 990 0 0.097 265 355 8
F.21 1.784 279 916 7 0.028 514 239 8
F.24 0.898 066 005 0 0.014 257 119 9
F.25 1.677 667 308 6 0.026 155 775 1
F.31 3.392 939 967 5 0.052 311 550 2
G.34 5.152 423 234 3 0.083 462 633 8
G.35 3.090 986 420 2 0.044 056 923 5
G.37 6.032 116 636 4 0.083 462 633 8
G.43 5.659 893 729 1 0.083 462 633 8
H.40 18.461 614 830 6 0.236 067 977 4
H.41 18.000 000 000 0 0.229 752 920 5
H.42 18.303 816 057 5 0.233 905 456 6
H.43 20.905 157 447 8 0.270 090 756 7
1.2 4.386 177 559 1 0.063 759 778 6
1.5 8.491 677 114 3 0.131 134 468 8
1.6 6.050 821 841 2 0.089 358 795 7
2.3 1.464 602 895 4 0.021 253 259 5
2.6 4.172 280 248 1 0.062 607 387 7
3.4 1.517 999 010 4 0.022 057 807 2
3.9 6.411 448 058 7 0.097 265 355 8
4.7 5.285 096 646 4 0.077 477 191 6
4.15 9.693 723 895 3 0.150 033 936 4
5.8 5.637 683 769 4 0.094 889 793 1
5.10 7.537 416 104 0 0.114 173 361 6
5.11 6.055 858 089 6 0.093 583 350 4
6. 9 3.176 823 040 3 0.048 195 189 5
6.10 4.610 950 352 8 0.072 125 502 6
7.15 9.732 301 447 7 0.149 302 498 3
8.11 7.149 662 286 8 0.114 056 959 4
8.17 5.454 720 361 5 0.092 274 018 5
9.13 2.738 243 780 9 0.043 498 389 4
9.15 4.386 177 559 1 0.068 776 992 6
10.12 2.029 086 362 6 0.033 010 149 9
10.13 2.671 200 742 6 0.040 540 965 8
11.12 2.640 881 070 3 0.040 897 423 7
11.16 3.132 150 742 0 0.049 844 718 9
12.14 2.875 640 202 7 0.044 115 614 4
12.16 2.470 208 624 9 0.041 468 360 1
13.14 1.021 937 630 8 0.016 614 906 3
13.15 2.842 130 414 4 0.043 498 389 4
14.15 2.784 253 526 4 0.042 506 519 1
15.18 8.300 774 799 5 0.129 299 756 3
15.20 6.199 817 397 8 0.103 165 489 0
15.21 6.645 371 907 7 0.103 165 489 0
15.23 5.553 615 682 9 0.098 116 210 5
15.27 7.255 972 925 4 0.121 278 305 9
16.17 7.761 243 907 0 0.130 495 168 4
16.21 4.273 238 778 0 0.079 911 644 1
16.22 5.335 169 839 4 0.087 831 396 2
17.19 2.733 554 118 1 0.054 237 307 2
18.20 3.339 948 337 2 0.049 388 112 2
18.26 3.934 435 560 7 0.065 247 584 2
19.22 4.939 325 954 6 0.086 622 133 5
19.34 8.078 762 845 4 0.141 995 113 9
20.23 3.809 713 949 5 0.055 217 588 0
20.26 4.162 659 787 3 0.063 759 778 6
21.24 2.126 983 210 6 0.031 879 889 3
21.25 2.235 962 523 5 0.038 693 493 5
22.24 2.960 694 112 2 0.048 423 244 7
22.34 7.947 355 749 0 0.121 380 004 3
23.27 3.858 641 165 3 0.055 217 588 0
23.29 2.714 082 869 3 0.043 878 903 3
24.34 9.227 644 881 6 0.135 045 378 3
25.27 2.028 465 913 4 0.030 872 704 6
25.28 1.252 042 605 4 0.020 869 129 2
26.32 3.535 409 062 8 0.051 582 744 5
27.28 1.596 079 196 2 0.022 750 897 4
27.29 4.340 706 639 4 0.063 759 778 6
