Sticky Note Math Beta: Don’t Let Your Trains Derail Your Plot

While I was reviewing this week’s Worldbuild Wednesday and I found myself more and more dissatisfied with it. Thus I’m putting it back into the oven and am going to take another stab at it over the next few weeks. I have a desire for a standard and it didn’t make it. Neither did the Sci-fi Saturday therefore I’m putting the main content on hold while I do some more research and develop them a bit more. Thus this week I’m going to do a beta test for one of the new series ideas I’ve been mulling over: Sticky Note Math. I don’t know if I want to keep that title. Let me know what you think of the title in the comments below.

When it comes to the first installment of Sticky Note Math I’m going to go over trains. Partly because I’ve been working though the Slonminman railroads and partly because they are one of the better examples of how a little bit of math can help the story teller understand what it is that the characters are interacting with in their world. This then provides the ability to then grant better descriptions, make better use of the scenarios on offer, and add in characterization to the world and side characters and locations.

I’ve gone over what makes trains work in a world before, to summarize there needs to be a meaningful distance between where goods, or people need travel on regular trips. Once that is set up the wall of specifics will hit. As questioning the gauge, size of cars, weight and everything will often raise the nerds from the internet. Any one of those nerds will go out of their way to highlight that the different gauges, style of car, and any other detail will influence or adjust what can actually do with the machines. Thus I find that trains are seldom described, or explained in any detail, as deciding on any detail will be wrong.

As the saying goes, “Your job is to lie to people for fun.”

Worldbuilding makes that lie more believable.

Which is I’ve developed the Sticky Note Math to build out trains. It starts with determining the gauge or width between the tracks. This is how we can determine the rails of the railway. The math I've settled on and use is the following.

Gauge in Inches = (35 ×Raidus of curve in feet) / Track speed in MPH²
Gauge in Milimeters = (7620 × Raidus of curve in meters) / Track speed in KPH²

These two numbers, track speed and radius of the curves should be for the main routes. Thus how fast and how tight are the curves. It will then spit out something that is more or less what is in line with real world situations. Generally if one needs to turn tighter at speed one will end up with a smaller track gauge. Now to determine what the absolute minimum curve can be, which is useful for those who need to interface rail with industry or have tracks running down city streets the following algebra will end up with this:

Minimum curve radius in feet = Gauge in inches / 1.85
Minimum curve radius in meters = Gauge in milimeters / 132

Those with algebra skills will realize that the formula can be translated and this simplistic version is at a reasonable shunting speed. If one wants to customize their track more go a head with the algebra. After that one has functionally everything they need to sanity check most routes and scenarios. Smaller gauge will mean smaller trains which is something that most people will notice even if they don't understand why. However with these two pieces we can go a step further, and determine how big the average car is going to be:

Car width = 2.2 (+/- 0.3) × G in inches or milimeters

Car length in feet = 30 + (0.000429 × Gauge in inches × track speed in MPH²)
Car length in meters = 9.1 + (0.000131 × Gauge in milimeters × track speed in KPH²)

This gives us the dimensions of the average car so that we can then decide how spread out or tightly packed people and things are within the train car. Thus giving something for the characters to think about, and perhaps reflect on while traveling in the trains. To figure out what the plausible number of people, or the tonnage of the cargo, in the cars would be we can plug in these equations after converting our width to feet or meters from the formula above.

Footprint = Width × Length

Seated Passengers ≈ Footprint in square feet / 9
Tons gross ≈ Footprint in square feet × 0.1

Seated Passengers ≈ Footprint in square meters / 0.84
Tonnes gross ≈ Footprint in square meters × 0.092

This will end up giving everything we could want about a train and it sanity checks fairly well provided one plugs in the correct scenario. As the AI chatbots are good at plugging and chugging this kind of math I asked one to invent a scenario and then run the numbers:

Highland Cross Line – Mountain narrow-gauge example

Inputs: R = 1500 ft (tight mainline curve), V = 40 mph

G ≈ (35 × 1500) / 40^2 # ≈ 32.8 inches → round to 33 inches (~2 ft 9 in gauge)

W ≈ 2.2 × 33 # ≈ 72.6 inches (~6 ft 1 in body width)

L ≈ 30 + 0.000429 × 33 × 40^2 # ≈ 52.6 feet (compounded length)

F = (72.6 / 12) × 52.6 # ≈ 318 sq ft footprint

Seated passengers ≈ 318 / 9 # ≈ 35 seated per coach
Gross freight tons ≈ 318 × 0.1 # ≈ 32 tons gross per freight car

The car would end up with a school bus style seating arrangement with 8.5 rows the other half row being a luggage rack. These would be smaller fixed benches where as on the train to the base of the mountain the same number of seats per row would be individual seats. More leg room, more cargo space, and more room for things such as arm rests or padding. Meaning our hero, having traveled to the base of the mountain in comfort is now crammed into what he sees as "a livestock car refit to pass the minimum human standards" before complaining that "these tiny trains don't have the space to provide for the better classes" as he is going to be the new lord and thus used to the comforts of reality, should he not die like the last four betrothed to this land's princess... His impending death aside, it shows us more about his character and is tied into this phase of the journey because of it's mode of transportation.

If you liked this jaunt into using math to rationalize bits of worldbuilding let me know. This is something that I do rather frequently, and I would enjoy working out the formulas for whatever topics you think would be helpful. Also let me know if you want a worksheet and the examples provided. For this beta run I'm going to copy/paste the full set of equations (with the conversions) below so that you can use them yourself.

Imperial (feet, inches, mph, tons, seated):
G ≈ (35 × R) / V² # inches
R ≈ (G × V²) / 35 # practical feet
W ≈ 2.2 × G # inches
L ≈ 30 + 0.000429 × G × V² # feet (compounded)
F = (W / 12) × L # sq ft
Seated ≈ F / 9 # passengers
Tons gross ≈ F × 0.1 # tons

Metric (metres, mm, km/h, tonnes, seated):
G ≈ (7620 × R) / V² # mm
R ≈ (G × V²) / 7620 # practical metres
W ≈ 2.2 × G # mm
L ≈ 9.1 + 0.000131 × G × V² # metres (compounded)
F = (W / 1000) × L # sq m
Seated ≈ F / 0.84 # passengers
Tonnes gross ≈ F × 0.092 # tonnes