Magical Physics (and Mathematics) part 1: Gaming the (spell level) System

This is the first part of a complex project I’m working on for a number of unrelated reasons (sort of a bet, sort of a dare, and then I got really intrigued…) that has to do with the real-world physics of D&D5e magic, and translating magic to physics equations. The goal is to be able to write something that looks like it could be a chapter out of a physics textbook, but for D&D magic-users. The first step, though, is to be able to come up with a way of representing magical power (in game terms, spell slots) as something that can be worked with (an equation or a constant or something). 

I started with a simple method that took the number and level of spell slots for a given character level and produces a single measure of magical power. I used the table for multiclass character spell slots by level (PH 165) for all my work, because I wanted something that did not require a specific class. This method should work for providing a measure of power for any class that can cast spells, including warlock, if you use the appropriate table. As if you were finding the spell slots of a multiclass caster, calculate the character level (C) for the purpose of using that table. 

Find the appropriate row for C on the table, and note the number of spell slots for each level of spell. Multiply the number of slots of a given level (S) times the level (L) of the slot for each spell level, and add that together. That gives the number of thaums (Þ) of magic that a caster of level C can call upon. Ignore cantrips when determining thaums — they require 0Þ to cast.

A note about units: the thaum, represented by the old English letter Þ (the thorn), is a unit of magic. It is the amount of magic required to cast one first level spell one time. The number of thaums required for a given spell is assumed to be equal to the spell level or the level of the slot (I don’t know how accurate that is, I just needed a simple assumption). A caster can produce a certain number of thaums per day, depending on their level and scaling with it. We are endeavoring to figure out that relationship.

Anyways. That gives the amount of magic at a given C, but how does it change over time? I calculated the thaum value for each level on the given table (1≤C≤20) and plotted it against C. T, the thaum value for a given level, is measured in thaums and is a function of C, the character level (measured in levels). The table of values of T is in this spreadsheet. 

I used a graphing calculator (TI-84 plus) to plot the known values of T and C, then used the calculator to run four different kinds of regression analyses (linear, quadratic, cubic, logistic) in an attempt to find a useful equation for relating T and C. The equations are also in the spreadsheet, under the column “Equations for lines of best fit”. 

The logistic equation works least well to match the points (4 of the points were the right pixel on the calculator screen), so I eliminated it. The cubic equation matched 7 points, the quadratic matched 9 points, and the line matched 5 points, so I determined that the quadratic was the most accurate equation to relate T and C (is this where the idea that wizards scale quadratically and fighters scale linearly came from? Further research clearly required). 

Using the equation T = 0.03C^2 + 3.87C - 4.04, I can now calculate a good enough value of T for any C. I did so in the spreadsheet as well. It’s a little low at the extremes, but it does well enough in the middle. 

Tune in next week, when I use this equation to relate spell levels and fire damage! (Or something. I plan on playing around with this more.)