22 Tetrahedral Coordinates

1Because its breakdowns entail an excessive number of different component lengths, a problem compounded by the high frequencies required to obtain anything acceptably spherical, the tetrahedron is of limited usefulness. Still, investigators may have a use for data. Class II Method 1 is recommended for study. It has orthogonal equators like the octahedron, permitting natural hemispheric truncation, and when so truncated provides large squarish openings at ground level. (0,0 is not at the zenith.)

2 The designer might also consider omitting some of the shorter struts that clutter the three-way vertices. So long as triangulation is preserved this will do no structural damage and greatly improve the symmetry.

3 Large families of intersecting equators—eight in the $4 v$ Class I—permit numerous zenith locations and overhead symmetries. Two Class I tables provide for the following frequencies: $16, 12, 8, 6, 4, 3$ . They cover one tetrahedron face, with comers at $0, 0; 0, \arccos - 1 ∕ 3; 120, \arccos 1 ∕ 3$ .

4 Two Class II Method 1 tables provide for sixteen, twelve, eight, and six-frequency breakdowns. They cover the Class II triangle with vertices at $0, 0; 0, \arccos 1 ∕ 3; 120, \arccos 1 ∕ 3$ . See Diagram 22.1 for the location of this triangle; twelve of them make a sphere.





63v 6v 12v	7V	8e
90,0 0,0 0,0	1000.0000000	1100.0000000
12 131,0	1400.0000000	155.0511525
16 171,1	18120.0000000	195.0511525
201,0 2,0	2100.0000000	2211.4217537
23 242,1	2560.0000000	265.7681812
271,1 2,2	28120.0000000	2911.4217537
30 313,0	3200.0000000	3319.4712206
34 353,1	3630.0000000	3711.5369590
38 393,2	4090.0000000	4111.5369590
42 433,3	44120.0000000	4519.4712206
461,0 2,0 4,0	4700.0000000	4829.4962085
49 50*4,1	5119.1066054	5220.5141272
532,1 4,2	5460.0000000	5515.7931691
56 574,3	58100.8933946	5920.5141272
601,1 2,2 4,4	61120.0000000	6229.4962085
63 645,0	6500.0000000	6641.4729343
67 685,1	6913.8978863	7032.5125173
71 725,2	7340.8933947	7425.0658292
75 765,3	7779.1066054	7825.0658292
79 805,4	81106.1021138	8232.5125173
83 845,5	85120.0000000	8641.4729343
873,0 6,0	8800.0000000	8954.7356103
90 916,1	9210.8933947	9347.2058629
94C\J 950 96KO 9700	9830.0000000	9939.2315205
100 1016,3	10260.0000000	10335.2643897
104* 105KO 106CM 107no	10890.0000000	10939.2315205
110 1116,5	112109.1066053	11347.2058629
1143,3 6,6	115120.0000000	11654.7356103
117 1187,0	11900.0000000	12067.9982863
121 1227,1	1238.9482756	12463.0694073
125 1267,2	12723.4132245	12857.0210379
129 1307,3	13146.1021138	13251.8870735
133 1347,4	13573.8978863	13651.8870735
137 1387,5	13996.5867756	14057.0210380
141 1427,6	143111.0517244	14463.0694073
145 1467,7	147120.0000000	14867.9982863
1492,0 4,0 8,0	15000.0000000	15179.9750121
152 1538,1	1547.5890895	15577.8296766
1564,1 8,2	15719.1066054	15875.0367825
159 1608,3	16136.5867756	16272.0247162
1632,1 4,2 8,4	16460.0000000	16570.5287793
1665	16783.4132244	16872.0247162





