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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
300.00346.41519.62 692.821,039.231,200.00

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 44.4448.7184.3788.89146.14177.78253.12355.56506.25711.11
Seventh 42.1951.3273.07102.64126.56205.28219.21410.56438.43
Eighth 59.2663.28109.61118.52189.84237.04379.69474.07759.37
Ninth 54.8068.4394.92136.85164.41273.71328.82547.41657.64
Tenth 47.4679.0182.20142.38158.02284.77316.05569.53632.10
Eleventh 41.1045.6271.1991.24123.31182.47246.61364.94493.23729.88
Twelfth 52.6761.65105.35106.79210.70213.57421.40427.15

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.