18 Class I Octahedral Coordinates

1Eight frequencies— $2, 3, 4, 5, 6, 8, 10, 12$ —are here combined in three tables. Pick the vertex labels for the frequency you are using out of the proper column at the left, and find the $ϕ$ and $Θ$ coordinates at the right.

2 Each table covers one octahedron face whose comers are 0,0; 0,90; 90,90. For spheres use only the symmetry triangle (Diagram 12.1). For ellipsoids of revolution (one cross-section circular) use the left half of the triangle. When both floorplans and rooflines are non-circular use the entire triangle.

3 If plans for coordinate rotation should require data on part of the #2 triangle (down-pointing below the equator) simply repeat $ϕ$ values from the #1 triangle and add 90 to $Θ$ values.

4 A peculiar symmetry of the octahedron permits all coordinates to be written in two notations: arctan and decimal degrees. ‘‘Arctan 3’’ (three keystrokes) is quicker to key on the calculator than ‘‘71.5650512’’ (ten keystrokes) and the entered number is identical. With expressions like ‘‘ $\arctan \sqrt{25 ∕ 32}$ ’’ the saving in keystrokes vanishes but the likelihood of error remains less, symbols on the keyboard being less easily confused than digits. But use the decimal forms if you prefer.

5 Coordinates were generated by an HP-65 program using Equations 12.1 and 12.2.

6 OCTAHEDRON CLASS I COORDINATES: Frequencies 12, 6, 3





73v 6v 12v	8V	9e
100,0 0,0 0,0	11 1200.0000000	13 1400.0000000
15 161,0 171,1	18 1900.0000000 2090.0000000	21arctan 1/11 5.1944289 22arctan 1/11 5.1944289
231,0 2,0 24 252,1 261,1 2,2	27 2800.0000000 29arctan 1 45.0000000 30 3190.0000000	32arctan 1/5 11.3099325 33arctan 72/10 8.0494670 34arctan 1/5 11.3099325
35 363,0 373,1 383,2 393,3	40 4100.0000000 42arctan 1/2 26.5650512 43arctan 2 63.4349488 44 4590.0000000	46arctan 1/3 18.4349488 47arctan 75/81 13-9527364 48arctan 75/81 13.9527364 49arctan 1/3 18.4349488
501,0 2,0 4,0 51 524,1 532,1 4,2 54 554,3 561,1 2,2 4,4	57 5800.0000000 59arctan 1/3 18.4349488 60arctan 1 45.0000000 61arctan 3 71.5650512 62 6390.0000000	64arctan 1/2 26.5650512 65arctan 75/32 21.5681294 66arctan 71/8 19.4712206 67arctan 75/32 21.5681294 68arctan 1/2 26.5650512
69 705,0 715,1 725,2 735,3 745,4 755,5	76 7700.0000000 78arctan 1/H 14.0362435 79arctan 2/3 33.6900675 80arctan 3/2 56.3099324 81arctan 4 75.9637566 82 8390.0000000	84arctan 5/7 35.5376778 85arctan 717/49 30.4987270 86arctan 713/49 27.2520337 87arctan 713/49 27.2520337 88arctan 717/49 30.4987270 89arctan 5/7 35.5376778
903,0 6,0 91 926,1 933,1 6,2 94 956,3 963,2 6,4 97 986,5 993,3 6,6	100 10100.0000000 102arctan 1/5 11.3099325 103arctan 1/2 26.5650512 104arctan 1 45.0000000 105arctan 2 63.4349488 106arctan 5 78.6900675 107 10890.0000000	109arctan 1 45.0000000 110arctan 713/18 40.3591005 111arctan 75/9 36.6992252 112arctan 71/2 35.2643897 113arctan 75/9 36.6992252 114arctan 713/18 40.3591005 115arctan 1 45.0000000
116 1177,0 1187,1 1197,2 1207,3 1217,4 1227,5 1237,6 1247,7	125 12600.0000000 127arctan 1/6 9.4623222 128arctan 2/5 21.8014095 129arctan 3/4 36.8698977 130arctan 4/3 53.1301024 131arctan 5/2 68.1985905 132arctan 6 80.5376778 133 13490.0000000	135arctan 7/5 54.4623222 136arctan 737/25 50.5799684 137arctan 729/25 47j1240113 138arctan 1 45.0000000 139arctan 1 45.0000000 140arctan 729/25 47.1240113 141arctan 737/25 50.5799684 142arctan 7/5 54.4623222
1432,0 4,0 8,0 144 1458,1 1464,1 8,2 147 1488,3	149 15000.0000000 151arctan 1/7 8.1301024 152arctan 1/3 18.4349488 153arctan 3/5 30.9637565	154arctan 2 63.4349488 155arctan 725/8 60.5037915 156arctan 75/2 57.6884668 157Aretan 717/8 55.5500980