27.30 2.437 750 969 9 0.037 477 057 6
28.30 1.842 834 648 7 0.029 781 311 4
28.31 3.828 255 534 4 0.054 636 089 7
29.32 3.639 808 757 0 0.051 582 744 5
29.33 3.550 887 184 3 0.054 237 307 2
30.36 3.957 799 773 0 0.060 690 002 1
30.37 4.851 712 665 6 0.070 997 556 9
31.34 6.984 061 429 0 0.101 300 881 3
31.35 4.186 768 877 9 0.061 745 409 2
32.33 3.665 833 058 3 0.051 582 744 5
33.36 4.425 128 163 0 0.059 854 422 2
33.38 4.675 036 786 5 0.063 759 778 6
33.39 4.216 785 519 4 0.060 639 152 9
34.35 4.803 812 441 4 0.074 954 115 2
35.37 4.589 200 885 1 0.067 040 998 7
36.37 3.187 918 800 3 0.043 311 066 7
36.38 1.960 609 485 2 0.028 653 932 1
37.38 3.889 512 261 9 0.054 237 307 2
37.43 6.359 720 886 9 0.087 757 806 6
38.39 4.499 294 987 5 0.060 639 152 9
38.42 3.934 940 035 4 0.055 217 588 0
38.43 7.761 243 907 0 0.103 165 489 0
39.40 4.610 950 352 8 0.060 639 152 9
39.41 1.964 628 743 8 0.027 118 653 6
40.41 4.172 280 248 1 0.054 237 307 2
41.42 3.377 507 040 6 0.043 878 903 3
42.43 7.434 809 922 8 0.098 116 210 5

Comb. VE & Icosahedron Surface Angular Data
Label Spherical Surface AnglePlane Angle
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.4 7.761 243 907 0 7.761 243 907 0
4.A.7 22.238 756 092 9 22.238 756 092 9
7.B.15 90.000 000 000 0 90.000 000 000 0
15.B.18 90.000 000 000 0 90.000 000 000 0
14.C.15 61.874 494 297 9 60.000 000 000 0
14.C.16 118.125 505 702 0 120.000 000 000 0
15.C.21 118.125 505 702 0 108.434 948 822 9
16.C.21 61.874 494 297 9 71.565 051 177 0
8.D.17 35.264 389 682 7 30.000 000 000 0
17.D.19 19.471 220 634 4 18.434 948 822 9
19.D.34 35.264 389 682 7 26.565 051 177 0
26.E.32 73.221 345 119 0 71.565 051 177 0
32.E.33 33.557 309 761 9 36.869 897 645 8
33.E.40 73.221 345 119 0 71.565 051 177 0
21.F.24 99.594 068 226 9 90.000 000 000 0
21.F.25 80.405 931 773 1 90.000 000 000 0
24.F.31 80.405 931 773 0 90.000 000 000 0
25.F.31 99.594 068 226 9 90.000 000 000 0
34.G.35 65.905 157 447 8 63.434 948 822 9
35.G.37 48.189 685 104 2 53.130 102 354 1
37.G.43 65.905 157 447 8 63.434 948 822 9
40.H.41 13.282 525 588 5 13.282 525 588 5
41.H.42 10.812 316 963 5 10.812 316 963 5
42.H.43 20.905 157 447 8 20.905 157 447 8
A.1.2 68.176 536 445 6 67.761 243 907 0
2.1.6 43.646 927 108 7 44.477 512 185 9
5.1.6 68.176 536 445 6 67.761 243 907 0
A.2.1 90.000 000 000 0 90.000 000 000 0
A.2.3 90.000 000 000 0 90.000 000 000 0
1.2.6 90.000 000 000 0 90.000 000 000 0
3.2.6 90.000 000 000 0 90.000 000 000 0
A.3.2 82.377 368 140 6 82.238 756 092 9
A.3.4 97.622 631 859 3 97.761 243 907 0