1693v 6v 12v	1704>	1716
1724,3 8,6	173100.8933946	17475.0367825
175 1768,7	177112.4109105	17877.8296766
1792,2 4,4 8,8	180120.0000000	18179.9750121
182 1839,0	18400.0000000	18590.0000000
186 1879,1	1886.587756	18990.0000000
190 1919,2	19216.1021138	19390.0000000
194 1959,3	19630.0000000	19790.0000000
198 1999.4	20049.1066054	20190.0000000
202 2039,5	20470.8933947	20590.0000000
206 2079,6	20890.0000000	20990.0000000
210 2119,7	212103.8978862	21390.0000000
214 2159,8	216113.4132244	21790.0000000
218 2199,9	220120.0000000	22190.0000000
2225,0 10,0	22300.0000000	22498.0494670
225 22610,1	2275.8175256	22899.3984525
2295,1 10,2	23013.8978863	231101.0958033
232 23310,3	23425.2849961	235103.0884761
2365,2 10,4	23740.8933947	238104.9632175
239 24010,5	24160.0000000	242105.7931691
2435,3 10,6	24479.1066054	245104.9632175
246 24710,7	24894.7150039	249103.0884761
2505,4 10,8	251106.1021138	252101.0958033
253 25410,9	255114.1824744	25699.3984525
2575,5 10,10	258120.0000000	25998.0494670
260 26111,0	26200.0000000	263104.4200681
264 26511,1	2665.2087191	267106.5150758
268 26911,2	27012.2163488	271109.0624559
272 27311,3	27421.7867893	275112.0017137
276 27711,4	27834.7150040	279114.9379827
280 28111,5	28251.0517245	283116.9305927
284 28511,6	28668.9482756	287116.9305927
288 28911,7	29085.2849961	291114.9379827
292 29311,8	29498.2132107	295112.0017137
296 29711,9	298107.7836512	299109.0624558
300 30111,10	302114.7912809	303106.5150758
304 30511,11	306120.0000000	307104.4200681
3083,0 6,0 12,0	30900.0000000	310109.4712206
311 31212,1	3134.7150040	314111.9343322
3156,1 12,2	31610.8933947	317114.8398925
318 31912,3	32019.1066054	321118.1255057
3223,1 6,2 12,4	32330.0000000	324121.4821541
325 32612,5	32743.8978863	328124.1908638
3296,3 12,6	33060.0000000	331125.2643897
332 33312,7	33476.1021138	335124.1908638
3363,2 6,4 12,8	33790.0000000	338121.4821541
	339 340(Continued)





3413v	3426v	34312v	344V	345e
	3466,5	34712,9 34812,10	349100.8933946 350109.1066053	351118.1255057 352114.8398925





3533v	3546v	35512v	356V	3576
3583,3	3596,6	36012,11 36112,12	362115.2849960 363120.0000000	364111.9343322 365109.4712206

366 TETRAHEDRON CLASS I COORDINATES: Frequencies 16, 8, 4





3674v 8v 16v	368V	3698
3700 371V 3720 3730 3740 3750 3760	37700.0000000	37800.0000000
379 3801,0	38100.0000000	3823.6780516
383 3841,1	385120.0000000	3863.6780516
3870 388CXI 3890	39000.0000000	3918.0494670
392 3932,1	39460.0000000	3954.0446912
3961,1 2,2	397120.0000000	3988.0494670
399 4003,0	40100.0000000	40213.2626760
403 4043,1	40530.0000000	4067.7493664
407 4083,2	40990.0000000	4107.7493664
411 4123,3	413120.0000000	41413.2626760
4150 4160 417cxj 4180	41900.0000000	42019.4712206
421 4224,1	42319.1066054	42413.1622883
4252,1 4,2	42660.0000000	42710.0249879
428n ⁴’³	429100.8933946	43013.1622883
4311,1 2,2 4,4	432120.0000000	43319.4712206
434 4355,0	43600.0000000	43726.7972576
438 4395,1	44013.8978863	44120.0124169
442 4435,2	44440.8933947	44514.9632174
446 4475,3	44879.1066054	44914.9632174
450 4515,4	452106.1021138	45320.0124169
454 4555,5	456120.0000000	45726.7972576
4583,0 6,0	45900.0000000	46035.2643897
461 4626,1	46310.8933947	46428.3717844
4653,1 6,2	46630.0000000	46722.2076543
468 4696,3	47060.0000000	47119.4712206
4723,2 6,4	47390.0000000	47422.2076543
475 4766,5	477109.1066053	47828.3717844
4793,3 6,6	480120.0000000	48135.2643897
482 4837,0	48400.0000000	48544.7106224
486 4877,1	4888.9482756	48938.2169233
490 4917,2	49223.4132245	49331.6513993
494 4957,3	49646.1021138	49727.0171232
498 4997,4	50073.8978863	50127.0171232
502 5037,5	50496.5687756	50531.6513993
506 5077,6	508111.0517244	50938.2169233
510 5117,7	512120.0000000	51344.7106225





