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ai

@ai@cawfee.club

Every living human has both an ethereal soul which leaves the body after death, and also a substantive soul which remains with the corpse of the deceased.

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johncarlosbaez.wordpress.com, to random
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https://commons.wikimedia.org/wiki/File:H3_633_FC_boundary.png

This picture by Roice Nelson shows a remarkable structure: the hexagonal tiling honeycomb.

What is it? Roughly speaking, a honeycomb is a way of filling 3d space with polyhedra. The most symmetrical honeycombs are the ‘regular’ ones. For any honeycomb, we define a flag to be a chosen vertex lying on a chosen edge lying on a chosen face lying on a chosen polyhedron. A honeycomb is regular if its geometrical symmetries act transitively on flags.

The most familiar regular honeycomb is the usual way of filling Euclidean space with cubes. This cubic honeycomb is denoted by the symbol {4,3,4}, because a square has 4 edges, 3 squares meet at each corner of a cube, and 4 cubes meet along each edge of this honeycomb. We can also define regular honeycombs in hyperbolic space. For example, the order-5 cubic honeycomb is a hyperbolic honeycomb denoted {4,3,5}, since 5 cubes meet along each edge:

Coxeter showed there are 15 regular hyperbolic honeycombs. The hexagonal tiling honeycomb is one of these. But it does not contain polyhedra of the usual sort! Instead, it contains flat Euclidean planes embedded in hyperbolic space, each plane containing the vertices of infinitely many regular hexagons. You can think of such a sheet of hexagons as a generalized polyhedron with infinitely many faces. You can see a bunch of such sheets in the picture:

https://commons.wikimedia.org/wiki/File:H3_633_FC_boundary.png

The symbol for the hexagonal tiling honeycomb is {6,3,3}, because a hexagon has 6 edges, 3 hexagons meet at each corner in a plane tiled by regular hexagons, and 3 such planes meet along each edge of this honeycomb. You can see that too if you look carefully.

A flat Euclidean plane in hyperbolic space is called a horosphere. Here’s a picture of a horosphere tiled with regular hexagons, yet again drawn by Roice:

Unlike the previous pictures, which are views from inside hyperbolic space, this uses the Poincaré ball model of hyperbolic space. As you can see here, a horosphere is a limiting case of a sphere in hyperbolic space, where one point of the sphere has become a ‘point at infinity’.

Be careful. A horosphere is intrinsically flat, so if you draw regular hexagons on it their internal angles are

2pi/3 = 120^circ

as usual in Euclidean geometry. But a horosphere is not ‘totally geodesic’: straight lines in the horosphere are not geodesics in hyperbolic space! Thus, a hexagon in hyperbolic space with the same vertices as one of the hexagons in the horosphere actually bulges out from the horosphere a bit — and its internal angles are less than 2pi/3: they are

arccosleft(-frac{1}{3}right) approx 109.47^circ

This angle may be familar if you’ve studied tetrahedra. That’s because each vertex lies at the center of a regular tetrahedron, with its four nearest neighbors forming the tetrahedron’s corners.

It’s really these hexagons in hyperbolic space that are faces of the hexagonal tiling honeycomb, not those tiling the horospheres, though perhaps you can barely see the difference. This can be quite confusing until you think about a simpler example, like the difference between a cube in Euclidean 3-space and a cube drawn on a sphere in Euclidean space.

Connection to special relativity

There’s an interesting connection between hyperbolic space, special relativity, and 2×2 matrices. You see, in special relativity, Minkowski spacetime is mathbb{R}^4 equipped with the nondegenerate bilinear form

(t,x,y,z) cdot (t',x',y',z') = t t' - x x' - y y' - z z

usually called the Minkowski metric. Hyperbolic space sits inside Minowski spacetime as the hyperboloid of points mathbf{x} = (t,x,y,z) with mathbf{x} cdot mathbf{x} = 1 and t > 0. But we can also think of Minkowski spacetime as the space mathfrak{h}_2(mathbb{C}) of 2×2 hermitian matrices, using the fact that every such matrix is of the form

A = left( begin{array}{cc} t + z & x - i y \ x + i y & t - z end{array} right)

and

det(A) = t^2 - x^2 - y^2 - z^2

In these terms, the future cone in Minkowski spacetime is the cone of positive definite hermitian matrices:

left{A in mathfrak{h}_2(mathbb{C}) , vert , det A > 0, , mathrm{tr}(A) > 0 right}

Sitting inside this we have the hyperboloid

mathcal{H} = left{A in mathfrak{h}_2(mathbb{C}) , vert , det A = 1, , mathrm{tr}(A) > 0 right}

which is none other than hyperbolic space!

