#cartesian system

[er-cryptid]

Coordinate Systems

-- Cartesian System = used for linear motion

-- Polar System = used for angles

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[sevenfactorial]

When the professor says the answers aren't meant to be long shortly after you write a full page for half a problem on an exam

[casbitchh]

in case you guys think i know math or anything, i need you to know that two of the tabs i have open right now are a google search for ‘sin(90) degrees’ and the wikipedia page for spherical coordinates

[lady-inkyrius]

Had an idea for a conlang in which instead of pronouns having the standard person/number setup (basically the cartesian product of {1, 2, 3} and {SG, PL}), it uses only the persons of the referents, not caring about how many there are (basically the powerset of {1, 2, 3}).

For example with this you would get a second person pronoun I'll gloss as just 2, but all it means is that the referents are a subset of the audience, independent of whether there are one or many people in that audience. However you'd also get another “second person” pronoun 2.3 which just means the referents include the audience but also people elsewhere (this is definitionally plural).

You'd end up with one third person pronoun (3, can be either plural or singular), two second person pronouns (see above), and four first person pronouns (1 – just singular, 1.3 – exclusive plural, 1.2 – partially inclusive plural, and 1.2.3 – inclusive plural). Additionally you'd also get an empty pronoun ∅, which could be used as an indefinite or impersonal pronoun like “one” or the dummy “it” (perhaps adding an animacy distinction to the system could help here).

It's interesting that the only thing you lose in this system compared to English at least is equivalents to the non-epicene third person singular pronouns “he/she/it”; because both “you” and “they” can refer to singular and plural they map directly to equivalents here. In first and second person you're purely gaining distinctions.

[squareallworthy]

Triangle Tuesday 7: the joy of coordinates, railroads, and you didn't see that coming

"There is no royal road to geometry," Euclid supposedly said to Ptolomey, and this is true. There is, however, a commuter rail line called analytic methods, and that's what I want to talk about today.

Everything I have discussed in this series so far falls under the heading of synthetic geometry -- that is, geometry done only using axioms such as Euclid's, without recourse to numerical formulas or coordinate systems. This is in contrast to analytic geometry, which most people are familiar with from the cartesian coordinate system of the plane.

Today, we're doing a different coordinate system, but the cartesian system provides a familiar jumping-off point. In the cartesian plane, we specify a point relative to two reference axes. The coordinates represent the perpendicular distances from the axes, and these distances are positive or negative depending on which side of the axes the point lies.

We are going to do something like that, but rather than orthogonal axes, we will take the sidelines of a triangle as our reference. This will give us three coordinates for only two dimensions, which seems redundant, but hold on and it will make sense in a bit.

First we'll look at the basics of this system, then I'll go through and find the coordinates for each of the triangle centers we've looked at so far, and we'll end with a bit of a surprise.

Coordinates for triangles

Why do I call this a commuter rail line? Because unlike a royal road, anyone can use it, and it can be a faster way to get to certain places. On the other hand, it doesn't always go exactly where you want, and we have to build some infrastructure before it becomes useful.

Let's work our way there by first adding some notation to our triangle. It's conventional in triangle geometry to use the uppercase letters A, B, and C for both the vertices and the size of the angles at the vertices, and lowercase letters a, b, and c for both the sides opposite the corresponding vertices and also the length of those segments.

This may seem liable to confusion in theory, but in practice context makes clear which sense is intended. If B or b appear in a drawing, they refer to points and lines. If they appear in a numerical expression, they refer to angle sizes or lengths.

Now let's look at a point P. We will label its feet on sides a, b, and c as Pa, Pb, and Pc, and the segments joining P to the feet as p1, p2, and p3. I don't know of any standard term for these segments. I'm just going to go with the obvious and call them legs.

And because we are all comfortable now with using the same symbol for a line segment and the length of that segment, I will also use p1, p2, and p3 as the signed distances from P to the sidelines of the reference triangle. The distance will be positive when P is on the same side of the sideline as the triangle, negative when P is on the other side, and zero when P is on a side line.

Here's P lying outside the reference triangle, with a negative p1 (in red) because it lies on the opposite side of line a from the triangle, a zero p2 because it is on line b, and a positive p3 (in blue) because it is on the same side of line c as the triangle.

Okay, so far so good. This is not too different from cartesian coordinates, except our axes aren't perpendicular and we have a seemingly unnecessary third coordinate. Two signed distances ought to uniquely identify a point, right?

Here's the difference that makes the third coordinate necessary: we are going to ignore absolute distances and only consider ratios between the coordinates. If we have a point with legs measuring p1, p2, and p3, we will consider any three numbers k*p1, k*p2, k*p3 to be the same coordinates (as long as k is not zero). We'll write this triple ratio as p1 : p2 : p3 in order to distinguish these coordinates from triples of cartesian coordinates in R^3.

This coordinate system is called homogeneous trilinear coordinates. (The homogeneous part comes from abandoning absolute distances and considering only ratios. If we kept to absolute distances, we would have exact trilinear coordinates. Hereafter I will just refer to the homogeneous version as trilinear coordinates, or trilinears.)

Why is this useful? Various reasons. One of them is that with one exception, any three real numbers specify a point. No point can lie on the outer side of all three side lines, so with exact coordinates, three negative numbers don't correspond to any point. But with homogeneous coordinates, we can multiply any coordinate triple by any nonzero real number k and it still represents the same point. Letting k = -1, we have x : y : z = -x : -y : -z. No problem with negatives.

But the main advantage of trilinears is that we can give a nice representation to important points of the triangle, and that lets us easily use analytic methods on them. The vertex A, for instance, has zero distance from sidelines b and c, so its trilinears are 1 : 0 : 0 (or equivalently any other nonzero real number for the first coordinate, but using 1 is simplest). Similarly vertex B is 0 : 1 : 0 and vertex C is 0 : 0: 1 . No point can lie on all three lines of the triangle at once, so 0 : 0 : 0 is the single coordinate that does not correspond to any point.

The incenter

Let's see how this works using the incenter as an example. The incenter is inside the triangle and equally distant from the three sides. With the three legs being equal, its trilinear coordinates are as simple as they can be:

The trilinears of the incenter are I = 1 : 1 : 1.

That's a very nice and neat expression, but not actually very informative. We already knew this about the legs of the incenters, because they are all radii of a common circle. Furthermore it's easier to understand "the incenter is the point that is equidistant from the sides" than "the incenter is the point with trilinear coordinates 1 : 1 : 1." So far our coordinate system hasn't shown us anything new or made anything clearer. Well, we're still building our commuter rail line. It will pay off after we complete more ground work.

But while we are at this stop, we might as well take note of the excenters. Here is excenter Ja, equidistant from the sidelines of the triangle, but on the negative side of line a and on the positive sides of lines b and c. So its trilinears are -1 : 1 : 1, and similarly for the other two excenters.

The centroid

The centoid is defined as the intersection of the medians, and the medians are defined as the lines that join the vertices and midpoints of the opposite sides. So let's start by finding the coordinates of the midpoints. The midpoint of side a, Ma, has first coordinate 0, so we just need to figure out the ratio of distances to the other two sides. To do that, we'll first take a look at the altitudes of the triangle.

Let hc be the altitude from C to c, and let △ be the area of triangle ABC. Then by the standard formula for area of a triangle,

△ = 1/2 * c * hc

hc = 2 △ / c.

Now look at the midpoint of side a. By similar triangles, the length of its leg to side c is half the length of hc, or △/c. By the same logic, the leg extending to side b has length △/b. That gives us the ratio we were looking for, so the trilinears are 0 : △/b : △/c. Since this is a ratio, se we can divide through by △ to get 0 : 1/b : 1/c, which is nice and clean.