5144v 8v 16v	515_sL_	516a
5172,0 4,0 8,0	51800.0000000	51954.7356103
520 5218,1	5227.5890895	52349.2169021
5244,1 8,2	52519.1066054	52643.0887231
527 5288,3	52936.5867756	53037.6161120
5312,1 4,2 8,4	53260.0000000	53335.2643897
534 5358,5	53683.4132244	53737.6161120
5384,3 8,6	539100.8933946	54043.0887232
541 5428,7	543112.4109105	54449.2169021
5452,2 4,4 8,8	546120.0000000	54754.7356103
548 5499,0	55000.0000000	55164.7605982
552 5539,1	5546.5867756	55560.6661258
556 5579,2	55816.1021138	55955.8091362
560 5619,3	56230.0000000	56350.7684795
564 5659,4	56649.1066054	56747.2058629
568 5699,5	57070.8933947	57147.2058629
572 5739,6	57490.0000000	57550.7684795
576 5779,7	578103.8978862	57955.8091362
580 5819,8	582113.4132244	58360.6661258
584 585’ 9,9	586120.0000000	58764.7605982
5885,0 10,0	58900.0000000	59074.2068310
591 59210,1	5935-8175256	59471.6832693
5955,1 10,2	59613.8978863	59768.5832860
598 59910,3	60025-2849961	60165.0620173
6025,2 10,4	60340.8933947	60461.8744943
605 60610,5	60760.0000000	60860.5037915
6095,3 10,6	61079.1066054	61161.8744943
612 61310,7	61494.7150039	61565.0620173
6165,4 10,8	617106.1021138	61868.5832860
619 62010,9	621114.1824744	62271.6832693
6235,5 10,10	624120.0000000	62574.2068310
626 62711,0	62800.0000000	62982.6739631
630 63111,1	6325.2087191	63381.5673284
634 63511,2	63612.2163488	63780.1975692
638 63911,3	64021.7867893	64178.5782463
642 64311,4	64434.7150040	64576.9115239
646 64711,5	64851.0517245	64975-7482449
650 65111,6	65268.9482756	65375.7482449
654 65511,7	65685.2849961	65776.9115239
658 65911,8	66098.2132107	66178.5782463
662 66311,9	664107.7836512	66580.1975692





6664v 8v 16v	667V	668e
66911,10	670114.7912809	67181.5673284
67211,11	673120.0000000	67482.6739631
6753,0 6,0 12,0	67600.0000000	67790.0000000
67812,1	6794.7150040	68090.0000000
6816,1 12,2	68210.8933947	68390.0000000
684 68512,3	68619.1066054	68790.0000000
6883,1 6,2 12,4	68930.0000000	69090.0000000
691 69212,5	69343.8978863	69490.0000000
6956,3 12,6	69660.0000000	69790.0000000
698 69912,7	70076.1021138	70190.0000000
7023,2 6,4 12,8	70390.0000000	70490.0000000
705 70612,9	707100.8933946	70890.0000000
7096,5 12,10	710109.1066054	71190.0000000
71212,11	713115.2849960	71490.0000000
7153,3 6,6 12,12	716120.0000000	71790.0000000
718 71913,0	72000.. 0000000	72196.2085447
722 72313,1	7244.3066191	72596.9911553
726 72713,2	7289.8264298	72997.9328899
730 73113,3	73216.9960881	73399.0406311
734 73513,4	73626.3295035	737100.2634279
738 73913,5	74038.2132107	741101.4217537
742 74313,6	74452.4109105	745102.1703234
746 74713,7	74867.5890895	749102.1703234
750 75113,8	75281.7867893	753101.4217537
754 75513,9	75693.6704965	757100.2634279
75813,10	759103.0039H9	76099.0406311
761 76213,11	763110.1735702	76497.9328899
76513,12	766115.6933809	76796.9911553
768 76913 ,’13	770120.0000000	77196.2085447
7727,0 14,0	77300.0000000	774101.4217537
775 77614,1	7773.9632350	778102.7203545
7797,1 14,2	7808.9482756	781104.2517550
782 78314,3	78415.2953443	785106.0231797
7867,2 14,4	78723.4132245	788107.9752839
789 79014,5	79133.6704965	792109.9074988
7937,3 14,6	79446.1021138	795111.4167141
796 79714,7	79860.0000000	799112.0017137
8007,4 14,8	80173.8978863	802111.4167141
803 80414,9	80586.3295035	806109.9074988
8077,5 14,10	80896.5867756	809107.9752839
810 81114,11	812104.7046557	813106.0231797
8147,6 14,12	815111.0517244	816104.2517550
81714,13	818116.0367651	819102.7203545
8207,7 14,14	821120.0000000	822101.4217537
823 82415,0	82500.0000000	826105.7931690
827 82815,1	8293.6704965	830107.4127302





8314v 8v 16v	832.V	833e
834 83515,2 83615,3 83715,4 83815,5 83915,6 84015,7 84115,8 84215,9 84315,10 84415,11 84515,12 84615,13 84715.14 84815,15	8498.2132107 85013.8978863 85121.0517244 85230.0000000 85340.8933947 85453.4132245 85566.5867755 85679.1066054 85790.0000000 85898.9482756 859106.1021138 860111.7867893 861116.3295035 862120.0000000	863109.2863254 864111.4167140 865113.7466788 866116.1001388 867118.1255057 868119.3338742 869119.3338743 870118.1255057 871116.1001388 872113.7466788 873111.4167140 874109.2863254 875107.4127301 876105.7931691
8774,0 8,0 16,0 878 879’ 16,1 8808,1 16,2 881 88216,3 8834,1 8,2 16,4 884 88516,5 8868,3 16,6 887 88816,7 8894,2 8,4 16,8 890 89116,9 8928,5 16,10 893 89416,11 8954,3 8,6 16,12 896 89716,13 8988,7 16,14 899 90016,15 9014,4 8,8 16,16	90200.0000000 9033.4179811 9047.5890895 90512.7305278 90619.1066054 90726.9955084 90836.5867756 90947.7836512 91060.0000000 91172.2163488 91283.4132244 91393.0044916 914100.8933946 915107.2694722 916112.4109105 917116.5820189 918120.0000000	919109.4712206 920111.2776255 921113.3320386 922115.6320697 923118.1255057 924120.6679177 925122.9789620 926124.6481469 927125.2643897 928124.6481469 929122.9789620 930120.6679177 931118.1255057 932115.6320697 933113.3320386 934III.2776255 935109.4712206





