§0. This page is a revisiting of the present author's previous work in track design for toy trains, specifically part 1 and part 2 of the 36-degree system within the track for toy trains site. Although the fundamentals are unchanged, many details are improved. The images have been completely redone, now using Scalable Vector Graphics.
In the years since the original work, it has become clear that the 36° system has remarkable potential for producing interesting, sophisticated track layouts for which it is relatively easy to find pieces that fit. Continued research has not found anything better.
Throughout this report, angles are measured in degrees, and distances in millimeters. Integer approximations are frequently used as a shorthand for exact values, although the angle of 36 degrees is exact. Full mathematical details are on another page.
§1. A near-standard of track design has evolved in the wooden toy train track industry (examples), with many plastic tracks exhibiting similar specifications. Although track segments may be straight or curved, long or short, their cross-section usually approaches these specifications:
Some manufacturers round the corners of the cross section, as in figure 1B. The essential dimensions are the same.
Note that the tracks do not have anything like the rails of a real railroad.
The radius of curves is rarely less than 100mm, because toy train cars of the typical design would tend to bind in anything tighter. Cars that are longer than average, as well as those with four axles, benefit from articulated trucks to ease passage around curves.
This report does not cover the matter of connecting one piece of track to the next, because that is independent of the track system's geometry. To allow room for a connecting mechanism, this report recognizes a practical minimum of 40mm for the length of a straight track segment.
§2. Figure 2A is a very simple track layout. There are ten equal pieces, each a 36° circular arc with a radius of 222mm measured to the centerline of the track. Although one piece touches the next, a slight gap is drawn to show where one piece ends and the next begins. The lighter-colored areas represent the grooves.
In figure 2B, several 72° segments have replaced 36° segments yielding an equivalent layout. In some complicated pieces, the 72° segment becomes necessary, but the underlying geometry of the layout is not affected.
|figure 2A||figure 2B|
A number appearing in a curved segment represents its radius; in a straight segment, its length.
Figure 2C includes two instances of the dead-end piece, in black, symbolized by a U-shaped groove.
§3. Figure 3A is a figure-eight with nine curved segments per lobe. The radius of the curved segments defines the length of the straight segments.
Figure 3B is a figure-eight with seven curved segments per lobe.
Figure 3B is useful in making the terminology more specific. The layout contains 16 segments (14 curved, 2 straight), but 15 pieces (14 blue with one curved segment, 1 red with two straight segments). In other words, one piece might contain multiple segments; if so, they are not detachable from one another. No segment is ever a member of more than one piece.
Pieces with one segment are colored blue, and crossings (necessarily containing multiple segments) are red. The dead-end is black to symbolize that it has zero segments: train cars cannot travel on it. More colors will appear later.
The radius of 222mm (more precisely, 221.595mm) was selected so that the lengths of straight segments as induced above (namely 144mm and 610mm), when rounded to an integer, would coincide with the famous Fibonacci sequence. The exact figures lie in the sigma sequence.
of the sigma sequence
|σ(8)||21||= 13 + 8|
|σ(9)||34||= 21 + 13|
|σ(10)||55||= 34 + 21|
|σ(11)||89||= 55 + 34|
|σ(12)||144||= 89 + 55|
|σ(13)||233||= 144 + 89|
|σ(14)||377||= 233 + 144|
|σ(15)||610||= 377 + 233|
|σ(16)||987||= 610 + 377|
Highlighted in this and subsequent tables are the values most likely to be needed.
Figure 3C, using switches colored in green, adds straight segments to figure 3A.
The switch shown in magnification is nearly an extreme:
|• 144mm is almost the shortest length …|
|• 36° is almost the smallest angle …|
|• 40mm is almost the largest track width …|
|• 222mm is almost the largest radius …|
|… that will allow the straight and curved segments to clear each other.|
Similar comments apply to the 36° crossing, but the 72° crossing requires an explanation. The segment length α(10) = φ10 ÷ √5 ≈ 55.004mm is not quite long enough to allow the segments to completely clear, because the required length is 40 cot36° ≈ 55.055mm. Thus the crossing does not work in theory. Yet the shortage (about 51 micrometers) is well below manufacturing tolerances for toy train track, so the piece should still work. Alternatively, a manufacturer could simply adopt a track width of 39.962mm rather than 40.000mm; or reject this 55mm crossing entirely.
Figure 3D is a modification of figure 3B. The straight segment of length 322mm comes from the tau sequence explained in the next section.
