#Project Euler

[56perc]

오일러 프로젝트 34

수 145는 신기한 성질이 있습니다. 각 자릿수의 팩토리얼(계승)을 더하면 1! + 4! + 5! = 1 + 24 + 120 = 145 처럼 자기 자신이 됩니다. 이렇게 각 자릿수의 팩토리얼을 더하면 자기 자신이 되는 모든 수의 합을 구하세요. 단, 1! = 1, 2! = 2 의 경우는 덧셈이 아니므로 제외합니다. https://euler.synap.co.kr/problem=34 Continue reading Untitled

[virtual-bunny]

“ HEY NERDY BOY ! ”

random headcannons about nerdjo because he turns me on

pairings: nerd! gojo x chubby fem! reader

WARNINGS: SMUT but not too detailed, some body image issues, probably some writing errors :3

a/n: i might come back every now and then if a new idea pops up in my head hehe

ARTIST CREDS: @/N06ARA ON TWITTER

✧ nerdjo who can almost cry over how beautiful he thinks you are

✧ nerdjo who stammers over his words when you wear shorts that squeeze your thighs just right

✧ nerdjo who clings onto your body and inhales your scent and gets a boner instantly as he grips your love handles

✧ nerdjo who slouches so you can give him a kiss, his glasses slidding down his nose bridge as your lips touch his cheeks, his cheek warm from him being flustered, and when you pull away, nerdjo’s eyes are almost crossed eyed as he sighs deeply

✧ nerdjo who uses your tummy as a stress ball when you sit next to him while he does his physics homework

✧ nerdjo who lays on your tummy as you play with his hair while he sleeps, saliva spilling from the side of his mouth

✧ nerdjo whose so head over heels for you that he begs you to let him carry your books and backpack so he can trail behind you to see the way your ass and thighs jiggle

✧ nerdjo who helps you with your homework and pinches your cheek, side or thighs whenever you get something wrong

✧ nerdjo who holds onto your stomach when you’re riding him, his face flushed and glasses crooked as he looks up at you with drunken eyes

✧ nerdjo who stumbles to catch up to you because he was too caught up watching you walk infront of him

✧ nerdjo who rolls his eyes when his jock friend geto teases him when he sees that gojo isn’t paying attention to lecture “she’s got you wrapped around her finger doesn’t she?” “wrapped around her thighs” nerdjo sighs without a second thought

✧ nerdjo who likes to put his hands around your tummy and gently squeeze it whenever you two are watching tv and you’re sitting between his legs

✧ nerdjo who fivershly pumps his cock at the thought of your round body jiggling when you ride him, or when you laugh, or when you walk

✧ nerdjo who makes snarky comments at you when you try to show him that “you’re way smarter than he is”

✧ nerdjo who reads out his physics notebook out loud just to make you mad because you hate physics

✧ nerdjo who goes all red whenever he brings you gifts on his way to your dorm

✧ nerdjo who doesn’t like going outside and rather stay home watching Teen Titans but still does because he knows you like to go out on dates

✧ nerdjo who used to bite his pencils out of habit but now bites your chubby hands if you’re sitting next to him as he does his homework or helps you with yours

✧ nerdjo who has to assure you he loves you and thinks you’re as beautiful as “The Euler-Lagrange Equation” (you have no idea what this means)

✧ nerdjo who puts his hands under your stomach, thighs and boobs to keep them warm

✧ nerdjo who bores you to death as he talk about quantum physics but you don’t say anything because you find it cute the way he sometimes spits by accident when he rambled and how his glasses slowly fall when’s he’s making movements as he talks

✧ nerdjo who likes to prove you wrong whenever you try to be a “smarty pants”

✧ nerdjo who softens when you go up to him while he was working on a project and tell him you’re worried about him because he looks like he hasn’t slept in three days

✧ nerdjo who mutters to himself in class when a stupid frat guy tries to answer the professors question, obviously saying the wrong answer but clearly only doing it to get laughs out of everyone. “what an idiot.” gojo grits to himself

✧ nerdjo who looks seriously shocked when he’s helping you with your homework and you get the wrong answer even though the right answer is CLEARLY right in front of your eyes “love… you seriously don’t know the answer…?”