Repeating this for the other two sides, we get trilinears for the midpoints of the sides as:

Ma = 0 : 1/b : 1/c

Mb = 1/a : 0 : 1/c

Mc = 1/a : 1/b : 0

Now, in general, when we have two points, we can find an equation for the line that runs through them, and given two lines, we can find their point of intersection. The first method uses the fact that three points P, Q, and R with trilinears p1, p2, p3, etc. are colinear when this determinant is zero:

And by letting one of the points be variable x : y : z, we get this determinant

which gives us the general form of an equation for a line:

d x + e y + f z = 0.

Don't know what a determinant is? Don't worry! We don't need to use this method. I'm just mentioning it out of a sense of duty to completeness. In this case we're working with a median, which is a cevian, which makes things very convenient. Recall that a cevian is a line that runs through one of the vertices.

We worked out that for the point Ma, the legs to the b and c sides are in the ratio 1/b : 1/c. By similar triangles, any point P on this cevian will have legs to those sides in the same ratio. In other words, a point on the c-median must have triinears of the form u : 1/b : 1/c for some u.

The centroid G is one such point. It's also on the b-median, so it must also have trilinears of the form 1/a : v : 1/c. It's also on the c-median, so it must also have trilinears of the form 1/a : 1/b: w. So instead of working with the equations for the medians, we can get the trilinears directly from points known to lie on the medians. Therefore,

The trilinears of the centroid are G = 1/a : 1/b : 1/c.

For more completeness, here are equations for the medians:

y/b - z/c = 0

x/a - z/c = 0

x/a - y/b = 0

To find the intersection of two lines m and n with coefficients m1, m2, m3, etc., take this determinant:

The trilinears are then equal to the minor determinants:

If you are familiar with linear algebra and enjoy finding determinants, go ahead and calculate it using the coefficients above and you will get 1/a : 1/b : 1/c. Otherwise, don't worry about it.

The symmedian point

Let's now look at the symmedian point, which is the isogonal conjugate of the centroid. Recall that when P and Q are isogonal conjugates, the cevians of P are the reflections of the cevians of Q across the angle bisectors, and vice versa. In other words, at each vertex the cevians of P and Q make the same angle (in red) with the angle bisector (dashed line). Equivalently, they make the same angle (in blue) with the sides adjacent to the vertex. And that sets us up for our next shortcut:

Theorem: If a point P has trilinear coordinates p1 : p2 : p3, then its isogonal conjugate has coordinates 1/p1 : 1/p2 : 1/p3.

Proof: as the two green triangles are similar and the two blue triangles are similar, we have CP/PPa = CQ/QQb and CP/PPb = CQ/QQa. Then

PPa/QQb = CP/CQ = PPb/QQa

PPa/PPb = QQb/QQa.

Then given a ratio d : e : f, the ratio 1/d : 1/e : 1/f gives the desired reciprocal ratio between any corresponding pair.

With that result, and knowing that the centroid is 1/a : 1/b : 1/c, finding the coordinates of the symmedian point is simple:

The trilinears of the symmedian point are K = a : b : c.

Before moving on, we should take note of two things about our theorem on isogonal conjugates. First, the formula is undefined for any point with a zero coordinate, so points on the sidelines of the triangle don't have isogonal conjugates. Also, the incenter is its own isogonal conjugate, as are the three excenters, and they are the only points with this property.

The orthocenter

To find the coordinates of the orthocenter, let's go back and look at that altitude again.

Let h be the altitude from C to Hc. Draw leg hc1 from Hc to side a. Then the triangles to the right of h are similar and the two angles marked in red are equal. The length of segment hc1 is therefore h * cos B. In the same way, the blue angles are equal and hc2 = h * cos A. Thus the trilinears of Hc are cos B : cos A : 0.

This is an inconvenient form, though, because the first coordinate is in terms of B and the second in terms of A, which is the reverse of what we want. So we'll divide through by the factor cos A * cos B:

cos B / (cos A * cos B) : cos A / (cos A * cos B) : 0

= 1/ cos A : 1/cos B : 0

= sec A : sec B : 0.

That's the c-altitude. By symmetry, the a- and b- altitudes are:

0 : sec B : sec C

sec A : 0 : sec C

And conveniently, altitudes are also cevians, so we can use the same shortcut here that we used for the cetroid, and get the trilinears directly from these points.

The trilinears of the orthocenter are H = sec A : sec B : sec C.

If you want to play around with the determinant formula, you can go ahead and find the equations for the altitudes. I found it a little more convenient to use the forms cos B : cos A : 0, etc..

The circumcenter

Cevians are nice to work with, but not useful for the circumcenter, which isn't defined by cevians. It's defined by perpendiculars, and while there is a formula for perpendicular lines in trilinear coordinates, it's awful and I'd rather not use it. Sometimes the commuter rail line doesn't run where you want and it's easier to go a different way.

So we'll fall back on synthetic methods. The circumcenter O is the intersection of perpendicular bisectors of the sides, and is the center of the circumcircle. In particular, note that the segments OA, OB, and OC are radii of the circumcircle and equal.

Now consider angle ACB. It subtends the segment AB and has measure C. By our old friend the inscribed angle theorem, angle AOB = 2C. Segment OOc, one of the legs of O, bisects AB perpendicularly, so angle AOOc is half of AOB, equal to C.

Now by the definition of cosine,

cos AOOc = OOc/AO = cos C

OOc = AO cos C.

Similarly we can establish lengths of the other legs:

OOa = BO cos A

OOb = CO cos B.

Therefore the trilinears of O are BO cos A : CO cos B : AO cos C. But AO, BO, and CO are all radii of the circumcircle and equal, so

The trilinears of the circumcenter are O = cos A : cos B : cos C.

But look back at the orthocenter! The secant is the reciprocal of the cosine. And what did we just learn about reciprocal coodinates? Doesn't that mean -- yes, yes it does!

The circumcenter and the orthocenter are isogonal conjugates.

They've been conjugating away right under your nose this whole time and you never even noticed!

And this, finally, brings us a result that is much easier to establish with trilinears than with synthetic methods. There probably is a clever synthetic proof that O and H are isogonal conjugates, or even many proofs, but I would have to stare at a bunch of lines for a long time before I could come up with one. Using trilinear coordinates, we can establish this relationship at a glance. And that sort of quick trip that anyone can take is the reason we use the commuter railroad of analytic methods.

If you found this interesting, please try drawing some of this stuff for yourself! You can use a compass and straightedge, or software such as Geogebra, which I used to make all my drawings. You can try it on the web here or download apps to run on your own computer here.

An index of all posts in this series is available here.

[nostalgebraist]

Here are some fun / amusing / potentially-interesting facts about the process of writing and plotting Almost Nowhere, if anyone's curious.

Major spoilers for the whole of Almost Nowhere under the cut.

(There's really no way to spoiler-censor this material without rendering it incomprehensible. If you haven't read the book, do that first before reading this post.)

(1)

A large fraction of the book's eventual plot emerged from my attempts to patch a single, in-some-sense trivial continuity error I made while writing the very first chapter.

The Mooncrash section of that chapter ends with this sentence (emphasis added):

All parties were used to stillness, now, for the Mooncrash was nearly four years old.

And a few paragraphs later, in the opening of the Academy section, we get this (emphasis added again):

For (as everyone knows) the Shroud is upon us and while it tolerates the Academy — as it presently is, as it has been for the last eight years, a chrysalis, preparing itself step by minuscule step [...]

So: The Mooncrash is 4 years old. The Academy crash is at least 8 years old, and indeed older.

Yet the Mooncrash is also as old as the crash system itself! It was made by humans, during the period between the discovery of the anomalings and the mass-crashing of the human race. (This is only shown in the second chapter, but I had it in mind before then.)

How long has the human race been crashed, then? At most 4 years, and at least 8 years? How could that possibly be?

It would have been easy enough to just edit the chapter, but that's not how I do things. Restrictions, famously, breed creativity. I enjoy attempting to solve puzzles I have inadvertently created for myself, and many of my best ideas have been produced through this process.

It would also have been simple and easy to merely say: "OK, I guess time elapses at different subjective rates, in different crashes."

