Tang's DW

One virtue of my math education was that it was fairly proof heavy; I feel I can demonstrate or prove most of what I know. I may have mentioned this not extending to sin(A+B), which I had to figure out later as an adult. It also didn't include detailed "how do we calculate this shit?" for stuff like trig, log, fractional exponents. Once you hit calculus the answer becomes "Taylor series or something better", but before then?

Feynman talks about it somewhat in Chapter 22 of his Lectures, which is a cool read; he goes from defining multiplication in terms of addition, to numerically suggesting Euler's law, and showing how to build up log tables along the way. It's a heady ride. But he doesn't talk about trig tables.

I've long had my own thoughts about those, and finally wrote some code to try out the calculations.

Basic idea is that given sin(a) = y/r, cos(a) = x/r, and the Pythagorean theorem, there are two right triangles for which we know the values precisely. The 45-45 is trivial, while the 30-60 one can be gotten from sin(60) = cos(30) = sin(30+30) = 2*sin(30)*cos(30). From there, you can apply the half-angle formula as often as you please, to get very small angles, at which point you'll notice that sin(x) ~= (approximately equal to) x for small x. Thus you can approximate an angle like 1 degree which you can't otherwise[1] get to, then use double and addition formulas to build back up to say 10 degrees or anything else.

The question in my mind always was, how good is that? And the code finally answers the question. Sticking to half-angle stuff is *very* accurate, matching the built-in function to 15 decimal places. Using a sin(1 degree) approximation and building up to sin(15 degrees) is accurate to 4 places; starting from 0.125 degrees instead is accurate to 6 places. I don't know how practically good that is; one part in a million sounds pretty good for pre-modern needs -- like, at that point can you make or measure anything that precisely? -- but Feynman says that Briggs in 1620 calculated log tables to 16 decimal places.

[ETA: hmm, I'd been assuming the built-in function, presumably based on power series, was the most accurate version, but maybe I should view it as deviating from the exact roots of the half-angle approach. Or both as having small errors from the 'true' value. Especially as Python's sin(30) returns 0.49999999999999994, not 0.5. Of course, both approaches are using floating point. sin(60) = sqrt(3)/2 to the last digit, but cos(30) is slightly different.]

[1] The two exact triangles basically give you pi/2 and pi/3, and those divided by 2 as much as you want. But you can't get to pi/5 exactly.
a/2 + b/3 = 1/5 #a and b integers
15a + 10b = 6
5(3a+2b) = 6
(note that 2a+3b=5 is totally solvable.)

Though I guess a more valid approach would be
c(1/2)^a + d/3*(1/2)^b = 1/5 #all variables integers
c*5*3*2^b + d*5*2^a = 3*2^(a+b)
5(c*3*2^b + d*2^a) = 3*2^(a+b)
which still isn't soluble.

Likewise for getting to pi/9 (20 degrees):
c(1/2)^a + d/3*(1/2)^b = 1/9
c*9*2^b + d*3*2^a = 2^(a+b)
3(c*3*2^b + d*2^a) = 2^(a+b)

I have no specific memory of when I first heard of the words fanfic, or fan fiction. There must have been a moment, after going to college, but no memory. (Filk I can do: S had tapes, and lent them to me, so that's restricted to a few years, and I can kind of remember borrowing and copying them.)

I can place a very likely upper bound. In the spring of 1998 I discovered "Buffy the Vampire Slayer" (there's a story there for another time), coming in with Surprise and Innocence as they aired, and after Becoming, plowing through jtr's videotapes. Some time after that I was on the Buffy mailing list, and was introduced to the Buffy-Xena crossover fanfic "When Hellmouths Collide" (a mere 870K words, which I never finished). I'd also found a trove of Post-Becoming fic, which I am tickled pink to see is still online.

I'd also had my own ficcy ideas, perhaps some Post-Becoming but mostly about Amy the Underused Witch; meanwhile, my friend Liz went and started writing her own Season 3 scripts, I think to the exclusion of watching the actual show.

I had been in online fandoms before then: the Babylon-5 newsgroup, where I coined "battlecrab" for the Shadow vessels; Deryni newsgroup or mailing list (where I helped with the FAQ), Pliocene newsgroup or mailing list; my own fan pages for Vernor Vinge and Steven Brust; adopting a fan page for Robin McKinley; a SF newsgroup in general... but not a blip or hint of memory of fanfic from any of those. Vs. the eruption of fanfic, named or otherwise, from Buffy.

Funnily enough, I grew up reading a fair bit of what I now consider to be essentially, if not legally, fanfic: tie-in novels. Lots of Star Trek novels, back when Paramount gave them a very loose leash and lots of ideas or varieties could be explored (also back when they were more written by women). Some Brian Daley Star Wars tie-ins. The "Jack McKinney" (pseudonym for Brian Daley and another writer) Robotech novels. In fact, despite watching some DS9 later, and bits of other series, "Star Trek" for me is largely the old TOS novels that I read as a child and teen, with mental visuals coming from the covers and whatever my imagination could spin from those.

As for fanfic after Buffy, I don't have a clear memory. I suspect it went into remission for some years, like my interest in (or even memory of) filk. I had my own unwritten ideas, particularly about Hodgell's stories, but I'm not sure when I re-engaged with ficcy fandoms again. Bujold list? IU Anime? I dunno.

Now, of course, it's a fairly decent part of the fiction I read, ranging from tiny drabbles to massive epics, from smut to deep explorations and extrapolations.

Chrono-trivia: FF.net is from 1998, AO3 from 2008. I remember Latin-Sarah talking about AO3 while I was at IU, but didn't realize it was that late.