2.3.9 97.622 631 859 3 97.761 243 907 0
4.3.9 82.377 368 140 6 82.238 756 092 9
A.4.3 74.759 837 717 3 74.477 512 185 9
A.4.7 105.240 162 282 6 105.522 487 814 0
3.4.15 105.240 162 282 6 105.522 487 814 0
7.4.15 74.759 837 717 3 74.477 512 185 9
1.5.10 75.385 673 586 0 74.477 512 185 9
8.5.11 75.385 673 586 0 74.477 512 185 9
10.5.11 29.228 652 827 9 31.044 975 628 1
1.6.2 46.512 922 254 4 45.522 487 814 0
1.6.10 133.487 077 745 5 134.477 512 185 9
2.6.9 133.487 077 745 5 134.477 512 185 9
9.6.10 46.512 922 254 4 45.522 487 814 0
B.7.15 53.022 528 567 6 52.238 756 092 9
4.7.15 73.954 942 864 6 75.522 487 814 0
A.7.4 53.022 528 567 6 52.238 756 092 9
D.8.17 55.105 900 902 9 52.238 756 092 9
5.8.11 55.105 900 902 9 52.238 756 092 9
11.8.17 69.788 198 194 1 75.522 487 814 0
3.9.6 39.055 646 138 6 37.761 243 907 0
3.9.15 140.944 353 861 3 142.238 756 092 9
6.9.13 140.944 353 861 3 142.238 756 092 9
13.9.15 39.055 646 138 6 37.761 243 907 0
5.10.6 83.669 460 597 7 83.283 731 721 1
5.10.12 96.330 539 402 2 96.716 268 278 8
6.10.13 96.330 539 402 2 96.716 268 278 8
12.10.13 83.669 460 597 7 83.283 731 721 1
5.11.8 49.797 034 113 4 53.283 731 721 1
5.11.12 130.202 965 886 5 126.716 268 278 8
8.11.16 130.202 965 886 5 126.716 268 278 8
12.11.16 49.797 034 113 4 53.283 731 721 1
10.12.11 104.477 512 185 9 105.522 487 814 0
10.12.14 75.522 487 814 0 74.477 512 185 9
11.12.16 75.522 487 814 0 74.477 512 185 9
14.12.16 104.477 512 185 9 105.522 487 814 0
9.13.10 76.366 977 774 6 75.522 487 814 0
9.13.15 103.633 022 225 3 104.477 512 185 9
10.13.14 103.633 022 225 3 104.477 512 185 9
14.13.15 76.366 977 774 6 75.522 487 814 0
C.14.12 82.752 501 441 4 82.238 756 092 9
C.14.15 97.247 498 558 5 97.761 243 907 0
12.14.13 97.247 498 558 5 97.761 243 907 0
13.14.15 82.752 501 441 4 82.238 756 092 9
B.15.7 37.377 368 140 6 37.761 243 907 0
B.15.18 20.905 157 447 8 24.094 842 552 1
C.15.14 20.905 157 447 8 22.238 756 092 9
C.15.21 37.377 368 140 6 45.000 000 000 0
4.15.7 31.717 474 411 4 30.000 000 000 0
4.15.9 31.717 474 411 4 30.000 000 000 0
9.15.13 37.377 368 140 6 37.761 243 907 0
13.15.14 20.905 157 447 8 22.238 756 092 9
18.15.20 20.905 157 447 8 20.905 157 447 8
20.15.23 37.377 368 140 6 31.717 474 411 4
21.15.27 31.717 474 411 4 31.717 474 411 4
23.15.27 31.717 474 411 4 26.565 051 177 0
C.16.12 54.735 610 317 2 52.238 756 092 9
C.16.21 70.528 779 365 5 65.905 157 447 8
11.16.12 54.735 610 317 2 52.238 756 092 9
11.16.17 70.528 779 365 5 75.522 487 814 0
17.16.22 54.735 610 317 2 65.905 157 447 8
