DON JOSÉ "nice guy" who is also the protagonist, for some reason
CARMEN young woman from "Bohemia",
because heaven forbid anyone should try to
actually learn anything about the Roma
(hint: they're not actually from Bohemia)
ESCAMILLO bullfighter guy
MERCEDES friend of Carmen
FIAT friend of Carmen
(plus a half dozen other people I'm just going to leave out. cope.)

Continued from Part 3, what happens when there is no "parallel", the rules for circles aren't what you thought they were, and so on.

Napier's Rules

So how does trigonometry work in this world?

See, I belatedly realized that spewing walls of equations like this is not actually going to be much use when you're stuck in a rowboat in the middle of the North Atlantic having to navigate by the stars with no cell phone and no GPS. Because, chances are, you also have No Internet, and then my blog entries with their handy tables go to waste.

It would be far better if I can teach you how to derive these relationships instead, i.e., in a way that you might actually be able to vaguely remember while sitting in a boat in the middle of the North Atlantic.

But first I'm going to introduce a bit of gratuitous extra notation. Write

ᶜᵒθ

— pronounce it "co-theta" if you want — to mean (90° − θ). I do this because:

I can,

it's less typing,

it's way less degree vs. radian waffling, which I already do too much of, but also,

you get all of the following useful and amusing equivalences (no, really; read them aloud):

sin ᶜᵒθ = cos θ

cos ᶜᵒθ = sin θ

tan ᶜᵒθ = cot θ

(= 1/tan θ, in case you've forgotten)

cot ᶜᵒθ = tan θ

sec ᶜᵒθ = csc θ

(= 1/sin θ, because nobody ever remembers that one)

csc ᶜᵒθ = sec θ

You'd almost think they planned it this way.

Hopefully, it goes without saying that ᶜᵒ(ᶜᵒθ) = θ, except I had to go and say it, didn't I? (Damn.)

And now let's start with a right triangle, with vertices/angles and sides/lengths labeled a,b,c,A,B, the way you usually see it in trigonometry class, ( Collapse )

Continued from Part 2, exploring the benighted universe where "parallel" is Not a Thing.

How circles work

So, to review the weird things we've seen so far:

When we have a circle radius of 90°, otherwise known as a straight line, and we're traversing the circumference, i.e., measuring the total length along it as we sweep out 360° from the pole in the middle, we get 360° worth of path (phrasing it this way so that if this turns out we're on a projective plane rather than a sphere and what we're really doing is traversing the same 180° path twice, I won't have been lying to you), which, being 4 times the radius, is slightly less than one might have expected (2π being roughly 6.28).

If we attempt a circle of radius of 180°, we stay firmly nailed to the antipode of the center, our circumference traversal goes nowhere and thus we get a circumference of zero.

Meaning if we have to explain to the residents what "π" is, we're going to lose horribly. Best we can do is, "So: Circumference to radius? That's a ratio. It's literally all over the map. But as radius approaches zero, once you're under 90°, you'll notice the ratio is always getting bigger. If you work at it, you can prove that it's bounded and it converges to this weird transcendental number like e. And, no, don't ask us how we came up with this…"

We need to understand better how curved paths work. ( Collapse )

Continued from Part 1, in which we discover at least one consequence to doing away with the concept of "parallel lines". (and yeah, I did a bit of a George Lucas thing here; sorry…)

Let's talk about Area

Having noticed that isoceles right triangles give us a natural way to define/measure distances, we see that we can do area this way as well. That is, the area of ΔAPX is clearly the angle at P times some constant, which we may as well just take to be 1 if we haven't defined a unit of area yet. ( Collapse )

So, as part of my possibly-continuing "Geometry on Drugs" series, here is a prequel to my post on spherical geometry, which was more of a "hey, this is useful" post in which much there's a whole lot you're expected to take on faith. It was really more intended for the hardcore engineering type who needs to see that use case up front.

This version is going back to first principles, where we do the axiom wanking and you (hopefully) get a sense of why things turn out the way they do.

Also, this is the practice run before I launch into the Essence of Hyperbolic Geometry, so, … Onward …

The Geometry Axiom Everybody Hates

Start with this diagram and the inevitable question that comes up:

Start with a line ℓ and a point A not on it. How do you put a line through A that doesn't intersect ℓ?

(In other news, I am now convinced that the Unicode committee contained at least one disgruntled geometry teacher. How else to explain why there's this isolated script ℓ code point?)

We can drop a perpendicular from A meeting ℓ at some point X, and then it's obvious that the line you want (dotted) is the one perpendicular to XA. If you tilt it even slightly away from 90°, then it simply must intersect ℓ somewhere.

Space 9 if you need to brush up on spacetime diagrams and why simultaneity gets screwed up

Today's post is about Hyperbolic Geometry, wherein you learn what those "Warning, Evil, Don't Look" columns are about.
It's now safe to look; well okay, no it isn't, but too late! AHAHAHAHAHAHAHA.

Hyperbolic geometry is basically Geometry On Drugs and we know that's never going lead anywhere good.