1583v 6v 12v 1592,1 4,2 8,4 160 1618,5 1624,3 8,6 163 1648,7 1652,2 4,4 8,8	166V 167arctan 1 45.0000000 168arctan 5/3 59.0362435 169arctan 3 71.5650512 170arctan 7 81.8698977 171 17290.0000000	173 1740 175arctan /2 54.7356103 176arctan /17/8 55.5500980 177arctan /5/2 57.6884668 178arctan /25/8 60.5037915 179arctan 2 63.4349488
180 1819,0 1829,1 1839,2 1849,3 1859,4 1869,5 1879,6 1889,7 1899,8 1909,9	191 19200.0000000 193arctan 1/8 7-1250164 194arctan 2/7 15-9453959 195arctan 1/2 26.5650512 196arctan 4/5 38.6598083 197arctan 5/4 51-3401917 198arctan 2 63.4349488 199arctan 3.5 74.0546041 200arctan 8 82.8749836 201 20290.0000000	203arctan 3 71.5650512 204arctan /65/9 69.5895545 205arctan /53/9 67.6043377 206arctan /5 65.9051575 207arctan /41/9 64.8959097 208arctan /41/9 64.8959097 209arctan /5 65.9051575 210arctan /53/9 67.6043377 211arctan /65/9 69.5895545 212arctan 3 71.5650512
2135,0 10,0 214 21510,1 2165,1 10,2 217 21810,3 2195,2 10,4 220 22110,5 2225,3 10,6 223 22410,7 2255,4 10,8 226 22710,9 2285,5 10,10	229 23000.0000000 231arctan 1/9 6.3401918 232arctan 1/4 14.0362435 233arctan 3/7 23.1985905 234arctan 2/3 33.6900675 235arctan 1 45.0000000 236arctan 3/2 56.3099325 237arctan 7/3 66.8014095 238arctan 4 75.9637566 239arctan 9 83-6598082 240 24190.0000000	242arctan 5_ 78.6900675 243arctan /41/2 77.5454236 244arctan /17~ 76.3669778 245arctan /29/2 75.2856464 246arctan /13 74.4986405 247arctan /25/2 74.2068310 248arctan /TJ 74.4986405 249arctan /29/2 75.2856464 250arctan /I? 76.3669778 251arctan /41/2 77.5454236 252arctan 5 78.6900675
253 25411,0 25511,1 25611,2 25711,3 25811,4 25911,5 26011,6 26111,7 26211,8 26311,9 26411,10 26511,11	266 26700.0000000 268arctan 1/10 5.7105931 269arctan 2/9 12.5288077 270arctan 3/8 20.5560452 271arctan 4/7 29.7448813 272arctan 5/6 39.8055711 273arctan 6/5 50.1944289 274arctan 7/4 60.2551187 275arctan 8/3 69.4439548 276arctan 4.5 77.4711923 277arctan 10 84.2894069 278 27990.0000000	280arctan 11 84.8055711 281arctan /101 84.3175615 282arctan /E5 83.8096005 283arctan /73 83.3244079 284arctan /b5 82.9294449 285arctan /51 82.7037247 286arctan /61 82.7037247 287arctan /&5 82.9294449 288arctan /73 83.3244079 289arctan /85 83.8096005 290arctan /101 84.3175615 291arctan 11 84.8055711
2923,0 6,0 12,0 293 29412,1 2956,1 12,2 296 29712 3 2983,1 6,2 12*4 299 30012,5 3016,3 12,6 302 30312,7 3043,2 6,4 12,8 305 30612,9	307 30800.0000000 309arctan 1/11 5.1944289 310arctan 1/5 11.3099325 311arctan 1/3 18.4349488 312arctan 1/2 26.5650512 313arctan 5/7 35.5376778 314arctan 1 45.0000000 315arctan 7/5 54.4623222 316arctan 2 63.4349488 317arctan 3 71-5650512	318 31990.0000000 32090.0000000 32190.0000000 32290.0000000 32390.0000000 32490.0000000 32590.0000000 32690.0000000 32790.0000000 32890.0000000