21.16.22 54.735 610 317 2 48.189 685 104 2
D.17.8 90.000 000 000 0 97.761 243 907 0
D.17.19 90.000 000 000 0 103.282 525 588 5
8.17.16 90.000 000 000 0 82.238 756 092 9
16.17.19 90.000 000 000 0 76.717 474 411 4
B.18.15 69.295 188 945 3 65.905 157 447 8
15.18.20 41.409 622 109 2 48.189 685 104 2
20.18.26 69.295 188 945 3 65.905 157 447 8
D.19.17 70.714 240 615 3 58.282 525 588 5
D.19.34 109.285 759 384 6 121.717 474 411 4
17.19.22 109.285 759 384 6 121.717 474 411 4
22.19.34 70.714 240 615 3 58.282 525 588 5
15.20.18 117.845 319 767 1 110.905 157 447 8
15.20.23 62.154 680 232 7 69.094 842 552 1
18.20.26 62.154 680 232 7 69.094 842 552 1
23.20.26 117.845 319 767 1 110.905 157 447 8
C.21.15 24.607 404 490 3 26.565 051 177 0
C.21.16 47.722 788 402 6 42.529 791 375 0
F.21.24 24.607 404 490 2 26.565 051 177 0
F.21.25 47.722 788 402 6 42.529 791 375 0
15.21.25 107.669 807 107 0 110.905 157 447 8
16.21.24 107.669 807 107 0 110.905 157 447 8
16.22.19 106.385 139 235 6 95.659 893 729 1
16.22.24 73.614 860 764 3 84.340 106 270 8
19.22.34 73.614 860 764 3 84.340 106 270 8
24.22.34 106.385 139 235 6 95.659 893 729 1
15.23.20 80.650 296 457 4 79.187 683 036 4
15.23.27 99.349 703 542 5 100.812 316 963 5
20.23.29 99.349 703 542 5 100.812 316 963 5
27.23.29 80.650 296 457 4 79.187 683 036 4
F.24.21 55.812 316 963 5 63.434 948 822 9
F.24.34 124.187 683 036 4 116.565 051 177 0
21.24.22 124.187 683 036 4 116.565 051 177 0
22.24.34 55.812 316 963 5 63.434 948 822 9
F.25.21 51.897 039 986 4 47.470 208 624 9
F.25.28 128.102 960 013 5 132.529 791 375 0
21.25.27 128.102 960 013 5 132.529 791 375 0
27.25.28 51.897 039 986 4 47.470 208 624 9
E.26.32 48.657 434 208 8 45.000 000 000 0
18.26.20 48.657 434 208 9 45.000 000 000 0
20.26.32 82.685 131 582 1 90.000 000 000 0
15.27.23 49.117 607 890 8 52.622 631 859 3
15.27.25 92.761 991 300 3 84.847 576 765 6
23.27.29 38.120 400 808 7 42.529 791 375 0
25.27.28 38.120 400 808 7 42.529 791 375 0
28.27.30 49.117 607 890 9 52.622 631 859 3
29.27.30 92.761 991 300 3 84.847 576 765 6
25.28.27 90.000 000 000 0 90.000 000 000 0
25.28.31 90.000 000 000 0 90.000 000 000 0
27.28.30 90.000 000 000 0 90.000 000 000 0
30.28.31 90.000 000 000 0 90.000 000 000 0
23.29.27 61.319 519 174 1 58.282 525 588 5
23.29.32 118.680 480 825 8 121.717 474 411 4
27.29.33 118.680 480 825 8 121.717 474 411 4
32.29.33 61.319 519 174 1 58.282 525 588 5
27.30.28 40.908 063 754 0 37.377 368 140 6
27.30.36 139.091 936 245 9 142.622 631 859 3
28.30.37 139.091 936 245 9 142.622 631 859 3
36.30.37 40.908 063 754 0 37.377 368 140 6