Amusingly, I ended up doing that anyway! But for some reason, this avenue didn't occur to me at first. By the time I started asking myself whether to include this kind of effect, I already had a different solution in mind.

I spent a lot of time beating my head against the figurative wall, trying to resolve the 4-vs-8-year issue. The early parts of my AN notes are full of this stuff.

----

At some early point, I came up with the idea that the anomalings/shades would deal with troublesome crashes by "rebasing" them, rewriting their histories.

I didn't intend, initially, for this idea to take over the plot as much as it eventually did. It was just a fun idea that underscored the huge power differential between the anomalings and their captives, and felt in line with the Cartesian/Wachowskian themes of transcending a "fake"/illusory world, radically doubting one's own perceptions and memories, etc.

But, having stipulated that "rebases" were a thing, I hit upon the idea that they could be used to modify the total quantity of past (subjective) time inside a crash -- turning 8 years into 4, or vice versa, or whatever.

So, I could fix the problem by stipulating that one -- or both -- of the problematic crashes had already been rebased, in this way.

But why? And by whom?

----

Now, at this early stage, I also had the idea in mind that the character "Anne" would eventually escape from her crash, and that she would have a hand in various major events in the story -- including some events that had already occurred, relative to the "present" of the textual PoV.

But I didn't know, yet, what these interventions actually were.

(I put "Anne" in quotes, here, because in the very early stages I casually assumed that only the PoV Anne introduced in Chapter 1 would be a major character, and that her sisters were merely background material for her personal narrative, like the tower itself. Of course, in the process of thinking through the details of things, I realized that this assumption was needless and indeed counterproductive.)

As often happens when I'm plotting a story, I found that two unknowns slotted neatly into one another, each one providing a potential solution to the problem posed by the other.

We need something for "Anne" to do in the past. Something consequential, something that shows off her newfound agency -- but also something that obscures her role from view. Ideally, something kind of weird, esoteric, "advanced"; something that feels buried inside the deep, dark center of the backstory, which the reader will only "excavate" at the end of a long, strange journey.

And we need someone to rebase the Mooncrash.

That answers the "who?" question. But again -- why?

Well, it was already in the plan that Azad would join forces with Michael, when Michael went in search of his lost Anne. That Anne would meet Azad, as a result, and that it would be Azad who persuades her to return to Michael's crash.

I didn't, at the time, have much else planned for the Anne-Azad connection.

As originally conceived, the "Azad convinces Anne to return" scene was about Azad's uncertain loyalties, and about Anne's lack of exposure to other human beings (and to the power of words, as deployed by human beings with access to real human culture). That is, it merely served specific, separate purposes in the sub-stories of these two characters. There was no intent to set up, or develop, a thread connecting these sub-stories, making Azad a major character in Anne's arc and vice versa.

But that seems like kind of a shame, doesn't it? Why go to the trouble of preparing these characters, and bringing them into contact, if I didn't have anything for them to do together?

Anne and Azad.

We need someone to rebase the Mooncrash.

We need Anne to learn about real human culture, somehow, before she leaves. I knew that, already, though I didn't have a mechanism in mind.

(I also knew, by this point, that causing Azad's appointment as translator was another one of "Anne's" consequential moves. I had conceived of this, at first, as a relatively impersonal act, done only for its historical significance. Indeed, that would have been enough -- but the more the merrier, theme/motivation-wise.)

Problems paired up, interlocked, and became each others' solutions.

(1b)

As is obvious from the above, I didn't have the scenario planned out in very much detail when I wrote the first chapter.

At the time, the story had been gestating in my head for a while, but only as a bunch of vague inklings and intentions.

The proximate cause of writing-the-first-chapter was a sudden and unexpected burst of inspiration. I was riding the bus to a social event, and suddenly my mind was awash with crisp, never-before-glimpsed details about Anne and her tower, the Mooncrash, the Academy, Cordelia's blue dress -- all the stuff of Chapter 1. It felt like a crucial message was being beamed into my brain, VALIS-style, from the Muse / Higher Power.

I had an urge to bail on the social event, turn around, ride back home, and start writing immediately -- what if the magic went away, as suddenly as it had arrived? I resisted that urge and made a perfunctory appearance at the event, but then went back home and wrote as much as I could before falling asleep.

So, when I was writing that chapter, stuff like "four years" and "eight years" wasn't based on any single coherent picture, just vibes and vague inklings.

(I think 4 years probably sounded like the right amount of time for G&A to have been in the Mooncrash, character-wise. Meanwhile, Hector's ascension from the Academy had to be long enough ago that there would be no direct overlap between Hector and any of the current students. The "Bad Old Days" had to feel like something you'd only hear about in rumors, or from authority figures who probably weren't telling the full story.)

(2)

Like TNC before it, Almost Nowhere was originally conceived as relatively simple and straightforward story, only to become something much weirder and more complicated as I fleshed out the details.

As I said above, I only had a very vague "plan" at the outset of the writing process. But I kinda knew where I was going with it, in very broad strokes.

The original arc, insofar as it existed at all, was something like:

The bilateral / anomaling tension is introduced.

The bilateral PoV characters come to an understanding of their situation.

Many of the bilateral PoV characters join up with Hector Stein, who is already trying to defeat the anomalings and free humanity from the crashes.

Azad temporarily sides with the anomalings, and Anne temporarily returns to her captive state. But both them "come around" eventually.

Anne eventually triumphs over Michael, delivers a dramatic monologue castigating him for imprisoning her (etc.), and mounts a successful escape.

Shortly after Anne's escape, some (TBD!) resolution to the main conflict is achieved. Whatever it is, it is proposed/spearheaded by the bilateral faction (and specifically Anne herself), and it somehow exemplifies "the bilateral way of thinking/being."

The humbled anomalings conclude that "the bilateral way of thinking/being" has its advantages, both practically and morally.

So the story, as originally conceived, was much more straightforwardly about the "good" PoV humans fighting back against aliens.

It unabashedly took the bilateral side in the conflict, and it ended with a "beauty of our weapons" sort of moment in which the bilaterals are both victorious and righteous, and in which these two kinds of success are closely linked and almost merged.

I have to imagine that, even in counterfactual worlds where some things went differently, I never would have stuck to this version of the story all the way through.

Because, one way or the other, I would have eventually realized that.. like... this version of the story kind of sucks, right?

I mean, why go to the trouble of introducing these aliens, and trying to make them interesting, only to say "nah, actually these guys were just wrong, it's us and our existing 'ordinary' pre-conceptions that are right, and that's what the story was about all along"?

It would have been "inventing a guy to be mad at," as the saying goes.

Not a great foundation for a story. And the least interesting possible direction to go in, given this kind of setup.

It also presents a seemingly unresolvable tension, for the writer, about how to portray the distinctively "bilateral" nature of the bilateral side in the conflict.

If "bilateral" is as broad a category as the anomalings say it is -- if you and I and all of us, whatever other qualities we possess, participate equally in this sin -- then it's hard to strike a note of emotional triumph around the quality of "bilaterality" that doesn't feel wrong, vacuous, or bloodlessly abstract.

"Woo, yeah, humans are great!" I mean, are they? All of them? You don't get to say "well, only the good ones," here, or "in their ideals if not always their acts," or anything like that. Everyone is included in the relevant category, except for the guys-who-aren't that were invented for this specific story.

It's difficult to make this land properly, in the same way it would be difficult to write a story that inspires "carbon-based life pride" or "having-DNA pride" or the like in its reader.

So this version of the story was dead on arrival. And indeed, by the time I was thinking through the stuff chronicled in (1) above, this version of the story felt like a provisional placeholder, at best, in my mind.

Nonetheless, there are various echoes of it in the story I eventually landed on.

For example, in the original version of "Anne's" escape -- conceived in a much more straightforwardly positive way -- I had Anne reading "real" books in secret, drawing moral strength from them, and then including a bunch of literary quotes in her big dramatic monologue to Michael. (I took inspiration, here, from John the Savage reading Shakespeare in Brave New World.)

