19. Curved pieces. In the 36° scheme, each curved piece is one-tenth of a circle, as in figure 19-1.

Figure 19-1

The radius of these circular pieces is the usual 1 quintimeter. Two examples of layouts using only curved pieces follow; figure 19-3 is a rough analogue of figure 2-4.

Figure 19-2
Figure 19-3

20. Alpha series. In almost any kind of analysis involving the 36° angle, an important mathematical constant arises. This number is (√5 + 1) / 2, approximately 1.618034, and is frequently symbolized by the uppercase Greek letter Φ. It must not be confused with a closely related constant, (√5 - 1) / 2, approximately 0.618034, which is often written with lowercase Greek φ. Noteworthy is that Φ * φ = Φ - φ = 1. Most expressions that use the smaller phi can be written equally well with the larger, so to minimize confusion in this report the larger phi (1.618034) will be used exclusively.

There are four essential figure eights that can be formed in the 36° system, and they need to be studied because they induce straight segments of inevitable lengths. Two of them will be taken up in this section, and the others in section 21.

Figure 20-1 contains the first to be examined, with nine curved segments per lobe. Depicted in red as always, the crossing has two straight segments intersecting at an angle that may equally well be regarded as 36° or 144°, and each segment has a length of 130mm (= 400 * tan18°). Meanwhile, figure 20-2 displays the oval-eight version, each straight segment within a switch having a length of 210mm (= 200 / cos18°).

Figure 20-1	Figure 20-2

A second figure eight has seven curved segments per lobe. Because the 551mm (= 400 * tan54°) crossing in figure 20-3a might be impracticably long, figure 20-3b shows an alternate disposition. Both of these crossings involve a 72° (equivalently, 108°) angle.

Figure 20-3a	Figure 20-3b

Figure 20-4 is the added-oval version of figure 20-3. Although the overall layout is symmetric, the implementation is not.

Figure 20-4

One switch happens to have a straight segment of 129.97mm -- this distance is close to the minimum of 126.49mm (= sqrt (220^2 - 180^2)), under the assumption of a 200mm radius and a 40mm track width. A straight segment any shorter would not be able to clear the curved segment, making attachment to other track pieces difficult or impossible. From another point of view, the track width must be no greater than 42.23mm (= 400 * (1 - cos36°) / (1 + cos36°)), or the same problem will occur.

A similar consideration determines that for a 36° crossing piece, as in figure 20-1, the minimum length is 123.11mm (= 40 * (tan54° + sec54°)). Also, for a 72° crossing piece, as in figure 20-3, the minimum length is 55.06mm (= 40 * (tan18° + sec18°)).

Table 20-A, which is similar in principle to table 4-A, includes the numbers appearing so far. This alpha series shares an important feature with any geometric sequence where the ratio is Φ: each entry is the sum of the two previous entries.

Table 20-A: Selected values of the Alpha series
As an integer	To two places	Exactly
19	18.96	400 * tan18° * Φ-4
31	30.68	400 * tan18° * Φ-3
50	49.64	400 * tan18° * Φ-2
80	80.32	400 * tan18° * Φ-1
130	129.97	400 * tan18° * Φ0
210	210.29	400 * tan18° * Φ+1
340	340.26	400 * tan18° * Φ+2
551	550.55	400 * tan18° * Φ+3
891	890.81	400 * tan18° * Φ+4

Below are some fundamental triangles the edges of which are in the ratio Φ. These drawings correspond to figure 4-3 in the 45° system. In figure 20-5, the corners of each triangle have 3, 4, and 3 circular segments respectively; in figure 20-6 the numbers of segments are 4, 2, and 4. In the first of each sequence are given the angles of the gray filled-in triangle.

Figure 20-5a	Figure 20-5b	Figure 20-5c	Figure 20-5d

Figure 20-6a	Figure 20-6b	Figure 20-6c	Figure 20-6d

21. Beta series. Below are the two remaining figure eights. In figure 21-1 the layout has eight segments in each lobe, while figure 21-2 pictures one with six segments. Unlike the figure eights in the previous section, these offer no way to add an oval in the manner of figure 7-1 or figure 13-2. In this regard, the layouts below correspond to figure 6-3 and figure 14-6.

Figure 21-1	Figure 21-2

The distances are found in the beta series of table 21-A, where again each entry is the sum of the two previous.

Table 21-A: Selected values of the Beta series
As an integer	To two places	Exactly	Alpha series addends
69	68.61	400 * tan36° * Φ-3	19 + 50
111	111.01	400 * tan36° * Φ-2	31 + 80
180	179.61	400 * tan36° * Φ-1	50 + 130
291	290.62	400 * tan36° * Φ0	80 + 210
470	470.23	400 * tan36° * Φ+1	130 + 340
761	760.85	400 * tan36° * Φ+2	210 + 551
1231	1231.07	400 * tan36° * Φ+3	340 + 891

Perhaps surprising is that each beta term is the sum of two next-to-consective alpha terms, as illustrated in the fourth column of table 21-A; this is because Φ^-1 + Φ⁺¹ = tan36° / tan18°. As a result, the beta-series values of figures 21-1 and 21-2 can be decomposed into the alpha-series values of figures 21-3 and 21-4. This convenient relationship does not exist between the secondary and primary series of the 45° system.

Figure 21-3

Figure 21-4a
Figure 21-4b

Figure 21-3 uses the same crossing piece as figure 20-3b, and figure 21-4b uses that of figure 20-1.