wrog

Relativity (6 of n): How to Fake Gravity

2026-02-14T08:15:27Z

Special Relativity continued from part 5, in which we learned about translating coordinates, the unit-later hyperbola, and velocity angles

Or you can start from Part 1

Today's topic is constant acceleration, which, in Special Relativity, is much weirder than you might expect. ( Read more... )

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Relativity (5 of n): About coordinate translations and velocity angles

2026-02-11T10:42:46Z

My Special Relativity series continued from part 4.

Today's topic is how to translate Moving People coordinates to Stationary People coordinates or to other Moving People and how it's really like a rotation except where it isn't, and how there's this thing that's like a rotation angle but different.

A Surveying Problem

So I needed a survey of my backyard to find out where everything is. The ISO (Stationary) People who I was originally going to hire were busy that day and also expensive, so I had the Rotated (Stationary) People do it, instead. ( Read more... )

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Relativity (4 of n): The Storrow Drive Theorem

2026-01-16T01:58:19Z

My explanation of Special Relativity, trying to keep it geometric, continued from part 3 wherein I describe the Interval between two events, a kind of "spacetime measurement" that everyone agrees on and that, depending on the trajectory between the events, can represent either a (proper) distance or a (proper) time lapse.

It is time to expand our universe a bit.

A New Dimension

Let's try adding a dimension. We move from the John Hancock Tower to the Flatiron Building (which, if you haven't seen it, is this mostly flat thing standing on end, hence the name). Each floor is a single south-to-north corridor of offices.

( Read more... )

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Relativity (3 of n): The Interval

2026-01-08T08:27:37Z

My explanation of Special Relativity, trying to keep it almost completely geometric, continued from part 2 wherein I describe

what the Moving People spacetime grid looks like when mapped out by the Stationary People,
how all we have left to do is figure out their unit spacing, and
how there can exist Intermediate People who see the Moving and the Stationary People moving in opposite directions at the same speed.

We now take a moment to introduce, out of left field, a new definition that will turn out to be very useful:

Give a pair of events, separated by a distance $Δz$ and elapsed time $Δt$ , the interval between them is defined as the quantity ${Δz}^{2} - {Δt}^{2}$

which looks like a squared distance except for an annoying minus sign,
which seems to depend on whose coordinates we're using (Stationary People vs. Moving People vs. somebody else), but we will now show that:

For any given pair of distinct events, the interval between them is an invariant

( Read more... )

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Relativity (2 of n): Meet the Moving People

2026-01-03T11:19:33Z

My revamped explanation of Special Relativity, trying to keep it completely geometric and not invoking Walls of Math, continued from here where I have now completely beaten to death the concept of "Stationary" and it's time to

Meet the Moving People

Actually, we're going to be particular about who we associate with. We want Moving People who will likewise be able to say that they don't think of themselves as moving, meaning they can't be accelerating or spinning, either. Which leaves having them coast at some constant velocity that is non-zero (because otherwise they'd be stationary and One of Us) and upwards (just to pick a direction).

We also want them moving slower than lightspeed (STL), for Reasons.

( Read more... )

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Relativity redux (1 of n): E pluribus immobiles

2026-01-03T10:52:28Z

(yeah, ok, I don't actually know Latin)

So, here's another go at explaining Special Relativity. I remain annoyed at how few people really get it, even amongst avid SF consumers, entirely too accustomed to generations of SF writers papering over FTL issues with technobabble, 'cause we need that galactic empire, don'tcha know.

Also, it's long past time to dump poorly motivated 1930s pedagogy that real physicists abandoned long ago (e.g.,. "Your mass increases as you go faster", "What? why?", "Fuck you, it just does" [spoiler alert: just No; forget you ever heard that. And if, in 2026, anyone is still trying to teach it that way, someone needs to sit them down for A Talk]).

All we need is basic geometry you knew or could have learned about in 6th grade plus a bit of algebra (up to Pythagorean Theorem). I think I can get by without using a single square-root sign.

But there will be fewer handwaves this time. Because we have to be clear why things have to Not Be The Way You Expected and not leave wiggle room. Here goes:

The speed of light as a constant

Shine a flashlight off of a moving boat. How fast does the light go?

( Read more... )

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Kepler Orbit Cheat Sheet

2023-12-18T11:27:18Z

(why I should not be allowed to write textbooks, part 342)

This is mainly because I wanted to have all of the formulas in one place. Also curious to see how much I can compress the derivation and still have it vaguely make sense. Also I wanted to learn more MathML.