And I had the idea that "Anne," being an autodidact, would read omnivorously without making culture-bound distinctions familiar to you and me; that her selection of quotes, in the monologue, would put low culture alongside high culture, infamous books alongside famous ones, etc.; and as a particular case, that it'd be fun if -- before going on to quote Shakespeare and co. -- she began the whole thing by quoting Ayn Rand.

And that one idea stuck, even if the rest of it didn't.

(Or, consider how the idea of "a powerful move in the conflict that exemplifies the bilateral way of thinking/being" actually crops up multiple times in the finished story, right up to its last scenes. One can see traces of it in the "trick" that obsesses Michael, in the use of autobiographical writing to build up nostalgium, and in Annabel's improved crash design.)

(3)

I came up with the Mirzakhani Mechanism relatively late, in between writing Chapter 13 and writing Chapters 14-15 (in which the MM is introduced).

The MM was a product of looking back at the sci-fi elements that already existed in the story, like crashes and rebases, and trying to invent some single underlying explanation that covered all of them in a relatively parsimonious way.

This basically "worked," I think -- it certainly worked better than I had been expecting, after playing the dangerous game of "write a bunch of weird stuff and hope you'll be able to explain it all later." (I remember talking to one reader who was shocked that I hadn't had the MM in mind from the very beginning, which was flattering.)

It also had unintended consequences that kinda took over the story, but largely in a good way.

Earlier, I had planned to have the post-rebase crash timelines "screened off" from the outside world somehow, so that rebasing a crash wouldn't mess up the timeline of the outside world. But, once I'd fixed the idea that "rebasing is an MM event" in place, I realized that this wasn't consistent with the way MM events were meant to work. Instead, the exposition in Ch. 15 directly implies the stuff about rebases that Grant realizes much later in Ch. 41.

Once I'd noticed this, it was obvious that it was extremely important, and I re-incorporated it into the broader plot.

On a related note, I eventually decided that the account of the anomalings "going backward in time to our era" in Ch. 15 didn't really make sense. This meant I needed a different, more viable way anomalings and bilaterals to exist at the same point in time.

This line of thought, along with several others (like "what happened to all the nonhuman organisms?" and "which parts of the MM multiverse are real?"), eventually led me to invent Everywhere-Heaven and the beasts.

That happened right at the start of 2022, between Chapters 21 and 22.

It quickly became clear that the E-H/beasts stuff could be put to a lot of valuable use in story's third act, which was largely a worrying blank space in my head (even at this point!). From thereon out, I worked on fleshing out the third act behind the scenes while writing the second.

Not coincidentally, Chapter 22 contains a ton of E-H-related foreshadowing, and also some hints that human scientists (like Aidan in Ch. 15) had never fully understood the anomalings.

The use of Maryam Mirzakhani, a real (and recently deceased) mathematician, was a weird choice and arguably one in poor taste. All I can really say in defense of it is that it came to me suddenly, and had a number of properties that fit the vibe of the part of the story in which it appeared, and I have a policy of "going with my gut" when it suggests such things to me.

I felt similarly about this choice and another thing introduced in Ch. 15, the nuclear attack intended to kill scientists. Both of these things underscored the fact that the story took place in an alternate reality. And both felt sort of "edgy," "too dark," "too close to the real world" compared to the tone of the story so far. But I wanted to take the story to new places in the coming acts -- "darker," "more real" places -- and something felt right about introducing these elements at this exact point, as signposts providing an indication of where things were headed.

(4)

The phrase "NOWHERE TO HIDE" was originally "NO MERCY," in my notes.

And the abbreviation "NM" for "NO MERCY" was used throughout my notes for Nowhere-To-Hide related stuff, e.g. "NM Annes."

This wasn't the product of much thought, just the first thing that came to mind that had roughly the correct vibe. I almost immediately concluded that I'd have to replace "NO MERCY" with something else in the work itself, since it would seem like an Undertale reference that I didn't intend to make.

"Moon" was originally just a placeholder name -- a shorthand for "the 'NM Anne' who rebased the Mooncrash." But I liked the idea of actually using it, once it had occurred to me.

The corresponding placeholder name for A11 was "Ling," as in "linguist" (but also an actual name).

(5)

I went through 3 different outlines of the third act.

Really, there was a first outline, which was really bad, and then there were two slightly-different versions of a very different outline that mostly corresponds to the finished draft.

The first, bad outline was amusingly titled "notes-satisfying-ending.txt", because I explicitly used this post about "satisfying endings" as a guideline while writing it.

(To be clear, I don't think the linked post was to blame for the badness of that first outline. I didn't ultimately find the post very helpful as writing advice, but the "satisfying ending" outline wasn't even a "satisfying ending" in the post's own terms, and was also bad in unrelated ways.)

I don't want to go into much detail about the bad outline. It was really bad, and also really different from what eventually occurred. It's honestly a pretty embarrassing document.

A lot of the key ideas were there (E-H, etc.), and the very end of the story was roughly the same. But it had a ton of needless flaws that I later corrected. Various existing character arcs and motivations were dropped and never picked up, or suddenly diverted in some new and unfruitful direction; way too much time was spent on getting characters and objects from point A to point B, or otherwise sort of rambling about in a way that didn't matter in the end; it included a lot of whimsical "fun ideas" that weren't necessary and would have added clutter to an already very full canvas; etc.

I never got to the point of building a chapter-by-chapter version of this outline, but I'm sure it would have much longer than the existing third act, also.

The existing third act is pretty long, but it was actually the result of an aggressive pruning and tightening process.

If the "satisfying-ending" outline had a single greatest flaw, it was terrible pacing. Lots of slack, lots of empty space, and when big things did happen, they came out of nowhere, not really prompted by what came immediately before them.

The next draft of the ending resulted from taking the raw materials of "satisfying-ending," purging all the dross, re-thinking all the obviously flawed stuff, and then trying to rearrange the pieces in front of me in a way that was maximally "tight" and interconnected, with questions and tensions introduced and then resolved in a rapid-fire manner, and without any major thread "sitting around in the background" long enough to feel stale, or get forgotten.

That outline was in a file called "notes-good-end.txt."

Much later, I tightened up the plan even further, merging some things that were originally in separate chapters. This was in a file called "notes-true-end.txt", and -- true to its name -- was the version reflected in the book itself.

So there was "satisfying-ending," which sucked; "good-end," which was good; and "true-end," which was slightly better.

(I realize the multiplicity of the ending, and the account of deliberate "tightening" etc., is in apparent tension with my recent account of working by direct inspiration.

There are a few things I can say about this tension.

For one, it really is true that the third act of AN was more deliberately reasoned-out, and less directly-inspired, than some of the earlier stuff. This is kind of inevitable: you don't get to do anything after an ending, that's what an ending is, and so you have to deliberately try to make the final act of a story fully work as a thing unto itself, rather than writing checks in the hope of cashing them at some later point.

And separately, I do think the final version of the ending feels "more real," "more true to the work" than the satisfying-ending draft.

I think I was aware, even while composing "satisfying-ending," that it felt off and wrong in some ways. But it was only after going through the exercise of creating a complete ending -- some sort of complete ending -- that I was able to look back and say "OK, this fits, but this doesn't fit," and distill something that actually felt right.)

[omegaphilosophia]

The Philosophy of Consciousness, Subconsciousness, and Unconsciousness

The study of consciousness, subconsciousness, and unconsciousness is central to philosophy, psychology, and neuroscience. Philosophers and scientists have long debated the nature of the mind, self-awareness, and the layers of mental activity that influence behavior, perception, and cognition. Here's an overview of the three concepts:

1. Consciousness

Definition: Consciousness refers to the state of being aware of and able to think about one’s environment, existence, thoughts, and sensations. It is the subjective experience of the mind, or what is often called "phenomenal experience"—what it feels like to be you at any given moment.

