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°:

• If the sides of a triangle are in the ratio 1:1:φ, the angles are 36°, 36°, and 108°.
• If the sides of a triangle are in the ratio 1:φ:φ, the angles are 36°, 72°, and 72°.

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 sequenceused for the lengths of straight segments;millimeters are assumedcompare the Fibonacci sequence exactly to threeplaces as aninteger sigma sequenceaddends 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 sequenceused for the lengths of straight segments;millimeters are assumedcompare the Lucas sequence exactly to threeplaces as aninteger tau sequenceaddends sigma sequenceaddends 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 sequenceused for the radii of circular segments;millimeters are assumed exactly to threeplaces as aninteger lambda sequenceaddends length of36° arc sagitta of36° 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:

kappa sequence
κ(11)λ(11) ÷ √5≈ 23.394
κ(12)λ(12) ÷ √5≈ 37.853
κ(13)λ(13) ÷ √5≈ 61.247
κ(14)λ(14) ÷ √5≈ 99.100
κ(15)λ(15) ÷ √5≈ 160.348
κ(16)λ(16) ÷ √5≈ 259.448
κ(17)λ(17) ÷ √5≈ 419.796
κ(18)λ(18) ÷ √5≈ 679.245

mu sequence
μ(11)λ(11) × √5≈ 116.972
μ(12)λ(12) × √5≈ 189.265
μ(13)λ(13) × √5≈ 306.237
μ(14)λ(14) × √5≈ 495.502
μ(15)λ(15) × √5≈ 801.740
μ(16)λ(16) × √5≈ 1297.242
μ(17)λ(17) × √5≈ 2098.981
μ(18)λ(18) × √5≈ 3396.223

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:

• tan54° = tan18° + sec18°
• 1 + sin18° = cos18° cot36°
• 2 φ2 tan18° sin36° = 1
The abundance of trigonometric identities is the reason that, when the layout designer needs tracks to fit together exactly, they so often will. Each row of the following table contains five equivalent expressions, the last an approximation.

 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 0.32492 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 1.37638 tan72° cot18° √(5 + 2√5) φ+1.5 5+0.25 3.07768 sec18° csc72° 1⁄5 √(50 − 10√5) 2 ÷ √(φ + 2) 1.05146 sec36° csc54° √5 − 1 2 φ − 2 1.23607 sec54° csc36° 1⁄5 √(50 + 10√5) 2 ÷ √(3 − φ) 1.7013 sec72° csc18° √5 + 1 2 φ 3.23607

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 φcompare 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