3296,5 12,10	330’arctan 5 78.6900675	331 33290.0000000
333 33412,11	335arctan 11 84.8055711	336 33790.0000000
3383,3 6,6 12,12	33990.0000000	340 34190.0000000

342
343OCTAHEDRON CLASS I COORDINATES: Frequencies 8, 4, 2





3442\>	3454v	3468v	V		347 348e
3490,0	3500,0	3510,0	00.0000000		00.0000000
		3521,0		35300.0000000	354arctan 1/7	3558.1301024
		3561,1		35790.0000000	358arctan 1/7	3598.1301024
	3601,0	3612,0		36200.0000000	363arctan 1/3	36418.4349488
		3652,1	366arctan 1	36745.0000000	368arctan /1/18	36913.2626760
	3701,1	3712,2		37290.0000000	373arctan 1/3	37418.4349488
		3753,0		37600.0000000	377arctan 3/5	37830.9637565
		3793,1	380arctan 1/2	38126.5650512	382arctan /1/5	38324.0948426
		3843,2	385arctan 2	38663.4349488	387arctan /1/5	38824.0948426
		3893,3		39090.0000000	391arctan 3/5	39230.9637565
3931,0	3942,0	3954,0		39600.0000000	397arctan 1	39845.0000000
		399M	400arctan 1/3	40118.4349488	402arctan /5/8	40338.3288181
	4042,1	4054,2	406arctan 1	40745.0000000	408arctan /1/2	40935-2643897
		4103	411arctan 3	41271.5650512	413arctan /5/8	41438.3288181
4151,1	4162,2	4174,4		41890.0000000	419arctan 1	42045.0000000
		4215,0		42200.0000000	423arctan 5/3	42459.0362435
		4255,1	426arctan 1/4	42714.0362435	428arctan /17/9	42953.9601066
		4305,2	431arctan 2/3	43233.6900675	433arctan /13/9	43450.2378408
		4355,3	436arctan 3/2	43756.3099325	438arctan /13/9	43950.2378408
		4405,4	441arctan 4	44275.9637566	443arctan /17/9	44453.9601066
		4455,5		44690.0000000	447arctan 5/3	44859.0362435
	4493,0	4506,0		45100.0000000	452arctan 3	45371.5650512
		4546,1	455arctan 1/5	45611.3099325	457arctan /13/2	45868.5832860
	4593,1	4606,2	461arctan 1/2	46226.5650512	463arctan /T	46465.9051575
		4656,3	466arctan 1	46745.0000000	468arctan /g/2	46964.7605982
	4703,2	4716,4	472arctan 2	47363.4349488	474arctan /5	47565.9051575
		4766,5	477arctan 5	47878.6900675	479arctan /13/2	48068.5832860
	4813,3	4826,6		48390.0000000	484arctan 3	48571.5650512
		4867,0		48700.0000000	488arctan 7	48981.8698976
		4907,1	491arctan 1/6	4929.4623222	493arctan /37	49480.6641409
		4957,2	496arctan 2/5	49721.8014095	498arctan /29	49979.4802651
		5007,3	501arctan 3/4	50236.8698976	503arctan 5	50478.6900675
		5057,4	506arctan 4/3	50753.1301024	508arctan 5	50978.6900675
		5107,5	511arctan 5/2	51268.1985905	513arctan /29	51479.4802651
		5157,6	516arctan 6	51780.5376778	518arctan /37	51980.6641409
		5207,7		52190.0000000	522arctan 7	52381.8698976





5242v 8\>	525V	5260
527o 528co 529o 530o 531CXJ	532 53300.0000000	534 53590.0000000
536 5378,1	538arctan 1/7 8.1301024	539 54090.0000000
5414,1 8,2	542arctan 1/3 18.4349488	543 54490.0000000
545 5468,3	547arctan 3/5 30.9637565	548 54990.0000000
5502,1 4,2 8,4	551arctan 1 45.0000000	552 55390.0000000
554 5558,5	556arctan 5/3 59.0362435	557 55890.0000000
5594,3 8,6	560arctan 3 71-5650512	561 56290.0000000
563 5648,7	565arctan 7 81.8698976	566 56790.0000000
5682,2 4,4 8,8	569 57090.0000000	571 57290.0000000

573
574OCTAHEDRON CLASS I COORDINATES: Frequencies 10, 5, 2





5752v	5765v	577lOv			578 579e
5800,0	5810,0	5820,0	583 58400.0000000		00.0000000
585 5861,0 5871,1			588 58900.0000000 59090.0000000		arctan 1/9 591arctan 1/9		5926.3401917 5936.3401917
	5941,0 5951,1	5962,0 5972,1 5982,2	599arctan 1	60000.0000000 60145.0000000 60290.0000000	603arctan 604arctan 605arctan	6061/4 607/1/32 6081/4	60914.0362435 61010.0249879 61114.0362435
612 6133,0 6143,1 6153,2 6163,3			617arctan 1/2 618arctan 2	61900.0000000 62026.5650512 62163.4349488 62290.0000000	arctan 3/7 623arctan /5/49 624arctan /5/49 625arctan 3/7		62623.1985905 62717.7154723 62817.7154723 62923.1985905
	6302,0 6312,1 6322,2	6334,0 6344,1 6354,2 6364,3 6374,4	638arctan 1/3 639arctan 1 640arctan 3	64100.0000000 64218.4349488 64345.0000000 64471.5650512 64590.0000000	646arctan 647arctan 648arctan 649arctan 650arctan	6512/3 652✓5/18 653/2?9 654✓5/18 6552/3	65633.6900675 65727.7913057 65825.2394018 65927.7913057 66033.6900675
6611,0 6621,1		6635,0 6645,1 6655,2 6665,3 6675,4 6685,5	669arctan 1/4 670arctan 2/3 671arctan 3/2 672arctan 4	67300.0000000 67414.0362435 67533.6900675 67656.3099325 67775.9637566 67890.0000000	679arctan 680arctan 681arctan 682arctan 683arctan 684arctan	6851 686✓17/25 687✓13/25 688✓13/25 689✓17/25 6901	69145.0000000 69239.5097123 69335.7957599 69435.7957599 69539.5097123 69645.0000000
	6973,0 6983,1 6993,2 7003,3	7016,0 7026,1 7036,2 7046,3 7056,4 7066,5 7076,6	708arctan 1/5 709arctan 1/2 710arctan 1 711arctan 2 712arctan 5	71300.0000000 71411.3099325 71526.5650512 71645.0000000 71763.4349488 71878.6900675 71990.0000000	720arctan 721arctan 722arctan 723arctan 724arctan 725arctan 726arctan	7273/2 728✓13/8 729✓574 730✓97F 731✓57T 732✓13/8 7333/2	73456.3099325 73551.8870735 73648.1896851 73746.6861434 73848.1896851 73951.8870735 74056.3099325
		7417,0 7427,1	743arctan 1/6	74400.0000000 7459.4623222	746arctan 747arctan	7487/3 749✓37/9	75066.8014095 75163.7476249