Philosophical Theories:

Dualism (René Descartes): Descartes famously proposed that the mind and body are two fundamentally different substances. According to Cartesian dualism, the mind is immaterial, and consciousness is a non-physical property of the mind. The body, on the other hand, operates like a machine.

Materialism/Physicalism: Materialists argue that consciousness arises from the brain's physical processes. According to this view, consciousness is a product of neuronal activity, and there is no separate, immaterial mind. Contemporary neuroscientific approaches align with this view, seeking to explain how brain activity correlates with conscious experience.

Phenomenology (Edmund Husserl, Maurice Merleau-Ponty): Phenomenologists focus on the first-person experience of consciousness. For them, consciousness is always consciousness "of" something (intentionality), and they explore how the mind structures experience.

Hard Problem of Consciousness (David Chalmers): Chalmers distinguishes between the "easy" problems of consciousness (understanding brain functions) and the hard problem, which is explaining why and how physical processes in the brain give rise to subjective experiences, such as the sensation of color or pain.

Panpsychism: This is the view that consciousness is a fundamental and ubiquitous feature of the universe, meaning that all matter has some degree of conscious experience, not just humans or animals.

2. Subconsciousness

Definition: The subconscious refers to mental processes that occur just below the level of conscious awareness. These processes influence thoughts, behaviors, and perceptions without being actively noticed by the individual.

Philosophical Perspectives:

Freudian Subconscious: Sigmund Freud introduced the concept of the subconscious (often used interchangeably with "preconscious" and "unconscious" in his early work). For Freud, the subconscious includes thoughts and desires that are not currently in conscious awareness but can become conscious when triggered (e.g., through memory or slips of the tongue).

Dual-Process Theories: Modern cognitive psychology divides thought into two systems: System 1 (fast, automatic, subconscious thinking) and System 2 (slow, deliberate, conscious thinking). Subconsciousness is often associated with System 1, where many decisions and impressions are made without conscious deliberation.

Carl Jung’s Collective Subconscious: Jung expanded on Freud's idea of the subconscious with the collective unconscious, a layer of the unconscious mind shared by all humans, filled with archetypes and universal symbols.

3. Unconsciousness

Definition: The unconscious refers to mental processes, desires, and memories that are entirely outside of conscious awareness and typically inaccessible to introspection. In psychological theory, the unconscious is thought to hold repressed feelings, unresolved conflicts, and primitive desires.

Philosophical and Psychological Perspectives:

Freudian Unconscious: Freud proposed that the unconscious mind is a repository for desires, fears, and memories that are too painful or socially unacceptable to acknowledge consciously. These repressed elements of the mind influence behavior in subtle and sometimes disruptive ways.

Id, Ego, and Superego: In Freud's structural model of the psyche, the id represents unconscious primal desires, the ego navigates reality, and the superego represents moral standards. The unconscious mind contains both the id and parts of the superego.

Jungian Unconscious: For Carl Jung, the unconscious mind is divided into two parts: the personal unconscious, which is unique to the individual, and the collective unconscious, a shared repository of human experience. The collective unconscious holds archetypes, symbols, and motifs that recur across cultures and history.

Philosophical Issues with the Unconscious: Some philosophers question whether it makes sense to speak of unconscious mental states. If a thought or desire is not accessible to conscious awareness, can it truly be said to be "mental"? This challenges traditional notions of mind and cognition.

Key Questions in the Philosophy of Consciousness, Subconsciousness, and Unconsciousness:

What Is the Nature of Conscious Experience? Philosophers debate whether consciousness can be fully explained through physical processes or whether something irreducible remains. The hard problem of consciousness remains one of the most pressing and unsolved issues in philosophy.

To What Extent Do Subconscious and Unconscious Processes Influence Behavior? How much of our decisions and perceptions are shaped by thoughts and feelings outside of our awareness? Psychological experiments have demonstrated that subconscious cues can powerfully affect behavior, challenging the belief in fully rational decision-making.

Is the Unconscious Real? Philosophical skepticism exists about whether unconscious thoughts and desires are truly "thoughts" if they cannot be directly experienced or known. Others argue that the unconscious is a necessary concept for understanding repressed feelings and psychological disorders.

Relationship Between the Three:

Consciousness represents active awareness, decision-making, and self-reflection.

Subconsciousness includes processes just below the level of awareness, such as habits, reflexes, or memories that can be brought into consciousness.

Unconsciousness involves deeper, hidden aspects of the mind, inaccessible to conscious introspection but influential in shaping desires, emotions, and behaviors.

The philosophy of consciousness explores self-awareness, subjectivity, and the mind-body problem. Subconsciousness refers to mental processes that influence behavior outside of immediate awareness. Unconsciousness deals with repressed desires and memories that operate beyond conscious thought. Each concept has rich philosophical implications for understanding the mind, free will, identity, and the nature of human experience.

[chronotsr]

No. 5 - D2, Shrine of the Kuo-Toa (August 1978)

Author(s): Gary Gygax Artist(s): David C. Sutherland III (Cover), David A. Trampier Level range: Average of 10, preferrably party size 7+ players Theme: Underground exploration Major re-releases: D1-2 Descent into the Depths of the Earth, GDQ1-7 Queen of the Spiders

I'm almost speechless. This is the most 1e module cover to ever have 1e'd. It is perfection. The way the combat is perfectly perpendicular to the step pyramid. The bondage gear fishman who has a complete fishhead so you 100% understand he's a fishman. Lobster mommy saluting the troops. It's just….it's what dreams are made of.

So I'm already in love with this module, deeply and irrationally in love with it, before breaking the cover. If you're BORING you might prefer the later Jim Roslof cover art that's got lame things like technical proficiency. Ugh. The shit I have to put up with.

Anyway, there's a lot to talk about with D2! It's a lot of firsts for an official TSR product, and critically it's a lot of GOOD firsts.

It's the debut of the Kuo-Toa, one of the most fun groups of people in D&D! It's the first module that doesn't presume the enemy will be inherently aggressive! It's got a lot of negotiation and learning! The only good type of gnomes debuts with the Svirfneblin! This model of "alien settlement where you are not instantly attacked but you gotta learn the social rules and play along" is just the best. This will be done again in U2 and I adore U2. Yeah it's how it feels to go to a different country, especially one that doesn't speak your language, and just have everything be a little "off" compared to what you're used to, but. To me, it will always be The Autistic Experience. How well and quickly can you learn these bizarro social rules you can't intuit and what's the fewest number of whacks to the head it takes to get there? How long can you swallow your complaints when you see stuff that's obviously cruel, but the people around you don't perceive it as cruel anymore because it's The Way Things Are and they will actively defend the cruelty of it?

Ok, ok, back to your regularly scheduled program.

Gary starts off this week's festivities by telling you to be toxic to your players:

Sometimes it feels like there's three Garys in a trenchcoat and they take turns writing the modules.

So D2 starts in the cave at the immediate end of D1 and, let me derail already by saying that I really, really hate old-style hex maps. I cannot follow them -- I don't mean I don't understand how you're supposed to follow them, I mean it's nearly impossible for me to follow the diagonal to the destination. Your coordinate here is R20. Here is your map. Follow the 20 axis diagonally upward and rightward until you intersect with the R row. Can you do it?

Personally, I can't. My eye cannot follow that straight line, it will get lost in the mix of blank identical hexes and occasional interest objects. I sat here trying to follow it for 5 minutes and I couldn't do it. I need a straightedge to do it. The correct answer is that if you follow the light blue area from the bottom right towards the top left, it's the hex up and left of the fourth fully black hex you run into -- the leftmost of the two touching black hexes. I tested this against a few guinea pigs and no-one else could mange it either. Later we will admit defeat and that this axial coordinate system for hexmaps is, uh, really fucking bad, and replace it with offset coordinates (or even better, double coordinates) which more closely resemble normal cartesian coordinates, and by extension are not Eye Strain Central. They have the downside of different eyestrain (tiny font) and that you literally cannot fit as many hexes on the page, but the point of a graphic is to communicate information and the axial coordinate hexmap is bad at that unless you're playing on a huge table with like, two DM screens.

