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:

• divide by 2
• divide by √3
• divide by 2 ÷ √3
• multiply by 2 ÷ √3
• multiply by √3
• multiply by 2

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
300.00346.41519.62 692.821,039.231,200.00

Performing the six operations on the first-generation values will generate the additional values listed in this second-generation table:

Second generation
150.00173.21200.00 259.81400.00450.00
800.00900.001,385.64 1,800.002,078.462,400.00

Applying the operations to the second generation gives these third-generation values that have not yet appeared:

Third generation
75.0086.60100.00 115.47129.90225.00
230.94389.71461.88 779.42923.761,558.85
1,600.002,771.283,117.69 3,600.004,156.924,800.00

Continuing:

Fourth generation
37.5043.3050.00 57.7464.9566.67
112.50133.33194.86 266.67337.50533.33
675.001,066.671,350.00 1,847.522,700.003,200.00
5,400.005,542.566,235.38 7,200.008,313.849,600.00

Fifth generation
18.7521.6525.00 28.8732.4833.33
38.4956.2576.98 97.43153.96168.75
292.28307.92584.57 615.841,169.131,231.68
2,133.332,338.273,695.04 4,676.546,400.009,353.07
10,800.0011,085.1312,470.77 14,400.0016,627.6919,200.00

In general, the nth generation contributes 6 × n new numbers. From here, only the feasible values are listed, approximately nine per generation:

 Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth 44.44 48.71 84.37 88.89 146.14 177.78 253.12 355.56 506.25 711.11 42.19 51.32 73.07 102.64 126.56 205.28 219.21 410.56 438.43 59.26 63.28 109.61 118.52 189.84 237.04 379.69 474.07 759.37 54.8 68.43 94.92 136.85 164.41 273.71 328.82 547.41 657.64 47.46 79.01 82.2 142.38 158.02 284.77 316.05 569.53 632.10 41.1 45.62 71.19 91.24 123.31 182.47 246.61 364.94 493.23 729.88 52.67 61.65 105.35 106.79 210.7 213.57 421.4 427.15

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.