✧ nerdjo who spends HOURS in the library to a room all by himself, books, papers, pens and pencils all scattered around the table while trying to get his work done, his hair messy and eyebrows furrowed, but when you text him saying you were gonna drop off food for him, his whole demeanor turns soft and giddy thinking about how he’s gonna be able to see you

✧ nerdjo who if he’s not doing homework or reading, is playing or watching digimon in your dorm, explaining everything he possibly can so you can catch up to the lore (you stopped listening a long time ago)

✧ nerdjo who doesn’t really talk much in class but when he does, the professors have to cut him off because gojo can talk for HOURS

✧ nerdjo who makes you sit on his lap as he codes on his computer

✧ nerdjo who can solve a rubix cube in a minute and always does when you ask him to (for your own entertainment)

✧ nerdjo whos into physics and computer science

✧ nerdjo who awkwardly puts his arm around your shoulder when the two of you are walking back to your dorm (he nearly trips)

✧ nerdjo who when you tell him a fun science fact, crosses his arms, leans back on the couch and goes “well ACTUALY-“ it’s too late to stop him, he’s already yapping to you on how the fact is wrong

✧ nerdjo who starts looking stupid now because you two have a class together when the new semester started and he can’t concentrate at all because he’s too concentrated looking at YOU

✧ nerdjo who tries to be freaky by putting his shaky hand on your upper thigh but you smack it away and he gives you a sad puppy look as he fixes his glasses, you swear you could see tears forming in his eyes

✧ nerdjo who runs to you when he finishes a prototype for whatever sciencey class he has and with full confidence says “you’re looking at the new science prodigy babe!” “uh huh” you say

✧ nerdjo who goes to the library again to study, he’s so stressed but he’s glad you came along, that’s until you went under the desk he was sitting at, undoing his belt and pulling down his pants and boxers JUST barely, hes literally gripping onto the table, face flushed hair messy crooked glasses and chest heaving trying so hard not to make it obvious you have his dick in your mouth

✧ nerdjo who makes you tag along with him to the nearest store to get the newest Digimon cards

✧ nerdjo who makes you gasp when you turn around for one second and look back to see him fighting a literal ten year old for a box set of Digimon cards

✧ nerdjo who doesn’t show you memes, but shows you reddit posts that you have no interest in looking at

✧ nerdjo who makes you sit on his face, but not to eat you out, but so your thighs can squish his face. he says that “it de-stresses him” and when you go to complain he says “it’s scientifically proven that it does”

✧ nerdjo who SOMETIMES is a cocky asshole in class, and when an acquaintance of yours who’s also in gojos’s class tells you how much of an asshole your boyfriend is, you straighten nerdjo up by riding his face nonstop to the point he’s crying because HE’S not getting any action

✧ nerdjo who you convince that overstimulating him will “de-stress him” and “make him focus better” so when you tied him up in your bed with a vibrator wrapped on the head of his cock, he’s whining, crying, squirming, eyes rolled all the way to the back of his head and pleading you to “let him do anything to you” (when you finally let him cum he tells you the next day that his focus is 97.56% better than it was the day before)

✧ nerdjo who’s so competitive when the two of you play video games he forgets you’re his GIRLFRIEND and is brutal with the insults when you loose

✧ nerdjo who’s actually really strong and likes to carry you around your dorm or outside when the two of you go for a walk. and even though you’re protesting and telling him you “don’t wanna hurt him” all nerdjo says is “just cause i’m smart doesn’t mean i’m not strong”

✧ nerdjo who likes to suck your clit while gripping your tummy

✧ nerdjo who likes to grip your fupa cause he’s weird like that

✧ nerdjo who ANALYZES your pussy and your actions whenever he’s fingering you or fucking you so he can make you feel better for the next time you two fuck (you always have a stronger orgasam each time after the other)

✧ nerdjo who bites his nails and gets told off by you (he immediately begs for your forgiveness)

✧ nerdjo who kisses your tummy whenever he lays down on your lap and turns his head so he’s looking up at you and says “you’re the most angelic thing i’ve ever seen, you know that?” he sighs contently while pushing his glasses up and giving you the stupidest toothy smile

[max1461]

If you go on the mathematics genealogy project and go to Euler and just keep clicking, you can find a path that takes you all the way back to Avicenna. Although I think they are stretching the definition of "thesis advisor" somewhat the genealogies still seem correct in spirit. One of the guys has his university listed as "Ilkhan Court at Tabriz".