Yes this rant should've gone in D1, mea culpa. In my defense, D1-2 is, basically one module in two parts, they're not really separable.

Here's the coordinate lined out for you, since I imagine many of you have the same issue:

So, now that I have a headache trying to read, we can get to the actual text of the adventure again. Now keep in mind that max movement rate is 1 hex per 1 inch of movement for the slowest member of the party (so like, your guy wearing platemail has 60ft of movement, 10ft to the inch: 6 hexes per day). This means you could hypothetically arrive at the final location as quickly as 22/6=4 days of gameplay, 3 if no one including hirelings wore plate. That is, if you beelined to D2 by sheer luck, never got lost, never got distracted, never got slowed down, never had to take a rest day. Which is good because the food in The Depths seems questionable.

The first segment of the adventure is mostly reprinted from D1 -- random tables and maps and the like. We do get the addition of everyone's favorite early DND trope: a slavery table! And also happilly we get some goopy guys to move your eyes away from that shit:

Which, is a lot more my speed. More goopy guys. It's a roper, actually, although I frankly didn't recognize it. It looks more like the monster from Dexter's Lab? Apparently Ropers have changed a lot in the last 50 years.

So it's all random tables teasing that we're going to end up arriving at a shrine soon. There is a special entry in the back for the new Kuo-Toa and Svirfneblin, and oddly the Svirfneblin don't get a header? We don't learn much. We know that they're natural elemental summoners, that they're "natural fighters", and that they live at some unstated cave somewhere. They like their stun gas darts, they "communicate with racial empathy" (which I guess means body language?) outside their own domains, deep gnomish at home, and underworld cant when they're trading, plus earth elemental-ese. So they learn a lot as kids. They love them some traps, too, basically they're the gnomish Rambos and I love them for it.

Meanwhile, our titular Kuo-Toa get a pretty standard write-up. Driven underground, human sacrifice, raiders, like their war parties. Their priests like their mancatchers, which are based on lobster claws, they spawn in pools, they can spontaneously generate lightning by holding hands (???), are too slippery to grab, can see both infrared AND ultraviolent, can see you moving through basically any magical means, immune to poison, paralysis, charming, sleep, and are resistant to magic missile and lightning. This is, very very weird. They are wildly powerful compared to their later versions, and the only upshot is that they're readily blinded by light spells. Apparently they go insane with such regularity that they have a dedicated social role to controlling or killing the crazed? Yeah these people are a piece of work.

We get a little setpiece moment here where, essentially, there's a rogue kuo-toa who will offer you a trip across the river for 10g. He only speaks kuo-toa and he'll sicc his giant fish on you if you don't say yes fast enough. In fact, a lot of ink is spilled on this little moment, which in all likelihood will be a brief conversation and some passing of money.

Before you get into the shrine proper, some svirfneblin offer to help you in the shrine if you go halfsies on treasure (with almost that exact wordchoice).

Finally, we end up in the shrine proper, which is keyed so let us enter Keyed Mode ™️

The whole area is lit by glow-in-the-dark lichens, which is a spooky way to reveal the lobster lady idol up on the pyramid

While the party can choose to politely integrate into the crowd and play along, there's lots of little things to harass them into nonconformity. Leeches, horrifying offerings, offerings of increasing amount, having to correctly pronounce nonsense names (Blibdoolpoolp????????), holding a live lobster, it's a good bit.

You can, in fact, visit the goddess, who will give you a boon (if you give an offering) or a geas (if you don't), which also grants you kuo-toa speech and also a mark of loyalty, which is neat. You can also encounter her if you fuck around in the prince's treasure room, so the odds of meeting her are actually pretty good! Note that this is pre-"Kuo-Toa believe their gods into existence" so in this case they are worshipping a (hypothetically) permanent, naturally-occurring deity. Being that this is 1e and she is a she, she is Extremely Naked. She is later called The Mother of Lusts, which is one hell of a title.

If you fail to get the priest-prince when you meet him, he actually has a pretty rock-solid escape plan and will come back with an army. So, probably whack him if possible. I really like when antagonists have the sense to piss off and come back armed, rather than pridefully stand and die. You get the sense that Va-Guulgh is priest-prince because he plans contingencies like this, whereas other Kuo-Toa simply vibe. That being said, the Kuo-Toa are apparently not equipped for a search, so it's pretty easy to ditch them.

Sigh.

We do not have a dramatic declaration of THE END anymore, which is a terrible shame. We instead get a more reasonable "This is the end of the section."

The magic of D2 is more in the play and less in the overview. Like, look at this map:

This is a pretty naturalistic map. This is just how you'd arrange a major structure, rather than the kind of nonsense layouts you see in a lot of early dungeons. I don't put much stock in "Gygaxian Naturalism", I think Gary presented pretty intensely game-y spaces and they only seemed naturalistic by 1970s published product standards, but nonetheless he was paving the way compared to some of the silliness you got in pre-G1 modules. This map is good, I think, in that it becomes super extremely obvious to the players from the moment you enter that they extremely do not want to provoke a full alarm -- this is a shrine where you want to kill as few Kuo-Toa as you can, and as many of those as you can behind closed-doors -- it's time to straight up bail if the alarm goes off because you are not beating the hundreds of guys here if you you provoke them up front.

We end with some rust monster art, my favorite monster that I never use because I think I'd get shanked if I did. See you next time in D3!

[beyond-mogai-pride-flags]

Statqueer Pride Flag

Estatoqueer or statoqueer (abbreviated to statqueer): queer statism. It's the counterpart of queer anarchism (anarqueer).

It's based on this flag template and anarchaqueer flag. Any spectrum is welcome to use it, from economically centrists to dichotomously extremists to moderates and beyond the political compass Cartesian coordinate system (such as in multidimensional spectrums and constellations).

Not to be confused with queer nationalism or queer totalitarianism. See also the midpoints: minarqueer/minqueer and grey-anarchoqueer. Inclusive of any -cracy and fluid viewpoints, use it as vaguely as you want.

[utilitycaster]

Bor'Dor/Prism is very fun but also, given that they are representing two ideological extremes of this party's absolutely fascinating moral cartesian coordinate system, I think we should look at the possibilities of Deni$e/Prism.

[hyperlexichypatia]

Y'know, one thing I honestly struggle a lot with as a disabled person is like... with certain things I experience, where do I draw the line between "there is a very literal physical set of phenomena occurring in my neurological and related bodily systems that is responsible for my experience of these physical and emotional/mental events" and "these events are being influenced or caused by external stimuli, often related to class dynamics and oppression" and even "said class dynamics and oppression are responsible for much of said physical phenonena within my bodily systems, as they are simply an interaction/reaction to said external stimuli".

To me, it's like the nuance between how "choice feminism" is used to shut down very genuine real critiques of misogyny and patriarchal beauty standards (which are so deeply rooted in racism and white supremacism), but have also been misused to ironically deny women any autonomy whatsoever, rather than striking a balance with "these systems are fucked, much of this stems from a place of coercion and misogyny, and not acknowledging that can feed into it, but as a marginalized class within that system women (and gender minorities in general) do still have the right to decide how they want to respond to that and informed consent still does matter" if that makes sense?

I feel like I'm not able to exactly address the root of the issue for myself with disability but I think it has some to do with the social model of disability, some to do with the way cartesian dualism has been used to divide the disabled community, and some to do with how I am both anti-psych and anti-phys (the institutions including the biased science as it currently exists, not say, medication or treatment).

It's like... okay I tend to look at my own disability through a mixed medical-social lens. In a "perfect" world, neither medical knowledge nor treatment would be gatekept. I'd be able to get medication for things like what are now labeled ADHD, POTS, and MCAS, including getting compounded medication without having to have an official diagnosis of MCAS (seriously, who does it hurt other than insurance company bottom lines to just make a med without corn or milk sugars/proteins or dyes as filler ingredients -_-).

