20 Class I Method 1 Icosahedral Coordinates

1Data are given for Class I icosa breakdowns of the following frequencies: $10, 8, 6, 5, 4, 3, 2$ . Pick the vertex labels for the fre-quency you are using out of the proper column at the left, and read values from the two columns at the right.

2 Each table covers two icosa faces: the #1 triangle whose corners are $0, 0$ ; 0, $\arctan 2$ ; 72, $\arctan 2$ ; and the downward-pointing #2 triangle beneath it whose comers are 0, $\arctan 2$ ; 72, $\arctan 2$ ; 36, ( $180 - \arctan 2$ ): see Diagram 12.3. For spheres use only the Symmetry Triangle shown in Diagram 12.2. For ellipsoids of revolution (one cross-section circular) use the left-hand half of both triangles as far down as the proposed truncation plane. For structures with two non-circular cross-sections you need a 90° span of 0 values as well. Diagram 20.1 shows how these can be extrapolated. But such structures have more tractable symmetries if rotated so that an edge, not a vertex, is at the zenith: see Chapter 15.

3 An HP-65 program using Equations 12.4 through 12.8 generated the values for Triangle #1. A second program using Equations 14.1, 14.2 then generated the Triangle #2 points by rotation.

45v lOv

10o

1100.0000000

1200.0000000

131,0

141,1

1500.0000000

1672.0000000

175.4084310

185.4084310

191,0 2,0

202,1

211,1 2,2

2200.0000000

2336.0000000

2472.0000000

2511.3716668

269.2416169

2711.3716668

283,0

293,1

303,2

313,3

3200.0000000

3322.3861776

3449.6138225

3572.0000000

3617.8315873

3714.9899922

3814.9899922

3917.8315873

40m m ro

41WWW

42MHO

43-t

44Xr UO MHO

4500.0000000

4616.0353713

4736.0000000

4855.9646288

4972.0000000

5024.6710680

5121.5724472

5220.3856407

5321.5724472

5424.6710680

555,0

565,1

575,2

585,3

59‘ 5,4

605,5

6100.0000000

6212.4463843

6327.7323015

6444.2676986

6559.5536157

6672.0000000

6731.7174744

6828.6099376

6926.8052365

7026.8052365

7128.6099376

7231.7174744

733,0 6,0

746,1

753,1 6,2

766,3 -

773,2 6,4

786,5

793,3 6,6

8000.0000000

8110.1562304

8222.3861776

8336.0000000

8449.6138225

8561.8437696

8672.0000000

8738.7638808

8835.8224702

8933.7590240

9033.0088450

9133.7590240

9235.8224702

9338.7638808

947,0

957,1

967,2

977,3

987,4

997,5

1007,6

1017,7

10200.0000000

1038.5725404

10418.7046980

10530.0743861

10641.9256139

10753.2953021

10863.4274596

10972.0000000

11045.6033616

11142.9496334

11240.8717741

11339.7156131

11439.7156131

11540.8717741

11642.9496334

11745.6033616

1184,0 8,0

1198,1

1204,1 8,2

1218,3

1224,2 8,4

12300.0000000

1247.4137289

12516.0353713

12625.7052785

12736.0000000

12852.0632820

12949.7664590

13047.8353485

13146.5286650

13246.0642057





1335v lOv		134e
1358,5 1364,3 8,6 1378,7 1384,4 8,8	13946.2947215 14055.9646288 14164.5862711 14272.0000000	14346.5286650 14447.8353485 14549.7664590 14652.0632820
1479,0 1489,1 1499,2 1509,3 1519,4 1529,5 1539,6 1549,7 1559,8 1569,9	15700.0000000 1586.5297024 15914.0192855 16022.3861776 16131.3846950 16240.6153050 16349.6138225 16457.9807145 16565.4702976 16672.0000000	16758.0265179 16856.1087309 16954.4163133 17053.1334950 17152.4366973 17252.4366973 17353.1334950 17454.4163133 17556.1087309 17658.0265179
1775,0 10,0 17810,1 1795,1 10,2 18010,3 1815,2 10,4 18210,5 1835,3 10,6 18410,7 1855,4 10,8 18610,9 1875.5 10,10	18800.0000000 1895.8333887 19012.4463843 19119.7952367 19227.7323015 19336.0000000 19444.2676986 19552.2047633 19659.5536157 19766.1666114 19872.0000000	19963.4349488 20061.8829712 20160.4665625 20259.3117249 20358.5495950 20458.2825256 20558.5495950 20659.3117249 20760.4665625 20861.8829712 20963.4349488