§4. Figure 4A is a figure-eight with eight curved segments per lobe. It naturally induces the 322mm straight segment first seen in figure 3D.
Figure 4B is a figure-eight with six curved segments per lobe.
A second sequence for the lengths, as measured in millimeters, of straight segments needs to be introduced. This tau sequence is closely related to the sigma sequence, because each tau can be decomposed into sigmas — but not vice versa.
of the tau sequence
|τ(6)||18||= 11 + 7||= 13 + 5|
|τ(7)||29||= 18 + 11||= 21 + 8|
|τ(8)||47||= 29 + 18||= 34 + 13|
|τ(9)||76||= 47 + 29||= 55 + 21|
|τ(10)||123||= 76 + 47||= 89 + 34|
|τ(11)||199||= 123 + 76||= 144 + 55|
|τ(12)||322||= 199 + 123||= 233 + 89|
|τ(13)||521||= 322 + 199||= 377 + 144|
|τ(14)||843||= 521 + 322||= 610 + 233|
|τ(15)||1364||= 843 + 521||= 987 + 377|
In figure 4C, which is a modification of 4A, the lengths of the crossing segments are 76mm and 144mm.
Figure 4D is a modification of 4C.
When each lobe has an even number of curved segments, as with figures 4C and 4D, the is no simple way to form an oval as was done in figures 3C and 3D.
§5. Numbers from the lambda sequence are used for the radii, measured in millimeters, of circular segments. Because roundoff error is sometimes apparent, the approximately-equal sign ≈ is employed.
of the lambda sequence
|λ(11)||52||≈ 32 + 20|
|λ(12)||85||≈ 52 + 32|
|λ(13)||137||≈ 85 + 52|
|λ(14)||222||≈ 137 + 85|
|λ(15)||359||≈ 222 + 137|
|λ(16)||580||≈ 359 + 222|
|λ(17)||939||≈ 580 + 359|
|λ(18)||1519||≈ 939 + 580|
A considerable variety of layouts can be produced if only one radius is chosen; for that purpose 222mm is recommended. Wooden train cars of conventional design can easily negotiate such a curve, and the arc length of 139mm is a convenient size. Further, a 72° curved segment, which might be required as part of a multi-segment piece, is not so long to be unwieldy.
On the other hand, figure 5 shows some other radii in use.
|figure 5A||figure 5B||figure 5C|
|figure 5D||figure 5E||figure 5F|
§6. Figures 6 illustrate an orderly method to create a variety of useful crossings and switches. All are based on a uniform arrangement of ten segments of indefinite length, drawn in gray.
Figure 6A shows the template alone, and with blue segments that will fit. The radius of 99mm does not appear in the lambda table above, but such a curve might be included in a very large set of pieces.
|figure 6A1||figure 6A2||figure 6A3||figure 6A4|
Figure 6B4 is the gauntlet, where the two segments do not ultimately cross, but where they overlap sufficiently that they cannot be used by two trains simultaneously. In this report, the gauntlet is classified as a crossing, and colored red.
|figure 6B1||figure 6B2||figure 6B3||figure 6B4||figure 6B5|
Figure 6C3 illustrates a triangular switch, made possible by the tight radius of figure 6A4.
|figure 6C1||figure 6C2||figure 6C3|
In figure 6D, the color brown indicates pieces that contain both a switch and a crossing. The astrino (Italian: "small star") of figure 6D3 risks practical difficulties, as there are so many grooves that there may remain too few non-grooved areas to properly guide train wheels.
|figure 6D1||figure 6D2||figure 6D3|
In figure 6E, the lengths and radii of segments have been multiplied by φ ≈ 1.618. Like 99mm, 160mm is not in the lambda table. Counterparts of all the pieces above exist, with figure 6E4 being a larger, less congested astrino (compare 6D3), and 6E5 (from 6C3) now having a hole in the middle.
Here is a subtle matter of coloration. The pieces in figures 6F1 and 6F2 are the same except for the length of the straight segment. However, when each straight segment is removed, revealed is either a gauntlet (6F3), or two disjoint segments (6F4). Thus 6F1 is brown because it qualifies as both a crossing and a switch; but 6F2, a multiple switch, is green because it lacks a crossing.
|figure 6F1||figure 6F2|
|figure 6F3||figure 6F4|
§7. The turntable is a staple of toy railroad layouts, although it has become rare in real-life railroads; it consists of a disk which rotates freely within a ring.
|figure 7A1||figure 7A2||figure 7A3|
The color brown was selected for all the track segments on a turntable, because the device combines the functions of a crossing and a switch. Yellow represents the areas of the turntable that a train would never roll on.
