Mathematical details of the 36-degree tracks.
§1. There is quite a detailed mathematical basis for why pieces in the 36° system can fit one another exactly in sophisticated, irregular layouts. It is presented on this separate page because it would be a lengthy interruption of the general description on the main page, and because a user of the 36° system does not need (and may not want) to know all the details. On the other hand, a manufacturer will certainly need this information.
§2. Everything starts with the positive square root of five. Written √5, it equals 2.236068 approximately.
From √5 arises the golden ratio. Written φ, it equals (1 + √5) ÷ 2 exactly, and 1.618034 approximately. Noteworthy is this relation: φ − 1 = 1 ÷ φ.
From φ follow angles that are multiples of 36°:
With an error of less than one-half percent, φ ≈ 5 tan18°. Although not pursued in this report, this approximation may be helpful to designers seeking to expand the catalog of pieces.
From φ also comes the Fibonacci sequence of integers; for the purpose here, a useful subsequence is:
… 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 …
Each number is the sum of the two previous numbers, and the sequence can be extended in either direction. An exact formula for the Fibonacci numbers is, for any integer n:
F(n) = (φn − (−φ)−n) ÷ √5
For instance, F(9) = 34.
Related to the Fibonacci sequence is the Lucas sequence:
… 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 …
As with the Fibonacci, each number is the sum of the two previous. An explicit formula is:
L(n) = φn + (−φ)−n
For instance, L(7) = 29.
§3. To establish the lengths of straight segments of track, the sigma sequence, with a formula related to that of the Fibonacci, has been chosen because it gives elegant results when other considerations are brought into play:
σ(n) = φn ÷ √5
The following table lists some useful values of σ(n). The numbers are used to define the lengths of straight segments.
| Some values of the sigma sequence used for the lengths of straight segments; millimeters are assumed compare the Fibonacci sequence | ||||
| exactly | to three places | as an integer | sigma sequence addends | |
| σ(8) | φ8 ÷ √5 | 21.010 | 21 | 13 + 8 |
| σ(9) | φ9 ÷ √5 | 33.994 | 34 | 21 + 13 |
| σ(10) | φ10 ÷ √5 | 55.004 | 55 | 34 + 21 |
| σ(11) | φ11 ÷ √5 | 88.998 | 89 | 55 + 34 |
| σ(12) | φ12 ÷ √5 | 144.001 | 144 | 89 + 55 |
| σ(13) | φ13 ÷ √5 | 232.999 | 233 | 144 + 89 |
| σ(14) | φ14 ÷ √5 | 377.001 | 377 | 233 + 144 |
| σ(15) | φ15 ÷ √5 | 610.000 | 610 | 377 + 233 |
| σ(16) | φ16 ÷ √5 | 987.000 | 987 | 610 + 377 |
| σ(n) = σ(n − 1) + σ(n − 2) | ||||
Highlighted in this and subsequent tables are the values most likely to be needed for ordinary toy track layouts.
Throughout this report, integer approximations are frequently used to represent exact values, with the pleasant result that the sigma values "look like" the Fibonacci values. Besides convenience, a further reason for the integral shorthand is that manufacturers of track pieces probably will not (and need not) achieve a tolerance of better than half a millimeter. Nonetheless, three decimal places are provided in the chart to aid anyone analyzing the figures.
§4. A second sequence for the lengths of straight segments needs to be introduced. This tau sequence is closely related to the sigma sequence, as indicated by the formulas at the bottom of the table below.
| Some values of the tau sequence used for the lengths of straight segments; millimeters are assumed compare the Lucas sequence | |||||
| exactly | to three places | as an integer | tau sequence addends | sigma sequence addends | |
| τ(6) | φ6 | 17.944 | 18 | 11 + 7 | 13 + 5 |
| τ(7) | φ7 | 29.034 | 29 | 18 + 11 | 21 + 8 |
| τ(8) | φ8 | 46.979 | 47 | 29 + 18 | 34 + 13 |
| τ(9) | φ9 | 76.013 | 76 | 47 + 29 | 55 + 21 |
| τ(10) | φ10 | 122.992 | 123 | 76 + 47 | 89 + 34 |
| τ(11) | φ11 | 199.005 | 199 | 123 + 76 | 144 + 55 |
| τ(12) | φ12 | 321.997 | 322 | 199 + 123 | 233 + 89 |
| τ(13) | φ13 | 521.002 | 521 | 322 + 199 | 377 + 144 |
| τ(14) | φ14 | 842.999 | 843 | 521 + 322 | 610 + 233 |
| τ(15) | φ15 | 1364.001 | 1364 | 843 + 521 | 987 + 377 |
| τ(n) = √5 σ(n)
τ(n) = τ(n − 1) + τ(n − 2) τ(n) = σ(n + 1) + σ(n − 1) σ(n) = τ(n − 2) + τ(n − 6) + τ(n − 10) + τ(n − 14) … | |||||
Each tau value is the sum of two sigma values, but each sigma value is not the sum of two (or finitely many) tau values. Because each tau can be decomposed into sigmas, manufacturers will probably concentrate on providing sigma-length segments, with only a limited selection of tau-length segments. Still, the tau values sometimes simplify understanding.