The 2 body problem

We have two bodies with mass. Gravity attracts them to each other. How do they move?

( Read more... )

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Getting Trilorne to Hover

2020-06-12T01:10:39Z

Well okay, I guess there is a way to get Trilorne to hover.

If we're willing to toss the usual definition of "North" (rotation axis points that way), we can put the sun over the equator, and then we can have it be tide-locked. Meaning this planet is basically Mercury but farther away so that only the stuff directly underneath the sun is getting fried to shit.

And maybe having an actual atmosphere will help, too, in various ways, though I can't imagine there not being freaky weather patterns, e.g., some kind of permanent cyclone storm around the solar-maximum point wherever it happens to be at the moment, but that won't damage the story too much because nobody ever goes to the Fire Lands anyway.

The Wall then runs around ( surprise )

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Arthur C. Clarke and the Projective Plane

2020-05-28T04:25:33Z

So there's this Arthur C. Clarke short story, "The Wall of Darkness" (1949). I read it as a kid and found it really haunting. Clarke does Haunting really well.

If you haven't read it already and want to go read it before I completely and totally ruin it, feel free.

(I found the whole thing by searching for "Trilorne" in google, which then gave me a google books hit; we'll see how much longer The Algorithm lets people do that).

But you've already had 70 years, so… onward…

( let the ruining begin... )

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Spherical Geometry 4 — Mommy, Where Does Trigonometry Come From?

2019-03-26T00:27:43Z

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
csc ᶜᵒθ = sec θ	(= 1/cos θ, and no, I don't know why reciprocals get these special names)
sec ᶜᵒθ = csc θ	(= 1/sin θ, because, seriously, WTFF?)

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, ( and then derive stuff about it )

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Spherical Geometry 3 -- A bit of Circular Reasoning

2019-03-24T23:34:39Z

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. ( and so ye shall... )

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Spherical Geometry 2 -- Deficits Matter

2019-03-24T11:56:19Z

Continued from Part 1, in which we discover at least one consequence to doing away with the concept of "parallel lines".

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. ( so let's do that ... )

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The Essence of Spherical Geometry, Part 1

2019-03-10T10:08:41Z

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.

Proof by diagram. We're allowed to do that, right? ( Read more... )

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Public Service Math Lesson: Hyperbolic Trigonometry

2018-07-10T21:45:08Z

a.k.a., Space 11: How to do Interstellar Navigation

Various antecedents you may want to have peered at first:

the Spherical Trigonometry lesson
Space 10 on consequences of Relativity (why clocks slow down, lengths change, and how FTL breaks things), or
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.

( bring on the drugs... )

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towards a truly chromatic Horn

2011-07-15T01:03:55Z

So here's a puzzler

Brief review of the physics of wind instruments

Your standard wind instrument is a pipe attached to some kind of sound source (lip or reed). Being of a certain length it resonates at particular frequencies and suppresses the others, ulimately being capable of producing tones that are multiples of a certain fundamental frequency f, 2f being an octave higher than f, 3f being a fifth above that and so on.

The main things to keep in mind about the multiples of f (the "harmonic series"):

going up by a particular interval means multiplying the frequency (or dividing the wavelength) by a particular ratio (2 ↔ octave, 3/2 ↔ perfect fifth, 5/4 ↔ major third, etc.…), and
intervalwise, the notes are getting closer together the higher you go.

Eventually, the notes get so close to each other that you can do scales of a sort -- this is how the natural (i.e., valveless) French horn works; the tube is so damned long that you're always playing in the 8f-16f range and then you just fudge the few notes that are out of tune with lip and hand-in-the-bell tricks. ... and likewise for why baroque trumpet parts are so insanely high-pitched.

But if you want to do anything useful down in the lower register, you have no choice but to mess with the length of the tube somehow, the two most popular methods being

(the woodwind solution) poke holes in the tube (and cover them with removable keys)
(the brass solution) insert valves or slides in the tube to change its actual length

We'll focus on the second solution since I really have no freaking clue what's going on with woodwinds. Oddly enough, the brass solution is simpler but it didn't come along until fairly late in the game. Not until the 19th century did metalworking technology finally get good enough that Heinrich Stölzel was able to construct the first actual valved instrument in 1814. Once this happened it didn't take long to catch on -- hence the 19th century explosion in the use of brass instruments in orchestral works (cf. Wagner, Mahler...).

So how do valves actually get used?

Let's first get clear what a valve is: ( Yes, we are going deep into the weeds. You knew this was going to happen )

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