Turntables can be made in various sizes, with three examples below (figures 7B-C, 7D-E, 7F-G). The overall diameter, measured across flats rather than across corners, must be carefully selected if the turntable will be easy to fit among the other track pieces. Some suitable values are 233mm, 322mm, and 377mm. In the illustrations, the diameter is represented by three digits arched across the top of the ring. (It was difficult to find a better location.)
The diameter of the disk is not critical, and in each example was arbitrarily chosen to give familiar-looking numbers. Presumably, the disk would be large enough that the segments on it would be long enough to hold a train car. That way, a car rolls onto the disk, the disk turns, and then the car rolls off in whatever direction.
In a complicated layout, it may be difficult to use all ten ports. Some hobbyists will deem it necessary to attach a dead-end piece to each unused port, while others will reject this policy as a needless formalism.
Figure 7B shows turntables whose outer diameter is σ(13) ≈ 233 and whose inner diameter is σ(12) ≈ 144. The disk happens to be the right size to accept any of the segment combinations of figures 6A through 6D. A manufacturer might even design turntables so that they accommodate interchangeable disks.
|figure 7B1||figure 7B2||figure 7B3|
Figure 7C shows some of the ways that the turntable as a whole can replace one or more plain segments, although the smaller radii are of little more than theoretical interest.
|figure 7C1||figure 7C2||figure 7C3||figure 7C4||figure 7C5|
|λ(15) ÷ √5|
|λ(12) ÷ √5|
Figures 7D and 7E are similar to figures 7B and 7C, except with a turntable of 322mm diameter.
|figure 7D1||figure 7D2||figure 7D3|
|figure 7E1||figure 7E2||figure 7E3||figure 7E4||figure 7E5|
|λ(14) × √5|
|λ(11) × √5|
Expanding on figure 7E3, figure 7E6 shows how the astrone (Italian: "large star"), an assembly of five ordinary crossings, is a drop-in replacement for the 322mm turntable. This crossing is often useful, but most frequently appears alone.
The turntable of outer diameter of 322mm is recommended if a collection of pieces will be limited to the radius 222mm. This is because it is equivalent to two curved segments (figure 7E3). Not pictured is the very large turntable of outer diameter σ(15) ≈ 610mm, which is equivalent to three curved segments of radius 222mm.
Figures 7F and 7G have a turntable of diameter 377mm. The 108° disk segment (7F4) approaches feasibility as the diameter of the disk grows.
|figure 7F1||figure 7F2||figure 7F3||figure 7F4|
|figure 7G1||figure 7G2||figure 7G3||figure 7G4||figure 7G5|
|λ(16) ÷ √5|
|λ(13) ÷ √5|
Using subsets of the astrino of figure 6D3, it is possible to assemble a ring around a 322mm-turntable (figure 7H).
Figure 7I1 shows a turntable of diameter τ(11) ≈ 199mm, which might be large enough for practical use. The diameter of the turntable of 7I2, σ(12) ≈ 144mm, is near the theoretical minimum for a ten-port turntable. To see this, observe that the crossing of 7I3 is a drop-in replacement for that turntable, and the crossing of 7I4 (which is like figure 3C3) is a subset of 7I3.
|drawn at larger scale than the turntables above|
|figure 7I1||figure 7I2||figure 7I3||figure 7I4|
Two turntables can be connected. In figures 8A and 8B, there is an odd number of curved segments between the turntables.
|figure 8A||figure 8B|
In figures 8C and 8D, the number of curved segments is even, possibly zero.
|figure 8C||figure 8D
smallest radius is 52mm;
radius 496mm is shown in part
Figure 8E places five turntables in a ring.
§9. An alternative to the turntable is the transfer table, which has a straight track that slides sideways. An example appears in figure 9A, placed in a context of other pieces to show the rationale for its dimensions. The center-to-center distance between parallel tracks is λ(12) ≈ 85mm. The color of segments is green, because the piece has switching functions, but displays little characteristic of a crossing.
This transfer table is 18° out of square because that makes it compatible with the sigma track lengths. Of course, it is possible to design transfer tables using other dimensions.
Figure 9B shows the slider in several different positions.
Figures 9C1 and 9C2 contrast the transfer tables of positive slant, as above, and negative slant. Figure 9C3 shows one way to connect them.
In figure 9D adjoin three transfer tables of differing dimensions, but the same slant.
§10. Figure 10 is an example layout.