Straight segment lengths in the thirty-degree system.
Version of 29 October 2010.
This page details the profusion of straight segment lengths in the 30-degree system. Recall the triangle of figure 28-4; if one side is of some given length, we might need to perform any of the six following operations to calculate the other two sides:
Starting with 600.00mm, these operations yield the values in the first generation table below. Numbers in the feasible range, arbitrarily defined as 40mm through 800mm, are on a green background, while others are on red.
First generation | |||||
---|---|---|---|---|---|
300.00 | 346.41 | 519.62 | 692.82 | 1,039.23 | 1,200.00 |
Performing the six operations on the first-generation values will generate the additional values listed in this second-generation table:
Second generation | |||||
---|---|---|---|---|---|
150.00 | 173.21 | 200.00 | 259.81 | 400.00 | 450.00 |
800.00 | 900.00 | 1,385.64 | 1,800.00 | 2,078.46 | 2,400.00 |
Applying the operations to the second generation gives these third-generation values that have not yet appeared:
Third generation | |||||
---|---|---|---|---|---|
75.00 | 86.60 | 100.00 | 115.47 | 129.90 | 225.00 |
230.94 | 389.71 | 461.88 | 779.42 | 923.76 | 1,558.85 |
1,600.00 | 2,771.28 | 3,117.69 | 3,600.00 | 4,156.92 | 4,800.00 |
Continuing:
Fourth generation | |||||
---|---|---|---|---|---|
37.50 | 43.30 | 50.00 | 57.74 | 64.95 | 66.67 |
112.50 | 133.33 | 194.86 | 266.67 | 337.50 | 533.33 |
675.00 | 1,066.67 | 1,350.00 | 1,847.52 | 2,700.00 | 3,200.00 |
5,400.00 | 5,542.56 | 6,235.38 | 7,200.00 | 8,313.84 | 9,600.00 |
Fifth generation | |||||
---|---|---|---|---|---|
18.75 | 21.65 | 25.00 | 28.87 | 32.48 | 33.33 |
38.49 | 56.25 | 76.98 | 97.43 | 153.96 | 168.75 |
292.28 | 307.92 | 584.57 | 615.84 | 1,169.13 | 1,231.68 |
2,133.33 | 2,338.27 | 3,695.04 | 4,676.54 | 6,400.00 | 9,353.07 |
10,800.00 | 11,085.13 | 12,470.77 | 14,400.00 | 16,627.69 | 19,200.00 |
In general, the nth generation contributes 6 × n new numbers. From here, only the feasible values are listed, approximately nine per generation:
Sixth | 44.44 | 48.71 | 84.37 | 88.89 | 146.14 | 177.78 | 253.12 | 355.56 | 506.25 | 711.11 |
---|---|---|---|---|---|---|---|---|---|---|
Seventh | 42.19 | 51.32 | 73.07 | 102.64 | 126.56 | 205.28 | 219.21 | 410.56 | 438.43 | |
Eighth | 59.26 | 63.28 | 109.61 | 118.52 | 189.84 | 237.04 | 379.69 | 474.07 | 759.37 | |
Ninth | 54.80 | 68.43 | 94.92 | 136.85 | 164.41 | 273.71 | 328.82 | 547.41 | 657.64 | |
Tenth | 47.46 | 79.01 | 82.20 | 142.38 | 158.02 | 284.77 | 316.05 | 569.53 | 632.10 | |
Eleventh | 41.10 | 45.62 | 71.19 | 91.24 | 123.31 | 182.47 | 246.61 | 364.94 | 493.23 | 729.88 |
Twelfth | 52.67 | 61.65 | 105.35 | 106.79 | 210.70 | 213.57 | 421.40 | 427.15 |
This shows how the 30° system differs substantially from systems previous studied. If either the primary series of the 45° system or the alpha series of the 36° system is extended, all new values are too small or large to be feasible, and hence may be ignored. On the other hand, in the 30° system, the supply of feasible-range segments is inexhaustible, making the system unwieldy.