21011,1	2113.4292316	21267.8508314
21311,2	2149.5794832	21566.6079548
21611,3	21716.4605636	21865.5245982
21911,4	22023.9792000	22164.7124135
22211,5	22331.9400304	22464.2746575
22511,6	22640.0599696	22764.2746575
22811,7	22948.0207000	23064.7124135
23111,8	23255.5394364	23365.5245982
23411,9	23562.4205168	23666.6079548
23711,10	23868.5707684	23967.8508314
2406,1 12,2	2416.9684505	24272.7969533
24312,3	24413.3994138	24571.8969574
2466,2 12,4	24720.4898913	24871.1600132
24912,5	25028.1002356	25170.6721118
2526,3 12,6	25336.0000000	25470.5008455
25512,7	25643.8997644	25770.6721118
2586,4 12,8	25951.5101087	26071.1600132
26112,9	26258.6005862	26371.8969574
2646,5 12,10	26565.0315495	26672.7969533





2675v 10v	268V	269e
27013,3 27113,4 27213,5 27313,6 27413,7 27513,8 27613,9 27713,10	27810.5948977 27917.2607131 28024.4947652 28132.1184790 28239.8815210 28347.5052349 28454.7392869 28561.4051023	28678.2204432 28777.6677097 28877.2528359 28977.0290522 29077.0290522 29177.2528359 29277.6677097 29378.2204432
2947,2 14,4 29514,5 2967,3 14,6 29714,7 2987,4 14,8 29914,9 3007,5 14,10	30114.2819224 30221.1287296 30328.4369994 30436.0000000 30543.5630006 30650.8712704 30757.7180776	30884.0101625 30983.7703158 31083.6117970 31183.5562062 31283.6117970 31383.7703158 31484.0101625
31515,5 31615,6 31715,7 31815,8 31915,9	32018.0000000 32124.9684505 32232.2819224 32339.7180776 32447.0315495	32590.0000000 32690.0000000 32790.0000000 32890.0000000 32990.0000000





3305v lOv	331I .	3326
33315,10 3348,3 16,6 33516,7 3368,4 16,8 33716,9 3388,5 16,10	33954.0000000 34021.7180776 34128.7463723 34236.0000000 34343.2536277 34450.2819224	34590.0000000 34695.9898376 34796.1303345 34896.1794105 34996.1303345 35095.9898376
35117,7 35217,8 35317,9 35417,10	35525.4051023 35632.4321744 35739.5678256 35846.5948977	359101.7795569 360101.9554615 361101.9554615 362101.7795569
3639,4 18,8 36418,9 3659,5 18,10	36629.0315495 36736.0000000 36842.9684505	369107.2030468 370107.3234344 371107.2030468
37219,9 37319,10	37432.5707684 37539.4292316	376112.1491687 377112.1491687
37810,5 20,10	37936.0000000	380II6.5650512





3812v 4v 8v	382*	383e
3840,0 0,0 0,0	38500.0000000	38600.0000000
387 3881,0	38900.0000000	3906.8485493
391 3921,1	39372.0000000	3946.8485493
3951,0 2,0	39600.0000000	39714.5454366
398 3992,1	40036.0000000	40111.8548751
4021,1 2,2	40372.0000000	40414.5454366
405 4063,0	40700.0000000	40822.9342438
409 4103,1	41122.3861776	41219.4026533
413 4143,2	41549.6138225	41619.4026533
417 4183,3	41972.0000000	42022.9342438
4211,0 2,0 4,0	42200.0000000	42331.7174744
424 4254,1	42616.0353713	42728.0116135
4282,1 4,2	42936.0000000	43026.5650512
431 4324,3	43355.9646288	43428.0116135
4351,1 2,2 4,4	43672.0000000	43731.7174744
438 4395,0	44000.0000000	44140.5007050
4425,1	44312.4463843	44437-0083770
445 4465,2	44727.7323015	44834.9242784
449 4505,3	45144.2676986	45234.9242784
453 4545,4	45559.5536157	45637.0083770
457 4585,5	45972.0000000	46040.5007050
4613,0 6,0	46200.0000000	46348.8895123
464 4656,1	46610.1562304	46745.8486935
4683,1 6,2	46922.3861776	47043.6469271
471 4726,3	47336.0000000	47442.8320885
4753,2 6,4	47649.6138225	47743.6469271
478 4796,5	48061.8437696	48145.8486935
4823,3 6,6	48372.0000000	48448.8895123
485 4867,0	48700.0000000	48856.5863995
489 4907,1	4918.5725404	49254.1031325
493 4947,2	49518.7046980	49652.0961412
497 4987,3	49930.0743861	50050.9544734
501 5027,4	50341.9256139	50450.9544734
505 5067,5	50753.2953021	50852.0961412
509 5107,6	51163.4274596	51254.1031325
513 5147,7	51572.0000000	51656.5863995





5172v	5184v	5198v		520e
5212,0	5224,0	5238,0	52400.0000000	52563.4349488
		5268,1	5277.4137289	52861.5115731
	5294,1	5308,2	53116.0353713	53259.8480232
		5338,3	53425.7052785	53558.6968610
5362,1	5374,2	5388,4	53936.OOOOOOO	54058.2825256
		5418,5	54246.2947215	54358.6968610
	5444,3	5458,6	54655.9646288	54759.8480232
		5488,7	54964.5862711	55061.5115731
		5518,8	55272.0000000	55363.4349488