I would still, however, be disabled. I do understand that that is covered under the social model's definition of "impairment", but I also take issue with the relabeling of disability. It seems oddly euphemistic in the same way that "differently abled" does - defining disability itself as only the social access barriers that cause people to be unable to live a satisfying, fulfilling life including with "impairments", to me ignores the reality of those of us with more severe symptoms and higher support needs.

As I saw someone say so well, "chronic pain is still gonna hurt." But it's also that the pain itself is still going to significantly impact my quality of life, even with full access to treatment and meds. My symptoms are still a physical reality within my body, causing distress and dysfunction, and disabling my ability to engage in certain activities - not just "impairing" said ability. It still would have a significant negative affect in my quality of life, outside of my control. And here I am referring to activities in again, a "perfect" world, where the only reason to do them is out of pure, uncoerced and uninfluenced desire.

There's also the way that rather than actually depathologizing what we label as mental illness, its ability to be profoundly disabling and its very nature as occurring physically for whatever reason has been ignored in order to be neuroableist and sanist against neurodisabled people.

I am firmly for self-labeling, and firmly against the forced labeling of any trait as "abnormal" and "unhealthy". I do think even for self-labeling, it's important to question the premise behind many labels and explore more deeply what you are actually using said labels to mean.

I have talked at length about what "demedicalizing DID", as an example, actually entails, and how it actually increases access to resources and treatment for those that want to pursue those things.

The labels themselves are social, even if they are categorizations for material experiences in many cases. The line gets blurrier with psychiatric labels, as the experience is essentially an internal abstraction of physical phenomena, and the categories themselves are significantly more arbitrary without that solidly material basis.

"Trauma" is used as a label to essentially put the onus of class oppression on marginalized people. Things like "dopamine" and "seratonin" at this point are little more than neurochemically-named horoscopes, in a discipline I already refer to as "the astrology to neurology's astronomy" - and noting here, that neurology and medicine in general are still themselves more partially social than the significantly more mathematical discipline of physics I compare them to.

I also have a reactivity to things that I perceive (or misperceive) as divorcing "mental illness" or "neurodivergence" from any kind of physical basis. It's why I think I was initially confused about your use of "pathologization" - my own bias causing me to struggle to see the difference between what ultimately are very clearly different understandings.

It was, "We should question the categorization of certain experiences as innately pathological, meaning abnormal, unhealthy, and to be suppressed," versus "Mental illnesses are diseases of an abstracted mindsoul, with no physical basis, and can therefore be 'overcome' with a minimum of effort and can never be profoundly or physically disabling".

Which, to be clear, I didn't at all think you were saying the latter. More that I struggled to recognize the former because I had not yet divorced "physical experience" from "specific pathologizing label" in my head, and ironically seeing it laid out so clearly threw such a wrench in that existing perception that I had to go in and decouple/detangle the two to get things going again.

Really, though, it seems it's once again about informed consent in a society where "choice" is so deeply and insidiously influenced by prevailing hegemonic attitudes. It's "if you're going to label yourself disordered, it is still good to question the very premise of that label".

In a world with no access barriers and oppression, I would still have some of the same physical experiences I have now. With unrestricted access to medical treatments that directly interface and alter my biochemical processes, I would still likely not be without what we now label as "symptoms". Certainly, assuming I grew up in that world, "trauma" might very well be an unrecognizable concept as to what it is considered in our reality - assuming it even existed in any meaningful way at all.

I guess it's just - where is that balance between acknowledging the extreme influence of current societal norms and ideas about disability, the way the very language we use to talk about them is steeped in those biases, misconceptions, and assumptions, and the way that a physical result is treated as the cause itself; with the existence of varied experiences of abstracted neurological phenomena, having/creating language some need to help define and understand ourselves, and those societal causes still engendering a physical result?

Is it in the connotation? Is destigmatizing the concept of disability and "disorder" (as meaning "causing distress and/or dysfunction as defined by the person experiencing it") and stopping their misapplication enough? I admit, when my disabled identity has been repeatedly denied by ableists, my instinct is to cling to the labels that say "yes, this is an experience that makes me not able to achieve my own personal desires and goals and causes me distress".

While the concept of "ab/normalcy" is deeply unhelpful and often harmful, there are times when I at least want to say "my material experience is not the same as yours despite your insistence that it is" to people who identify as abled, who have described their perception of my experience as something oppositional to my actual experience. Not abnormal, but not identical, either.

Maybe that's getting off in the weeds. I guess just... at what point does acknowledging my own experiences as a significantly overlapped venn diagram of innate physical and purely societal causes meeting at mixed causes and societal causes of physical results, cross the line into mislabeling societal forces as innate physical events.

Is it just divorcing them from the greater context of society? It is the reversal of causality? Is it the lack of acknowledgement that the way we label these experiences is inherently tied up in the social environment surrounding them? If I view some of my experiences as entirely unrelated to and uninfluenced by that social environment (at least to the extent that is possible), while still being a natural variation in neurology (or physiology) that is itself neutral overall regardless of how I experience it, is that enough?

Is viewing the things labeled as ongoing "trauma" and "mental illness" as a natural and rational response to class oppression a factor in depathologization, as long as we also acknowledge that the labels of "trauma" and "mental health" themselves carry meaning and bias and connotations that don't uphold or even contradict that belief?

I dunno. Maybe I'm just stuck in stubbornly not wanting to give up labels I've been forced to fight for because I'm scared to admit that I could have been fighting for not having them at all while still having my needs met in the first place. Maybe I am just struggling as someone who can't actually process or understand their own experiences without language to integrate a new understanding of that language into my paradigm. Maybe some parts of us are still reactively misinterpreting "question and examine how the language we use is a social construct and how that has been wielded against marginalized identities, including your own" as some sort of threat to our autonomy and in particular self-determination.

I may very well be afraid of and biased by those things, but the one thing I'm not scared of is examining that and admitting it's a possibility.

I guess I'm sending you this ask in particular precisely because you've articulated precise analyses of these subjects so clearly. I'd love to hear your thoughts, if you're willing.

Oh, this is the ask I thought I'd lost!

So, before I start, my own bodymindbrain is VERY compromised by COVID right now. I am living that physical illness affecting cognition life.

This is a great question and I'm really struggling to come up with a thoughtful response, because so much of it, I just don't know. I have no idea "how disabling" any of my disabilities would be if I lived in a society that accepted and accommodated differences, because I've never lived in a society like that and I can only vaguely imagine it. I know that part of that goal is making it so that the supports we receive (medical, social, or otherwise) aren't contingent on any particular label or any particular concept of "disorder." And also that they're never an excuse to infringe on someone's autonomy.

On the language of it all, I'm always struggling to refine my own use of language, but it's especially frustrating because any potentially radical/liberatory use of language gets co-opted and appropriated by pathologization (like "neurodiversity," "Mad," or "anti-psychiatry"). I've left so many groups that I thought were about rejecting pathologization that turned out to be about "recovery" or "healing" (which is fine for people who are interested in those things! I'm just not one of them!).

I also struggle with my own... parts of my mind/emotions that are distressing to me... and I struggle to find words for that that aren't part of the pathology paradigm like "mental health," and also aren't spiritual because that's not what I believe. I just don't know the words. I know that I have anxiety attacks and it sucks and I hate it, but I don't know good overarching terms for "The experience of having profoundly unpleasant unwanted emotional states" or "The attempt by various means to mitigate or remediate profoundly unpleasant unwanted emotional states."

Your question is better than my answer, and I'm sorry for that, but I'm glad you asked!