To be fair, Spherical Geometry is arguably also on drugs, but at least it's easier to explain in that, having had lots of experience with basketballs and whatnot, you already know what a sphere is. Having a concrete place for the "points" to live, I can then tell you

what "lines" are (great circles, or planes slicing the sphere through the origin / center of the sphere),

how to measure "distance" along a "line" segment (measure angle between endpoints from the center of the sphere),

how to measure "angles" between "lines" (the planes will intersect; there's an angle there; done), and

what "circles" are (they're um, circles, … or, if you like, planes that don't necessarily go through the origin, or cones coming out of the origin; whatever works for you),

and then you're basically good to go, ready to do all of the geometry/trigonometry you could ever want, once you've heeded my warnings that Certain Things Will Be Different (no such thing as "parallel", triangles add up to 180 plus area instead of just 180, do not feed them after midnight, etc…).

Unfortunately, the place where we're Doing Geometry today is this inside-out Hyperboloid Sheet Thing with a fucked up metric, … and if you've actually seen one of those in real life, I will be very surprised, especially since it's not something that can exist in ordinary 3D space. Oddly enough, it will end up relating to something you do have day-to-day experience with, namely (cue reverb and James Earl Jones voice)… Your Future,… but I'm not sure how much help that's going be in visualizing it.

So,... picking up where we left off, I, the intrepid hero, am sailing off to your right into the sunset, with my incredibly reliable gerbil keeping time for me. You, the diligent historian, will eventually reconstruct everything I'm seeing from all of the reports you'll get — from the cloud of NSA bugs that I'm flying through — into a big, happy space-time diagram in which:

My time axis, everything that is happening "Here" according to me, is — as everyone would reasonably expect because I'm moving — slanted away to the right from your own natural, obvious, and vertical notion of "Here",

My space axis, everything that is happening "Now" in my direction of motion, according to me, is, — as nobody expected prior to 1905, — slanted up from your own natural, obvious, and horizontal notion of "Now" and by the same angle,

that second item being what makes all the difference, ruins Galactic Empire stories, and happens to be the only thing you really have to remember about Relativity because it's enough to derive all of the other wacky effects you hear about.

If you're writing your Galactic Empire story and you find yourself needing to say, "Meanwhile, back on Earth,…" that, right there, means you are doing it all wrong.

So, wow, it's been 15 years. And yours truly is Not Actually Illegal in the Russian Federation (at least, not so far as I know and not yet).

But I'm still joining the Exodus to Dreamwidth because, well, fuck every last bit of that noise (also, fuck the ads, and maybe also fuck brad, while I'm at it...).

In other news, I learned enough of the API to fix the cross references in my journal to all point to Dreamwidth so that you can painlessly follow my various n-part series which might possibly be continuing; we'll see. (I may be sufficiently annoyed to change all of cross references on the LJ side as well, whee...)

Now I just need to turn off commenting on the LJ side (just c'mon over — or if you don't want to do the wholesale move, just create an account and claim your LJ OpenID (note that some of you who've already moved here still need to do that...)

(Still waffling on whether this should be Space Travel part 8 or Relativity part 1; we'll see... If you want to go back to part 1, that's here though none of the prior material really matters for this one.)

It's weird to me how everybody knows there's a problem.

How, if you are, say, Issac Asimov or George Lucas or Gene Roddenberry trying to write your Galactic Empire/Federation/Whatever story, you've got to do some kind of handwave about Relativity. We know this because we were taught at some early age that trying to go faster than lightspeed is Somehow Bad, even if nobody ever explains the details.

It's particularly annoying to me because many of the details actually could be explained without going beyond 6th grade math. If you get how right triangles work, that's pretty much all you really need for Special Relativity, except nobody seems to bother trying.

I suppose part of the problem here is the mystique of General Relativity. True story:

Arthur Eddington is at a meeting of Royal Society in 1919 where Sir J.J. Thomson (President) concludes a talk saying nobody's ever really stated in clear language what Relativity is about. The meeting disperses, Ludwig Silberstein (the author of one of the early books on relativity) comes up to Eddington and essentially says, "So you must one of three people in the world who actually get this stuff." Eddington demurs, but Silberstein pushes further, "C'mon, don't be modest," at which point Eddington replies, "Actually, I'm trying to think who #3 is..."

Then again, given the sheer number of people I've encountered who have Misner, Thorne, and Wheeler's Gravitation sitting on their bookshelves, that's clearly not the case anymore.

But if even Albert Fucking Einstein had to take five years off to get up to speed on Differential Geometry — which to some extent is just the Partial Differentiation Chain Rule on Steroids and working through all of the various consequences, but if you didn't get past high school calculus, even that much is going to be a bit hard to take — what hope is there for the rest of us ordinary mortals?

But it so happens that, for Special Relativity, you don't need differential geometry. So let's get started:

Why is the speed of light constant?

How did they even get this idea that the speed of light is always the same? That's the part They never explain. They figure there's no way anybody's going to get that without having the full-ass physics course, so they don't bother trying.

The short answer is that they've done the experiments and that's the way it actually is. Since that's unsatisfying, I'm going to try giving you the half-assed physics course instead: