Sunday, 6 October 2013

Author Explains: The Math

Pic by Errol of Debs and Errol

IN BRIEF: "Taylor's Polynomials" is a web serial which includes math, urban fantasy, pop culture, and wordplay. If you like some of those things, hopefully you're already on board. But if you need more convincing, let me break it down for you... post by post.


This first post is for the math geeks. Part of the reason I want you to be reading is to keep me honest! While I do research before I post, sometimes I go out of my depth - is the math correct?? Tell me!

Perhaps some of you have even been to this blog before - then left, not understanding what I was doing. This time, read on. I'll make the connections very explicit.


I) THE MATH



Personified Parabola
I am a high school mathematics teacher, and this is, first and foremost, a mathematical web serial. All of the characters are mathematical functions. But mathematical knowledge is not a prerequisite! The Modulus function runs an Absolute Value Bar, the Parabola has a curved hairband and three forms, and the Conic family is eccentric... if you get the joke, great! If not, they're just characters. This brings me to my first point:
If you don't know much math, this may be a fun way to learn something.

Which immediately leads to my second point:
If you DO know math, this may be a fun way to learn something new.

Are you a middle school math teacher wondering about high school? Are you a high school math student wondering about fractals? Are you a non-math educator wondering how you might go cross curricular? Are you NOT involved in education, but enjoy READING? Look, learning something isn't required! Yet all that stuff has been in here.

Plus I can be subtle about it. Here's more "points":


HAIRSTYLE GRAPHS



The "modulus" function
Every character has some aspect of their graph pictured within their hair. This is the main reason why I began by drawing for the serial myself. Hairstyles even became a minor plot point in Series 3.

This means that, merely seeing the characters, you will be exposed to what each of their graphs look like. With any luck, if you chance to graph a function later in your life, it will trigger a recollection - or maybe it's triggering a memory from your high school days.


Minkowski's ?(x)
I try to be really careful about maintaining this when designing new characters. For the step functions, Heaviside is zero until she takes on the identity of a particular function, so her right twintail is set "at zero", and the other is higher, "at one". I found Minkowski's question mark function on a graph from 0 to 1, hence the earrings of "0" and "1" each side of the hair. There was some debate as to whether the Conway Box function was a similar function, or an inverse; unable to find a graph, I reflected Minkowski's hairdo in the line y=x and made the situation somewhat ambiguous.

I challenge you to find a mistake.

The obvious problem here is that it's a visual gag, and I'm not able to draw every character for every episode! It's a web serial, not a web comic; I don't have the time nor the skill to produce the latter. So if you only come once or twice, you may never see the character drawn, and you won't necessarily pick up on this.

Come more often!


FAMILIES AND INVERSES


Something else you might notice after a dozen views. Visually, certain mathematical families share certain traits. The polynomials are all blonde. They all have a bow in their outfit to represent the existence of a y-intercept. Odd degree functions have short hair and green eyes. Even degree functions have long hair and blue eyes. Moving away from the polynomials, trigonometry are all brunettes, and have more wavy hair.


Character Pairings?
Gender doesn't even need the visual, thanks to using personal pronouns. Most of the key high school functions (parabola, sine, exponential) are female. Their inverses are MALE. There's a running gag in my world that a function and her inverse are destined to marry... which may give me the opportunity to comment on gay marriage in the future. I am being completely serious.


Trigonometric Reciprocals.
In degree mode. Do you see why?
Among other things, I like to think this helps to distinguish between Sine and Sin^-1, aka ArcSin - whom I've also dubbed 'Nis' (Sin backwards). They have different outputs, which is seen on their T-Shirts. Further, we know Cosecant ISN'T an inverse (except to ArcCsc), because she's female... she also wears a Reciprocal Tie.

Of course, there are some functions which are their own inverses, notably Line (female, to match the other polynomials) and the Reciprocal (male since XY=1 and XY implies male). More subtleties, Lyn loses her y-intercept bow when she goes standard form, as vertical lines have no (or infinite) y-intercept. Reci is also always "under some 1" - that being Hyper, the main hyperbola, of which XY is simply a special case. Oh, and speaking of the language I use...


MATHEMATICAL LANGUAGE


First, nicknames. I use "Cosecant" and "Csc" interchangeably the way normal stories might use "Candice" and "Candi" as meaning the same person. I've heard no complaints. (Seriously, SAY SOMETHING!) Second, I have to be extra careful not to use an english word that has a different meaning mathematically. Because that's the sort of thing you probably only notice when it's WRONG.


Circe.
More geek than greek.
For a simple example, I said the conics are eccentric - except for Circe. (She's simply a little power mad.) That's because mathematically, a circle has no eccentricity. For a more complicated example, I realized I couldn't have a "transformation sequence" to switch the quadratic between vertex and standard forms. Because mathematically, transformation means rotate, reflect, slide, or resize! Not exactly the meaning I wanted, namely it being the SAME character, expressed differently. Hence I went with a "form change" sequence instead.

In conclusion, ALL of this is meant to be a bit of math tutorial without shoving it in your face that THIS IS A MATH TUTORIAL, because that's no fun. At the end of the day, my hope is it's fun to read. I doubt you'll pick up on everything even if you do come for every update! Yes, that's another challenge.

But then, the math is merely the vehicle, with functions as the getaway drivers. It's all happening within a larger universe. Thus if you're not buying into the math "angle", let me explain to you about my many and various science fiction/fantasy/pop culture aspects. Next post.


Logarithm's base might look familiar.

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