F.31.28 42.393 817 630 8 47.470 208 624 9
F.31.34 137.606 182 369 1 132.529 791 375 0
28.31.35 137.606 182 369 1 132.529 791 375 0
34.31.35 42.393 817 630 8 47.470 208 624 9
E.32.26 58.193 899 980 6 63.434 948 822 9
E.32.33 121.806 100 019 3 116.565 051 177 0
26.32.29 121.806 100 019 3 116.565 051 177 0
29.32.33 58.193 899 980 6 63.434 948 822 9
E.33.32 24.712 012 949 5 26.565 051 177 0
E.33.39 94.702 438 036 8 95.152 423 234 3
29.33.32 60.585 549 013 5 58.282 525 588 5
29.33.36 94.702 438 036 8 95.152 423 234 3
36.33.38 24.712 012 949 5 26.565 051 177 0
38.33.39 60.585 549 013 5 58.282 525 588 5
D.34.19 36.000 000 000 0 31.717 474 411 4
G.34.35 36.000 000 000 0 31.717 474 411 4
19.34.22 36.000 000 000 0 37.377 368 140 6
22.34.24 18.000 000 000 0 20.905 157 447 8
24.34.31 18.000 000 000 0 20.905 157 447 8
31.34.35 36.000 000 000 0 37.377 368 140 6
G.35.34 78.221 767 845 4 84.847 576 765 6
G.35.37 101.778 232 154 5 95.152 423 234 3
31.35.34 101.778 232 154 5 95.152 423 234 3
31.35.37 101.778 232 154 5 84.847 576 765 6
30.36.33 95.165 347 482 1 95.659 893 729 1
30.36.37 84.834 652 517 8 84.340 106 270 8
33.36.38 84.834 652 517 8 84.340 106 270 8
37.36.38 95.165 347 482 1 95.659 893 729 1
G.37.35 30.153 386 188 0 31.717 474 411 4
G.37.43 54.367 004 351 7 58.282 525 588 5
30.37.35 95.479 609 460 1 90.000 000 000 0
30.37.36 54.367 004 351 7 58.282 525 588 5
36.37.38 30.153 386 188 0 31.717 474 411 4
38.37.43 95.479 609 460 1 90.000 000 000 0
33.38.36 70.528 779 365 5 69.094 842 552 1
33.38.39 54.735 610 317 2 58.282 525 588 5
36.38.37 54.735 610 317 2 52.622 631 859 3
37.38.43 54.735 610 317 2 58.282 525 588 5
39.38.42 54.735 610 317 2 52.622 631 859 3
42.38.43 70.528 779 365 5 69.094 842 552 1
33.39.38 64.828 737 828 0 63.434 948 822 9
33.39.40 115.171 262 171 9 16.565 051 177 0
38.39.41 115.171 262 171 9 16.565 051 177 0
40.39.41 64.828 737 828 0 63.434 948 822 9
E.40.39 77.378 583 519 2 76.717 474 411 4
H.40.41 77.378 583 519 2 76.717 474 411 4
39.40.41 25.242 832 961 5 26.565 051 177 0
H.41.40 90.000 000 000 0 90.000 000 000 0
H.41.42 90.000 000 000 0 90.000 000 000 0
39.41.40 90.000 000 000 0 90.000 000 000 0
39.41.42 90.000 000 000 0 90.000 000 000 0
H.42.41 79.722 778 671 6 79.187 683 036 4
H.42.43 100.277 221 328 3 100.812 316 963 5
38.42.41 100.277 221 328 3 100.812 316 963 5
38.42.43 79.722 778 671 6 79.187 683 036 4
G.43.37 60.000 000 000 0 58.282 525 588 5
H.43.42 60.000 000 000 0 58.282 525 588 5
37.43.38 30.000 000 000 0 31.717 474 411 4
38.43.42 30.000 000 000 0 31.717 474 411 4