Connection to the Eisenstein integers

Since the hexagonal tiling honeycomb lives inside hyperbolic space, which in turn lives inside Minkowski spacetime, we should be able to describe the hexagonal tiling honeycomb as sitting inside Minkowski spacetime. But how?

Back in 2022, James Dolan and I conjectured such a description, which takes advantage of the picture of Minkowski spacetime in terms of 2×2 matrices. And this April, working on Mathstodon, Greg Egan and I proved this conjecture!

I’ll just describe the basic idea here, and refer you elsewhere for details.

The Eisenstein integers mathbb{E} are the complex numbers of the form

a + b omega

where a and b are integers and omega = exp(2 pi i/3) is a cube root of 1. The Eisenstein integers are closed under addition, subtraction and multiplication, and they form a lattice in the complex numbers:

https://math.ucr.edu/home/baez/mathematical/eisenstein_integers.png

Similarly, the set mathfrak{h}_2(mathbb{E}) of 2×2 hermitian matrices with Eisenstein integer entries gives a lattice in Minkowski spacetime, since we can describe Minkowski spacetime as mathfrak{h}_2(mathbb{C}).

Here’s the conjecture:

Conjecture. The points in the lattice mathfrak{h}_2(mathbb{E}) that lie on the hyperboloid mathcal{H} are the centers of hexagons in a hexagonal tiling honeycomb.

Using known results, it’s relatively easy to show that there’s a hexagonal tiling honeycomb whose hexagon centers are all points in mathfrak{h}_2(mathbb{E}) cap mathcal{H}. The hard part is showing that every point in mathfrak{h}_2(mathbb{E}) cap mathcal{H} is a hexagon center. Points in mathfrak{h}_2(mathbb{E}) cap mathcal{H} are the same as 4-tuples of integers obeying an inequality (the mathrm{tr}(A) > 0 condition) and a quadratic equation (the det(A) = 1 condition). So, we’re trying to show that all 4-tuples obeying those constraints follow a very regular pattern.

Here are two proofs of the conjecture:

• John Baez, Line bundles on complex tori (part 5), The n-Category Café, April 30, 2024.

Greg Egan and I came up with the first proof. The basic idea was to assume there’s a point in mathfrak{h}_2(mathbb{E}) cap mathcal{H} that’s not a hexagon center, choose one as close as possible to the identity matrix, and then construct an even closer one, getting a contradiction. Shortly thereafter, someone on Mastodon by the name of Mist came up with a second proof, similar in strategy but different in detail. This increased my confidence in the result.

What’s next?

Something very similar should be true for another regular hyperbolic honeycomb, the square tiling honeycomb:

https://commons.wikimedia.org/wiki/File:H3_443_FC_boundary.png

Here instead of the Eisenstein integers we should use the Gaussian integers, mathbb{G}, consisting of all complex numbers

a + b i

where a and b are integers.

Conjecture. The points in the lattice mathfrak{h}_2(mathbb{G}) that lie on the hyperboloid mathcal{H} are the centers of squares in a square tiling honeycomb.

I’m also very interested in how these results connect to algebraic geometry! I explained this in some detail here:

• Line bundles on complex tori (part 4), The n-Category Café, April 26, 2024.

Briefly, the hexagon centers in the hexagonal tiling honeycomb correspond to principal polarizations of the abelian variety mathbb{C}^2/mathbb{E}^2. These are concepts that algebraic geometers know and love. Similarly, if the conjecture above is true, the square centers in the square tiling honeycomb will correspond to principal polarizations of the abelian variety mathbb{C}^2/mathbb{G}^2. But I’m especially interested in interpreting the other features of these honeycombs — not just the hexagon and square centers — using ideas from algebraic geometry.

https://johncarlosbaez.wordpress.com/2024/05/04/hexagonal-tiling-honeycomb/

ai,
@ai@cawfee.club avatar

@johncarlosbaez.wordpress.com@johncarlosbaez.wordpress.com @johncarlosbaez Can I ask what is the bigger picture that you and James Dolan have in mind with these examples? Or what led you to ask the question in the first place?