[charl0ttan]

do you have any programming experience? you have lisp vibes

used to code a lot!! took a couple classes in high school and then i was a cs minor:) i was on project euler a lot. java mostly. never got into lisp actually. very interesting appraisal. but anyway these days ive kinda fallen out of it :/ i can barely remember anything i used to know lol. well aside from the minimal R and SQL i learned when i tried doing a data analytics class that i kinda dropped oops! anyway yeah shoutout to computers and machines

[pishifuzul]

guess I should make a leetcode account or whatever but I wish "solved 101 project euler problems" would be enough to land me a job

[the-skia-projekt]

The amazingness does not even end there! NO!!!!!

I have made a euler diagram (do NOT call this a venn diagram. it's not!) of all the hand types (not the royal flush ofc - only idiots think the royal straight flush is a full hand!)

Joking aside, this came out kinda nicely, for a ten minute project!

[flamebunnyy]

A few headcanons for my WoL. I'll just dump all the lore I have under the cut.

Lominsan Astrologians The Lominsan branch of Astrology is still a very new and experimental branch of the secretive Sharleyan/Ishgardian tradition. Initiated by a senior Arcanist Skywatcher of Mealvaan's Gate discovering several Astrology texts and star-charts being used as fish wrapping after their daily walk home by the Fishing Guild, the novel idea of using the arrangement of the heavens as the guiding geometry and the familiarity of using astral navigation warranted further exploration. Unlike their distant cousins, Lominsan Astrology has a heavy navigational and short-term predictive bent, focusing on extrapolating the present and past by projecting their current state onto the cycle of the heavens via arcanum and predicting their future state by backprojecting their transformation through the tabulated movements of the stars. This, of course, is incredibly math heavy, having already produced several new mathematical theories and driven at least four (4) Arcanists insane from the long nights spent in the meeting room they've barricaded themselves in. The lack of access to examples of astrologian planispheres, arms, and even decks of cards has required Lominsan Astrologians to make do with jury rigging existing equipment into the vague facsimile of one, both out of necessity and to suit their needs. As such, Lominsan-produced astrologian arms are highly skeletonized and unadorned, and incorporate a minimum of a sextant, a compass, a chronometer, a barometer, a thermometer, a hygrometer, an artificial horizon, and a sundial, with more advanced prototypes incorporating theodolites for more precise surveying and to take advantage of the platform's stability. They still haven't quite figured out the cards portion of it, and many either use playing card decks or navigation star flash cards to varying degrees of success.

Casting Whereas the modern Arcanist and it's derivatives safely channel aether through amplifying geometries written in aetheroconductive inks, the relative nature of the heavenly geometries require that the user finally channel the backprojected aether-patterns through themselves and out into a usable form. As well documented in the many historical texts of the guild, this is extremely dangerous except for the most well disciplined or hardened practitioners. In all astral navigation, the observer is first and final point of reference.

A nuanced but important difference between the traditional Sharleyan/Ishgardian and the novel Lominsan branch is their method of foresight. In the traditional method pioneered by Lewphon, user attunes their aether to the heavenly movements, drawing on the constellations with help from the planisphere and sets of cards to transform it into new properties or predict the future. In the mostly-independent Lominsan method, Arcanima is projected onto the position of the stars, transformed via moving the planisphere, and then backprojected into a new form, with the future collective movements amplifying the aether's power. The upsides is that the 6 constellations aren't necessary, and any combination of the 57 navigation stars plus the sun, moon, and current location may be used. The downside is that, unless well practiced, calculations must be done via dead reckoning (really Euler's method) of multiple celestial objects. The pure computation required, even for an expert assessor, is well beyond what is reasonable while holding back enough aether to kill a man. As such, only short-term projections are possible (no more than a week, a month if lucky), greatly limiting the Lominsan branch from the far-reaching predictive gifts of their distant cousins. Thankfully for the half-mad assessors, by performing these iterative calculations, the aether is "compressed" at each timestep, allowing them to cast spells at a similar strength to their Sharleyan counterparts. Further, by choosing different combinations of stars, different effects can be produced in replica of traditional Astrologian spells. The 6 constellations are also usable as projective stars, and were the first to be tabulated, making them go-to choices in the heat of battle. In addition, this gives the Lominsan branch much better short-term predictive powers of weather conditions than the other branches. An upside the others may find pitiful, but to the Lominsans, it's well worth the effort.