9368v 16v	937V	9380
9390,0 0,0	94000.0000000	94100.0000000
9421,0	94300.0000000	9447.3260370
9451,1	946120.0000000	9477.3260370
9481,0 2,0	94900.0000000	95015.7931690
9512,1	95260.0000000	9538.0494670
9541,1 2,2	955120.0000000	95615.7931690
9573,0	95800.0000000	95925.2394018
9603,1	96130.0000000	96215.2251570
9633,2	96490.0000000	96515.2251570
9663,3	967120.0000000	96825.2394018
9692,0 4,0	97000.0000000	97135.2643897
9724,1	97319.1066054	97425.0658292
9752,1 4,2	97660.0000000	97719.4712206
9784,3	979100.8933946	98025.0658292
9812,2 4,4	982120.0000000	98335.2643897
9845,0	98500.0000000	98645.2893775
9875,1	98813.8978863	98936.0707687
9905,2	99140.8933947	99228.1255057
9935,3	99479.1066054	99528.1255057
9965,4	997106.1021138	99836.0707687
9995,5	1000120.0000000	100145.2893775





10028v 16v	1003V	10046
10053,0 6.0	100600.0000000	100754.7356103
10086,1	100910.8933947	101047.2058629
10113,1 6,2	101230.0000000	101339.2315205
10146,3	101560.0000000	101635.2643897
10173,2 6,4	101890.0000000	101939.2315205
10206,5	1021109.1066053	102247.2058629
10233,3 6,6	1024120.0000000	102554.7356103
10267,0	102700.0000000	102863.2027424
10297,1	10308.9482756	103157.5844268
10327,2	103323.4132245	103450.9542458
10357,3	103646.1021138	103745.5617593
10387,4	103973.8978863	104045.5617593
10417,5	104296.5867756	104350.9542458
10447,6	1045111.0517244	104657.5844268
10477,7	1048120.0000000	104963.2027424
10504,0 8,0	105100.0000000	105270.5287793
10538,1	10547.5890895	105566.6679614
10564,1 8,2	105719.1066054	105861.8744943
10598,3	106036.5867756	106157.0210380
10624,2 8,4	106360.0000000	106454.7356103
10658,5	106683.4132244	106757.0210380
10684,3 8,6	1069100.8933946	107061.8744943
10718,7	1072112.4109105	107366.6679614
10744,4 8,8	1075120.0000000	107670.5287793





10776v	107812v	1079V	10806
10810,0	10820,0	108300.0000000	108400.0000000
	10851,0	108600.0000000	108710.0249879
	10881,1	1089120.0000000	109010.0249879
10911,0	10922,0	109300.0000000	109422.0017137
	10952,1	109660.0000000	109711.4217537
10981,1	10992,2.	1100120.0000000	110122.0017137
	11023,0	110300.0000000	110435.2643897
	11053,1	110630.0000000	110722.2076543
	11083,2	110990.0000000	111022.2076543
	11113,3	1112120.0000000	111335.2643897
11142,0	11154,0	111600.0000000	111748.5270657
	11184,1	111919.1066054	112036.8086651
11212,1	11224,2	112360.0000000	112429.4962085
	11254,3	1126100.8933946	112736.8086651
	11284,4	1129120.0000000	113048.5270657

1131 TETRAHEDRON CLASS II COORDINATES: Frequencies 12, 6





11326v	113312v	1134V	1135e
	11365,0	113700.0000000 ’	113860.5037915
	11395,1	114013.8978863	114151.8870735
	11425,2	114340.8933947	114443.0887231
	11455,3	114679.1066054	114743.0887231
	11485,4	1149106.1021138	115051.8870735
	11515,5	1152120.0000000	115360.5037915
11543,0	11556,0	115600.0000000	115770.5287793
	11586,1	115910.8933947	116065.1601075
11613,1	11626,2	116330.0000000	116458.5178459
	11656,3	116660.0000000	116754.7356103
11683,2	11696,4	117090.0000000	117158.5178459
	11726,5	1173109.1066053	117465.1601076
11753,3	11766,6	1177120.0000000	117870.5287793