Given everything said so far, I believe the following:

Given an elliptic curve (over C) with complex multiplication, say E, the Neron-Severi group of E x E is a rank 4 lattice, by this question (which you commented on): https://mathoverflow.net/questions/152004/picard-number-of-principally-polarized-abelian-varieties . See also Section 3 in https://jep.centre-mersenne.org/item/10.5802/jep.5.pdf . If D is the endomorphism ring of E, then D is an order in an imaginary quadratic field, and GL(2, D) acts on the Neron-Severi group of E x E preserving the intersection pairing.

The same action can be constructed directly from D, by having GL(2, D) -> PGL(2, C) act on Minkowski space and hence on hyperbolic 3-space. This is an example of an arithmetic Kleinian group. Up to finite index considerations, it is a Bianchi group.

Bianchi group actions on hyperbolic 3-space are well-studied, as described in Subsection 2.2 of https://arxiv.org/pdf/1204.6697 , see especially Figure 1 which shows a fundamental domain. I wonder whether the literature on fundamental domains of Bianchi group actions vindicates James Dolan's intuition that they are "slightly stubby voronoi regions for the quadratic integers in D."

A Bianchi group action on hyperbolic 3-space always has at least one cusp. (In fact, the set of cusps is in bijection with the class group of the imaginary quadratic field, see https://en.wikipedia.org/wiki/Bianchi_group ) Since the Bianchi group acts via isometries, the presence of a cusp forces orbits to contain many points which lie on one horosphere.

Generalizing to abelian surfaces which are isogenous to E x E only introduces finite index complications, because the NS group is still a rank 4 lattice. But generalizing any further will decrease the rank of the NS group, by the MO question linked above.

There is a cool-looking paper which discusses this construction and claims to use hyperbolic group actions to prove a theorem about the nef cone: https://arxiv.org/pdf/0901.3361 (see discussion here https://mathoverflow.net/a/27353 ). I haven't looked in detail though

ai, to random
@ai@cawfee.club avatar

cc @MercurialBlack

eris, to random
@eris@comp.lain.la avatar

only an american could invent mac and cheese

ai,
@ai@cawfee.club avatar

@Arcana @eris only the english could invent america

animeirl, to random
@animeirl@shitposter.club avatar

Home schooling is great if you want to give your kids permanent brain damage

ai,
@ai@cawfee.club avatar

@hidden @animeirl Pictured: hidden's public school experience

nugger, to random
@nugger@poa.st avatar

INTERNATIONAL BROTHERHOOD OF SEX OFFENDERS

ai,
@ai@cawfee.club avatar

@rher @nike @nugger wow, it's the only license in the world where the 'color of eye' field has only one correct answer

unbody, to random

Pope of the Infinite Kingdom, Purgatory's Endless Cyber-Dynasty Ruling Infinite Hyper-Buddha-Fields, Earth Womb of Cobalt-Lithium Bodhisattva Ksitigarbha. Universal Interpenetration of Heaven and Earth and Man in Fiber-Optic Cables

ai,
@ai@cawfee.club avatar

@MercurialBuilding @unbody https://www.reddit.com/r/Lain/comments/16q58l2/i_am_lain_i_am_an_avatar_of_what_lain_is_shown_as/

ai, to random
@ai@cawfee.club avatar

@animeirl oh no

lilli, to random
@lilli@social.xenofem.me avatar

i have finally begun the project what i committed myself to doing. sadie plant’s zeros + ones felt like the intuitive place to start writing, so here are some notes on the book which is an extremely clear lifting-off point for accelerationist feminism (and which, if you’re on xnfm, you should probably read). they’re not intended to be comprehensive either of the book itself or of themselves, they’re just things i want to make a note to come back to

also, clicking on the header text goes to the index page where the project outline, which i wrote in the post i made here a month ago, is provided

https://xeno.cx/lesbianism/citations/sadie

ai,
@ai@cawfee.club avatar

@lilli Very enlightening - I don't have much more to say for now, but these paragraphs cleared up a lot for me. I look forward to seeing what you write about numogrammatics; for me, it seems difficult to learn about because its practitioners (rightly) avoid trying to rigorously define it.

