Playing on Railroad Tracks

This post is about turning a toy into a mathematical exploration

Willy Viv

8/8/20235 min read

A couple of years ago, I was helping my son build a wooden railroad track so he could drive his trains along them. I got bored of tracks that were elliptical-ish, and used twists and turns, throwing symmetry to the wind. I quickly noticed that some mid to late decisions created a challenge in making the track's end pieces connect. I wondered if some kind of mathematical equation could determine ahead of time if the two end-pieces would connect (and if not, which pieces to include or take out to make them match up).

The wooden tracks have several different kinds of pieces: straight (of various lengths: small, medium, and large), a short turn, a long turn, up/down hill, crosses (where the railroad crosses itself in a perpendicular way), short split (where the track splits and can either go straight or turn), long split.

This paragraph talks about how I thought about pursuing this exploration. First, though I wanted to eliminate symmetry, I thought I needed a line of some kind, this would represent the equal sign (the values on the left should equal the values on the right once all the proper terms--or pieces--had been combined). Additionally, turns could be positive or negative since when they are flipped around the track would send the train in a different (or symmetrically opposite direction, I suppose that's the best way to describe it). I'll define left turning tracks as the negative of that track, and right turning as the positive, and "south" of (or below) the starting point, would be negative for straight tracks. Straight tracks that are horizontal will have a different label.

I discovered, a website dedicated to building a track online. The benefit here is that I am not limited to the blocks I have at home (so if I needed a small turn and don't have one, I can use the online tool which has infinitely many). The website started me with -1 small turn (t1), +3 small turn (t1), 1 long horizontal straight track (h3), 2 small horizontal straight tracks (h1), +1 medium straight track (s2), -1 long straight track (s3), and 2 horizontal hills (one up one down) (m--for mountain). Looking at the setup (see image 1), I already know that I have assumed something I should not have, and I already have a new theory. First, my assumption, I think there will be configurations that work that do not fulfill my mathematical equation (though, I suspect if they do fulfill it, they will work). My new theory is that the degrees (of the turns) matter slightly more than the straight portions.

Image 1 (from Choo Choo World website linked above)

So I already know that if I combine the turns of the original setup (figure 1), I end up with 2 small right turns (direction, by the way comes from the direction of the train, it should not matter which direction the train is going as long as it is consistent), which will get the track turned all the way around to connect with the original point, however, order appears to matter here because if I add in the rest of the points here (see image 2). So my original idea clearly needs some work and I need to reconfigure my original plan (this is called "monster barring," redeveloping a conjecture so that it dodges monsters--or situations--where the math won't work, mathematicians do this all the time). But there is something else I notice, the station (which I think is approximately the length of a long straight track) is about equal to 2 small straights (s1), 1 medium straight (s2), and 1 mountain (m); figuring out what pieces are equivalent to a set of other pieces can be helpful for other explorations.

Image 2

I played around a little more and found that two small straights are equal to a medium one, each turn, regardless of size is a 90 degree turn. I began to think that both angles and lengths are equally important, and I am reminded of building a 2-dimensional geometric shape. So what do we need to keep in mind about geometric shapes? Typically, geometric shapes for K-12 education are closed (meaning the side lengths and angles are such that they connect, much like this train problem I tried to tackle). Inspired by the track from image 1, I created a geometric figure that could represent a train track.

Image 3 (a screenshot from Desmos Geometry calculator)

Image 3 depicts an interesting shape and the shape of the corresponding track is below (image 4). Image 3 includes angles and side measures, so if we are to build a track that corresponds, we'll need to figure out how long each track piece is worth. I fiddled around and found that the long straight pieces were about 4 times the length of a small one, and a medium was half the length of a long one (approximate because I was going on visual cues, it was hard to actually measure using this program). I used the small turns for all the 90 degree angles so length of the turns was only an issue at the end.

Image 4

While I leave for the reader the work of verifying the majority of my track layout, I must attend to the small straight tracks at the end of the (just prior to the station, and just prior to the final turn). What is different about how we draw geometric figures and laying track, is that the turn on a track adds to the total length and width of the shape, whereas when drawing geometric shapes, angles add nothing to the length or width. To accommodate the reality that trains do not pivot like our pencils can, we must add on additional track to make up for that reality (or we could have accounted for it at the beginning). So, how much extra distance does a small turn add? I leave that question up to the reader to ascertain.

Here is the final hiccup (and maybe something I will explore at a later date), the Choo Choo World exploration tool does not include split track options as described earlier, that is, a track component that allows the train to go straight OR turn. How much of a turn do these splits make (are they 90 degrees?) and what geometic shape possibilities do they allow?