7522v	7535v	754lOv	V			0
		7557,2	756arctan	7572/5	75821.8014095	759arctan	760/29/9	76160.8784319
		7627,3	763arctan	7643/4	76536.8698977	766arctan	767<25/9	76859.0362435
		7697,4	770arctan	7714/3	77253.1301024	773arctan	774/25/9	77559.0362435
		7767,5	777arctan	7785/2	77968.1985905	780arctan	781<29/9	78260.8784319
		7837,6	784arctan	7856	78680.5376778	787arctan	788✓37/9	78963.7476249
		7907,7			79190.0000000	792arctan	7937/3	79466.8014095
	7954,0	7968,0			79700.0000000	798arctan	7994	80075.9637566
		8018,1	802arctan	8031/7	8048.1301024	805arctan	806/25/2	80774.2068310
	8084,1	8098,2	810arctan	8111/3	81218.4349488	813arctan	814/10	81572.4515994
		8168,3	817arctan	8183/5	81930.9637565	820arctan	821✓17/2	82271.0681768
	8234,2	8248,4	825arctan	8261	82745.0000000	828arctan	829/S’	83070.5287793
		8318,5	832arctan	8335/3	83459.0362435	835arctan	836✓17/2	83771.0681768
	8384,3	8398,6	840arctan	8413	84271.5650512	843arctan	844/io	84572.4515994
		8468,7	847arctan	8487	84981.8698976	850arctan	851/25/2	85274.2068310
		8538,8			85490.0000000	855arctan	8564	85775.9637566
		8589,0			85900.0000000	860arctan	8619	86283.6598082
		8639,1	864arctan	8651/8	8667.1250164	867arctan		86882.9294449
		8699,2	870arctan	8712/7	87215.9453959	873arctan	$	87482.1787645
		8759,3	876arctan	8771/2	87826.5650512	879arctan		88081.5212868
		8819,4	882arctan	8834/5	88438.6598083	885arctan	886Al	88781.1236049
		8889,5	889arctan	8905/4	89151.3401917	892arctan	893AT	89481.1236049
		8959,6	896arctan	8972	89863.4349488	899arctan		90081.5212868
		9019,7	902arctan	9037/2	90474.0546041	905arctan		90682.1787645
		9079,8	908arctan	9098	91082.8749836	911arctan	912/U5	91382.9294449
		9149,9			91590.0000000	916arctan	9179	91883.6598082
9192,0	9205,0	92110,0			92200.0000000			92390.0000000
		92410,1	925arctan	9261/9	9276.3401918			92890.0000000
	9295,1	93010,2	931arctan	9321/4	93314.0362435			93490.0000000
		93510,3	936arctan	9373/7	93823.1985905			93990.0000000
	9405,2	94110,4	942arctan	9432/3	94433.6900675			94590.0000000
9462,1		94710,5	948arctan	9491	95045.0000000			95190.0000000
	9525,3	95310,6	954arctan	9553/2	95656.3099325			95790.0000000
		95810,7	959arctan	9607/3	96166.8014095			96290.0000000
	9635,4	96410,8	965arctan	9664	96775.963’7566			96890.0000000
		96910,9	970arctan	9719	97283.6598082			97390.0000000
9742,2	9755,5	97610,10			97790.0000000			97890.0000000

979

980 Sphere covered with

981 Class II triangles.

982 Two Class II triangles

983 and halves of two others .... rotated 45° = Octa face.