In principle could be created a upsilon sequence, multiplying each tau value by √5, but little practical need for it has yet arisen. In the other direction, an rho sequence could be defined, dividing each sigma value by √5.
§5. Appearing in the lambda sequence of the table below are circular radii compatible with the sigma and tau lengths for straight segments given above.
| Some values of the lambda sequence used for the radii of circular segments; millimeters are assumed | ||||||
| exactly | to three places | as an integer | lambda sequence addends | length of 36° arc | sagitta of 36° arc | |
| λ(11) | φ11 sin36° ÷ √5 | 52.312 | 52 | 32 + 20 | 32.868 | 2.560 |
| λ(12) | φ12 sin36° ÷ √5 | 84.642 | 85 | 52 + 32 | 53.182 | 4.143 |
| λ(13) | φ13 sin36° ÷ √5 | 136.953 | 137 | 85 + 52 | 86.050 | 6.703 |
| λ(14) | φ14 sin36° ÷ √5 | 221.595 | 222 | 137 + 85 | 139.232 | 10.846 |
| λ(15) | φ15 sin36° ÷ √5 | 358.549 | 359 | 222 + 137 | 225.283 | 17.549 |
| λ(16) | φ16 sin36° ÷ √5 | 580.144 | 580 | 359 + 222 | 364.515 | 28.394 |
| λ(17) | φ17 sin36° ÷ √5 | 938.693 | 939 | 580 + 359 | 589.798 | 45.943 |
| λ(18) | φ18 sin36° ÷ √5 | 1518.837 | 1519 | 939 + 580 | 954.314 | 74.337 |
| λ(n) = λ(n − 1) + λ(n − 2) λ(n) = σ(n) sin36° | ||||||
Here are formulas for some of the non-lambda radii that appear:
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It is important to remember that plenty of layout variety is possible if non-turntable circular segments are limited to the sole radius λ(14) ≈ 222. The radii of segments within disks will be whatever they will be.
§6. In the table below are algebraic forms for trigonometric functions of the angles being used here; they are frequently helpful in verifying obscure-looking identities such as:
| sin18° | cos72° | 1⁄4 (√5 − 1) | 1⁄2 (φ − 1) | 0.309017 |
| sin36° | cos54° | 1⁄4 √(10 − 2√5) | 1⁄2 √(3 − φ) | 0.587785 |
| sin54° | cos36° | 1⁄4 (√5 + 1) | 1⁄2 φ | 0.809017 |
| sin72° | cos18° | 1⁄4 √(10 + 2√5) | 1⁄2 √(φ + 2) | 0.951057 |
| tan18° | cot72° | 1⁄5 √(25 − 10√5) | φ−1.5 5−0.25 | 0.324920 |
| tan36° | cot54° | √(5 − 2√5) | φ−1.5 5+0.25 | 0.726543 |
| tan54° | cot36° | 1⁄5 √(25 + 10√5) | φ+1.5 5−0.25 | 1.376382 |
| tan72° | cot18° | √(5 + 2√5) | φ+1.5 5+0.25 | 3.077684 |
| sec18° | csc72° | 1⁄5 √(50 − 10√5) | 2 ÷ √(φ + 2) | 1.051462 |
| sec36° | csc54° | √5 − 1 | 2 φ − 2 | 1.236068 |
| sec54° | csc36° | 1⁄5 √(50 + 10√5) | 2 ÷ √(3 − φ) | 1.701302 |
| sec72° | csc18° | √5 + 1 | 2 φ | 3.236068 |
Next is a listing of powers of φ, in other words the tau sequence, written in a way different from the tau table above. Note how each can be simplified to a linear function of φ.
| Powers of φ compare the tau sequence | |||
| φ−4 | −3φ + 5 | (7 − 3√5) ÷ 2 | 0.14590 |
| φ−3 | 2φ − 3 | (−4 + 2√5) ÷ 2 | 0.23607 |
| φ−2 | −φ + 2 | (3 − √5) ÷ 2 | 0.38197 |
| φ−1 | φ − 1 | (−1 + √5) ÷ 2 | 0.61803 |
| φ0 | 1 | 1 | 1.00000 |
| φ1 | φ | (1 + √5) ÷ 2 | 1.61803 |
| φ2 | φ + 1 | (3 + √5) ÷ 2 | 2.61803 |
| φ3 | 2φ + 1 | (4 + 2√5) ÷ 2 | 4.23607 |
| φ4 | 3φ + 2 | (7 + 3√5) ÷ 2 | 6.85410 |
| φ5 | 5φ + 3 | (11 + 5√5) ÷ 2 | 11.090170 |
| φ6 | 8φ + 5 | (18 + 8√5) ÷ 2 | 17.944272 |
| φ7 | 13φ + 8 | (29 + 13√5) ÷ 2 | 29.034442 |
| φ8 | 21φ + 13 | (47 + 21√5) ÷ 2 | 46.978714 |
| φn | F(n)φ + F(n−1) | (L(n) + F(n)√5) ÷ 2 | |