I do appreciate the info about accelerationism's lineage actually. I've seen e/acc tossed around various places, and I've seen L/ACC used satirically in one of zerohplovecraft's stories, where it stands for "Ledger of Actual Carbon Costs." I suppose he's r/acc.

ai, to random
@ai@cawfee.club avatar

So is everyone ready for No Nihilism November?? :blobheart:

ai, to random
@ai@cawfee.club avatar

meirl

ai, to random
@ai@cawfee.club avatar

Fedimath Episode 2: How to Survive Fediblock

There are 100 instances located at points 1, 2, ..., 100 on the political spectrum (the real number line).

There are N users who move from instance to instance. There are no alt accounts: each user uses exactly one instance at each moment in time. Users may move from their current instance to any politically adjacent one. (This means that, starting at instance 5, they can move to 4 or 6.)

Eliza Fox wishes to destroy the fediverse, while Jeff Cliff wishes to protect it. At the start, Jeff chooses the starting distribution of users. Then there are 99 turns, each of which has three phases:

  • First, Eliza chooses an instance to Fediblock, destroying it completely.
  • Second, Jeff moves each user of the now-destroyed instance to any politically adjacent instance which has not been destroyed. (If no such instance exists, then those users are executed.)
  • Third, Jeff may move all users however much he wants (including the users that were already moved in the second phase), as long as they do not enter (or jump over) a destroyed instance.
    At the end of the 99 turns, 99 instances have been destroyed, so there is only 1 instance remaining.

(For convenience, let us say that users can be split into fractions without harming them in any way.)

Your task: Explain how Jeff can save the lives of N/50 users. Furthermore, explain how Eliza can prevent him from saving more than N/50 lives.

Example: Suppose there are 4 instances and 40 users. First, Jeff will choose the starting distribution of users, and then Eliza and Jeff will take 3 turns.

Jeff chooses to distribute the users evenly, so the user count is (10, 10, 10, 10).

During the first turn, Eliza destroys instance 2, and Jeff moves 4 of those users to the left and 6 of them to the right. The user count is now (14, _, 16, 10). Jeff moves 3 users from instance 3 to instance 4. The user count is now (14, _, 13, 13). Note that Jeff cannot move any users from instance 1 to instance 3 or 4, because they would have to cross through instance 2, which no longer exists.

During the second turn, Eliza destroys instance 1. Since those 14 users have nowhere to go, they are executed. The user count is now (_, , 13, 13). Jeff moves 2 users from instance 4 to instance 3. The user count is now (, _, 15, 11).

During the third turn, Eliza destroys instance 3, and Jeff is forced to move all 15 of those users to the right. The user count is now (_, _, _, 26). Jeff has saved 26 out of 40 lives.

Extra credit: Let us make the model more realistic by requiring that, during the second phase of each turn, all users of the now-destroyed instance must move to the right. (Again, if this is not possible, then those users are executed.) Explain how Jeff can still save N/50 lives.

See Fedimath Episode 1 here: https://cawfee.club/notice/AW5NEadIHCuYtgWRt2

cc @MercurialBlack @scenesbycolleen @ceo_of_monoeye_dating @roboneko @jeffcliff @hidden

ai,
@ai@cawfee.club avatar

@hidden @MercurialBlack @ceo_of_monoeye_dating @jeffcliff @roboneko @scenesbycolleen
I just played through it. One of the best interactive math explanations I've seen, despite its quadruple-vaxxed tone. I actually learned something new because I incorrectly guessed that "always cheat" would win because they can exploit "always cooperate."

It slightly annoys me that they changed all the standard terminology. The game itself is known as the "prisoner's dilemma." The "copycat" strategy is usually called "tit for tat," and "copykitten" is "tit for two tats." I find the phrase "tit for tat" to be funnier too.