[birdofmay]

What the f Tumblr! 😂

[ID: a Tumblr ad with the headline "Kinky or gentle? What is your sex type?" and a graph below with apparently random groups of two zodiac signs each that are placed on a Cartesian coordinate system without labels. The pairing is Taurus and Sagittarius, Aries and Taurus, Gemini and Virgo, Leo and Scorpio, Cancer and Libra, Sagittarius and Pisces, Capricorn and Aquarius, and lastly Scorpio and Aquarius. Below that is a "take test" button. /end ID]

(In case you're curious, I took the liberty of clicking on it and you have to pick a gender before starting the test - of course the only two options are male and female, which makes this ad completely unsuitable for Tumblr, as always)

[coming-up-with-usernames-is-hard]

PhD Blog Week 8

Courses

CFT: More messing around with correlators, radial quantisation, lots of things that are almost vertex operator algebras without ever saying that

Diff Top: Main complaint this week is that he keeps mentioning bundles without defining them, but I finished the assignment so that's good

Lie Algebras: Root systems, mostly just linear algebra, should finish the classification this week

Talks

Integrable Systems Seminar: Spherical double affine Hecke algebra, still don't really know what it is but despite that the talk was actually one of the most understandable seminars I've been to

Algebra Seminar: More Yangians, still don't know what a Yangian is, followed for about ten minutes before getting lost. The preseminar was good, but didn't help follow the main seminar

Example Showcases: Mostly pretty good this week. The first was on geometric applications to number theory, simple start showing that some equation has no solutions over the integers, then the schemes appeared and I was lost. This seems to be a common occurrence. I should learn what a scheme is. Second talk was on Feynman diagrams and how they arise in QFT, it was a good explanation of a lot of physics in just half an hour, and it was nice to be back on familiar ground after the étale fundamental group. Third one was on the Witt and Weyl algebras, a good explanation but I felt that the talk lacked a conclusion. The final talk was on topological data analysis, very interesting application of things that I don't know that well, and a little bit of rep theory.

Reading Groups

Complex Geometry: The Weyl group part 2, I missed part 1 but it still mostly made sense

Infinity Categories: Really interesting this week. We looked at how to define monoidal categories, the approach taken was to move away from the notion of a biniary product and phrase symmetric monoidal categories as co-cartesian fibrations, then the monoidal structure is naturally induced from maps in FinSet_*, and replacing finite sets with ordered sets gives a non-symmetric structure. Then this (much more complicated) definition generalises to infinity categories.

Categories: The 2-category of categories, after infinity-categories 2 feels like a very small number! I've also agreed to give a talk in a couple of weeks so I need to brush up on universal properties

Supervisor Meeting

Finally heard from my second supervisor as we discussed Wronskian solutions to differential equations, not my favourite area of maths but hopefully once it's related to the algebra I won't have to deal with the differential equations much

Teaching

Two tutorials this week, the work sheet was harder this week as they've just started graph theory, the final question was to prove that the utilities problem has no solution which was a pretty big ask

[bad-dagon]

riemann spheres as a fundamental type, pt.1 basics

ive been thinking on and off about riemann spheres for a while now, a couple weeks really, and so far i think there's some utility to them as a building block of a type system of some kind for a joke/toy computer language

background: riemann spheres are a neat tool in complex analysis where we imagine a sphere whose equator intersects the complex plane, and every number on the complex plane is representable by a point on the sphere such that a line is projected from the north pole and through that point onto the complex plane. naturally, this means that the north pole is P(∞) and the south pole is P(0). see below how that would look like with other unit points of the complex numbers

a neat thing the riemann sphere allows us is to define meaningful division by zero so now z/0 = ∞ clean and simple! and also its inverse, z/∞ = 0 is well behaved as well. this simplifies doing complex analysis but stereographic projection is an absolute bitch to work with turns out, and doing arithmetic on points on the sphere is a mess because it's not a linear mapping (it's continuous though so that's fine)

if we're dealing with ONLY real numbers in relation to a circular slice of the riemann sphere, it already starts looking like a mess; for any number n∈R its projective cognate on the circle is located at (2n/n²+1, n²-1/n²+1). on the real riemann sphere though? zoo wee mamma

an arbitrary z∈C, represented as a point (x,y) on the complex plane, has to walk through a rather ugly mapping (related to the previous one) to find its point on the sphere; more accurately, given the coordinates (x,y) of the point on the plane, the point on the sphere is located at (2x/1+x²+y², 2y/1+x²+y², -1+x²+y²/1+x²+y²), which is godawful in spherical-to-polar coordinate terms, this is much simpler; for a polar pair (r,θ) the point on the unit sphere representing it is (φ,ξ) = (2*arctg 1/r, θ); and conversely projecting from the riemann sphere is also dead simple, given the zenith-azimuth pair (φ,ξ), (r,θ) = (ctg φ/2, ξ). of course, translating from polar to real coordinates is ALSO dead simple; x = r * cos θ, y = r * sin θ. if a computer system were to store complex numbers (or any coordinate on a 2d plane really), it makes sense to store them in terms of spherical coordinates of a riemann sphere, since this makes infinity well-behaved as a unit (zenith = 0, azimuth = literally who gives a fuck) and is surprisingly efficient. knowing that the zenith is ∈[0,π] and azimuth ∈[0, 2π] can allow for some formidably dumb optimisations that can save on space and ensure granularity. compared to storing them as 2d cartesian or polar coordinates, this provides the benefit of having neither number be larger than like 6.3, so an underlying/backing type that can offer great precision in this small range would be more efficient numerically than, say, floating points which have insane baggage and gaps

or iunno i'll look into that one a bit later, i'm just kind of furious right now that i rederived stereographic projection on my own when the formulas were right fucking there if id only just googled for them

[scottishcommune]

Systems theory is certainly useful in explaining the operation of systems, especially ones so structured as to lend themselves to systems theory analysis, just as the equations of physics can explain any phenomenon that can be reduced to the terms of physics. What serious people in ecological philosophy have to ask themselves is whether evolution, let alone self-transcendence, can really be reduced to “dynamics,” “interdependent fluctuations, “feedback mechanisms” — or even “inputs” and "outputs" — that do not differ in principle from the Newtonian orientation toward phenomena or from La Mettrie’s eighteenth-century description of human beings as machines. If there is anything developmental or evolutionary (as distinguished from merely kinetic) about a systems theory “paradigm,” it is simply that some relatively homeostatic phenomena, conceived precisely as systems, may be replaced with other, hopefully complex systems. In either case, despite the imagery that Capra tries to form in the reader’s mind, we cannot properly speak of one mechanism being qualitativey transformed into another. If the essential problem of organic development is reduced at all its levels to “feedback loops” and “fluctuations,” our thinking has not advanced beyond Cartesian and Hobbesian mechanism, however lavishly we speak of the “coevolution of an organism plus its environment,” of “wholeness,” or of Taoist sagacity and Franciscan theology.

- Murray Bookchin, Thinking Ecologically: A Dialectical Approach

[ananke-xiii]

I find self-proclaimed rational people making fun of "fanatical" people into tarots and/or astrology etc so amusing! Because, to me, they're exactly the same type of people, i.e. people who see an object or a field of study and confuse the object with the means. A tarot deck by itself doesn't mean anything. A birth-chart doesn't have anything to say about the future. The meaning is in the relationship with these things, because they're means to discover yourself ("know yourself" always yes, but first you have to meet yourself and it's a whole journey). They're just ways that have been used before and for a looooong time and so yes, energetically speaking they're more charged but you can make up your own system and it'd be the same cause the object doesn't matter in itself. As a matter of fact, "you" as subject are not that important after all. Anybody who's ever started the so-called "spiritual" journey knows that it's definitely not all light&positivity, more like tears&depression if I have to be honest. So yes, some stuff cannot be approached lightly because there is a risk but it's not the "evil" and "devilish" things people usually think of. Sometimes I think we know of the discoveries of contemporary physics but, inside ourselves, in our dark dephts, we're still rather Cartesian in the way we think about the world. There is no object, ergo there is no subject. It's always about connection.