■ --- ■ ■ - (Triangle # 2) ---,
554 5559,1	5564.3045403	55769.0387834
558 5599,2	56012.1989876	56167.6108929
562 5639.3	56421.1768230	56566.4786457
566 5679,4	56830.9587180	56965.8438654
570 5719,5	57241.0412821	57365.8438654
574 5759,6	57650.8231770	57766.4786457
578 5799,7	58059.8010124	58167.6108929
582 5839,8	58467.6954597	58569.0387834
5865,1 10,2	5878.7723551	58875.4545635
589 59010,3	59117.0666553	59274.5700194
5935,2 10,4	59426.2676986	59573.9549430
596 59710,5	59836.0000000	59973.7329627
6005,3 10,6	60145.7323015	60273.9549430
603 60410,7	60554.9333447	60674.5700194
6075,4 10,8	60863.2276449	60975.4545635
610 61111,3	61213.3560643	61382.5366786
614 61511,4	61621.9478021	61782.1599429
618 61911,5	62031.2307464	62181.9488415
622 62311,6	62440.7692536	62581.9488415
626 62711,7	62850.0521979	62982.1599429
630 63111,8	63258.6439357	63382.5366786
6343,1 6,2 12,4	63518.0000000	63690.0000000
637 63812,5	63926.7723551	64090.0000000
6416,3 12,6	64236.0000000	64390.0000000
644 64512,7	64645.2276449	64790.0000000
6483,2 6,4 12,8	64954.0000000	65090.0000000





6512v	6524v	6538v	6541	6556
		65613,5	65722.6439357	65897.4633215
		65913,6	66031.4749784	66197.6447358
		66213,7	66340.5250216	66497.6447358
		66513,8	66649.3560643	66797.4633215
	6687,3	66914,6	67027.2276449	671104.5454366
		67214,7	67336.0000000	674104.7101762





6752v	6764v 6777,4	6788v 67914,8	680V 68144.7723551	6826 683104.5454366
		68415,7 68515,8	68631.6954597 68740.3045403	688110.9612167 689110.9612167
6904,2	6918,4	69216,8	69336.0000000	694116.5650512





6953v	6966\>	697V	6986
6990,0	7000,0	70100.0000000	70200.0000000
	7031,0	70400.0000000	7059.3247035
	7061,0	70772.0000000	7089.3247035
7091,0	7102,0	71100.0000000	71220.0767513
	7132,1	71436.0000000	71516.4722107
7161,1	7172,2	71872.0000000	71920.0767513
	7203,0	72100.0000000	72231.7174744
	7233,1	72422.3861776	72527.2237351
	7263,2	72749.6138225	72827.2237351
	7293,3	73072.0000000	73131.7174744
7322,0	7334,0	73400.0000000	73543.3581976
	7364,1	73716.0353713	73839.1034177
7392,1	7404,2	74136.0000000	74237.3773682
	7434,3	74455.9646288	74539.1034177
7462,2	747■ 4,4	74872.0000000	74943.3581976
	7505,0	75100.0000000	75254.1102453
	7535,1	75412.4463843	75550.6513527
	7565,2	75727.7323015	75848.4869490
	7595,3	76044.2676986	76148.4869490
	7625,4	76359.5536157	76450.651352.7
	7655,5	76672.0000000	76754.1102453
7683,0	7696,0	77000.0000000	77163.4349488
	7726,1	77310.1562304	77460.9162275
7753,1	7766,2	77722.3861776	77859.0080312
	7796,3	78036.0000000	78158.2825256
7823,2	7836,4	78449.6138225	78559.0080312
	7866,5	78761.8437696	78860.9162276
7893,3	7906,6	79172.0000000	79263.4349488





7933v 6v	794V	795e
7967,1 7977,2 7987,3 7997,4 8007,5 8017,6	8025.7778072 80316.7344818 80429.3546281 80542.6453719 80655.2655182 80766.2221928	80871.0914502 80969.4828700 81068.5073635 81168.5073635 81269.4828700 81371.0914502
8144,1 8,2 8158,3 8164,2 8,4 8178,5 8184,3 8,6	81911.8185857 82023.3461239 82136.0000000 82248.6538761 82360.1814143	82480.1168545 82579.4443283 82679.1876831 82779.4443283 82880.1168545
8299,3 8309,4 8319,5 8329,6	83318.0000000 83429.8185857 83542.1814143 83654.0000000	83790.0000000 83890.0000000 83990.0000000 84090.0000000
8415,2 10,4 84210,5 8435,3 10,6	84424.1814143 84536.0000000 84647.8185858	84799.8831455 848100.0928406 84999-8831455
85011,5 85111,6	85230.2221928 85341.7778072	854108.9085499 855108.9085499
8566,3 12,6	85736.0000000	858116.5650512

859
860Triangle #1

861Triangle #2