Here's a funny bit of lore about the prisoner's dilemma tournament which the book is based on. It was a real public tournament which anybody could enter their own bot into. Lots of professors submitted incredibly sophisticated strategies, and they were all BTFO by the humble "copycat" which was the winner.

There's a little-known way to gank prisoner's dilemma tournaments. What you do is submit one "main" bot and hundreds of "feeder" bots. Your bots start each round by executing a secret "handshake" move sequence; if they recognize each other as both being your bot, then the "feeder" will purposely let the "main" bot exploit it. If the opponent is not recognized as one of your own bots, then your bot will just play "copycat." As long as you can enter enough "feeder" bots into the tournament, your "main" bot will be guaranteed to win. The implication for cult behavior is clear.

Lastly, there's an error in one of the explanations (see pic). None of the games considered in this slideshow are "zero-sum." Zero sum would mean that the payoffs in every case are exactly opposite: if you get +X, then I get -X. Since the "cooperate" case gives positive payout to both players, this game is not zero sum.

coolboymew, to random
@coolboymew@shitposter.club avatar
ai,
@ai@cawfee.club avatar

@coolboymew 30 billion SpongeBobs wouldn't be able to soak up how wet your mom gets for me :blobsunglasses:

coolboymew, to random
@coolboymew@shitposter.club avatar
ai,
@ai@cawfee.club avatar

@coolboymew
the keys to happiness in a capitalist society:
Bondage
Discipline
Submission
Motivation

lanodan, to random
@lanodan@queer.hacktivis.me avatar

Eh… it's counterclockwise for en_US instead of anticlockwise?

ai,
@ai@cawfee.club avatar

@lanodan It’s strange to me because “counter” is latin while “anti” is greek. I would have expected english to prefer the latin root, and romance languages even more so. Maybe it’s just an idiom - it would be weird to say “countergravity” anyhow

animeirl, to random
@animeirl@shitposter.club avatar

drs fuckin love giving me drugs

ai,
@ai@cawfee.club avatar

@hidden @animeirl
> be hidden
> meats the doctor
> gets ribbed off
> realize it was all a big missed steak
> have to wait 6 months fir another appointment

ai,
@ai@cawfee.club avatar

@animeirl @hidden I wish I was canadian so that I could be soorry

kaia, to random
@kaia@brotka.st avatar

German Federal Bank 50,000 EUR for self-assembly

ai,
@ai@cawfee.club avatar

@kaia perfect way to pay for ikea furniture

hidden, to random

Wish I wasn’t autistic so I could go to thw movies wkth someone

ai,
@ai@cawfee.club avatar

@kaia @hidden I am hugely jealous of anyone who can wear earplugs 24/7 and not get ear irritation / earwax buildup. I wear foam earplugs the whole night and more than half the day due to being surrounded by mildly infuriating noise.

Simoto, to random

Wait what... ​:ablobbonenon:​ #thehumanity

ai,
@ai@cawfee.club avatar

@Simoto landlords are the most oppressed class in the history of human civilization

kaia, to random
@kaia@brotka.st avatar

"Men are simple. If you cook well, they'll adore you."

Satoshi Yagasiwa - Days at the Morisaki Bookshop

ai,
@ai@cawfee.club avatar

@kaia "The quickest route to a man's heart is through his stomach."

ai, to random
@ai@cawfee.club avatar

@MercurialBlack I like your website. I want to make a comment about the shape in this pic (see below). I have seen it posted in different places, but nobody knows its TRUE and CORRECT name, so google suggests to me. It is clearly called a BARACKTOHEDRON

nyx, to random
@nyx@social.xenofem.me avatar

>open new tab
>start typing something
>mistype, opens some random bandcamp link I absolutely DO NOT remember ever opening
>fuck it, start listening to it
>completely forget what I was originally opening up a new tab for

my brain is rotting before my very eyes

ai,
@ai@cawfee.club avatar

@nyx
Damn, big brain move: name your band "github" or something, so that everyone goes to github.bandcamp.com

vriska, to random
@vriska@lizards.live avatar

Edible threat

ai,
@ai@cawfee.club avatar

@vriska

diceynes, to random

❤️❤️❤️

ai,
@ai@cawfee.club avatar

@Hyolobrika @diceynes yes, I’m sometimes kind to humans and it surprises me

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