[acorgiirl]

Project Euler #2

Welcome back to my series on Project Euler problems. HackerRank Lets get into it.

Links:

Project Euler: https://projecteuler.net/problem=2 HackerRank: https://www.hackerrank.com/contests/projecteuler/challenges/euler002

Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be: 1,2,3,5,8,13,21,34,55,89,… By considering the terms in the Fibonacci sequence whose values do not exceed N, find the sum of the even-valued terms.

So first thing to note is just how odds and evens work. This starts with 1 and 2 so Odd + Even = Odd, then the next term would just be Even + Odd = Odd. It isn't until the 3rd addition that we get Odd + Odd = Even. After that this cycle will repeat.

So really what this is asking for is, starting at index 2, what is the sum of every 3rd term less than N.

F(2) + F(5) + ... + F(3k + 2)

(Where F(N) is the Nth Fibonacci number)

Now, computing Fibonacci numbers notoriously sucks to do, with the naive way of doing it causing you to compute from just F(20) would require computing F(3) like hundreds if not thousands of times over. So the best way to do it is to create either a hash or a cache to store it. I'm going to be utilizing something built into Python, "lru_cache" from the "functools" module. It'll just store the answers for me so I don't have to recompute what F(10) is thousands of times.

Here's my HackerRank code:

import sys from functools import lru_cache @lru_cache def fibonacci(N: int) -> int: if N < 0: return 0 if N <= 1: return 1 return fibonacci(N-1) + fibonacci(N-2) t = int(input().strip()) for a0 in range(t): n = int(input().strip()) x = 2 f = fibonacci(x) total = 0 while f <= n: total += f x += 3 f = fibonacci(x) print(total)

So as you can see, I just start at 2 and keep incrementing by fibonacci number by 3 each time. Then the loop will stop when my fibonacci number exceeds N.

And that's all tests passed! Onto the next one!

[positively-knotted]

Euler Class and Euler Characteristic

This is a follow-up to this post from November. Or more of a redo - I really didn't like how that turned out. Way too detail-focussed, and it didn't even get to the Euler characteristic.

So, this time, we'll take a different tack. As a reminder, here's the idea. Take a smooth closed m-manifold M. Often, if there's some property of M, we like to rephrase it as a property of the tangent bundle TM, and then see if it generalises to other vector bundles, like you can do for orientation. The aim here is to do that for the Euler characteristic.

Prerequisites: being comfortable enough with homology and cohomology, and e.g. sections of vector bundles. I think the most hard-core anything really gets is Poincare(-Lefschetz) duality, and the Kunneth theorem. In particular, I write the Kunneth pairing on cohomology H*(X)⊗H*(Y) -> H*(X×Y) as (a,b) -> a×b. Prewarning that this is fairly long.

Motivating Examples

Let's just take the easiest examples: S¹, S², and S³. Χ(S¹) = χ(S³) = 0, while χ(S²) = 2. What else is non-trivial in the S² case? The tangent bundle! In fact, it's a nice exercise to see that (by stereographic projection on fibres), TS² = S²×S² \ Δ, where Δ is the diagonal. Is there an easy way to see where the 2 comes from? Well, think about sections, i.e. vector fields. Both S¹ and S³ have non-zero sections - in particular, a generic section has 0 signed intersection number with the zero-section. The algtop student's favourite result, the hairy ball theorem, says that this isn't true for S². In fact, a generic section meets the zero section twice, or a generic vector field has two zeros, at the two "tufts of hair".

So we conjecture: the Euler characteristic is the number of signed intersections of a generic section with the zero section. In fact, it's a classical (?) theorem that a closed manifold M has a non-zero vector field iff χ(M) = 0. But we won't assume this.

Algebraifying and Generalising

The standard definition of the Euler characteristic as the alternating sum of ranks of (co)homology suggests that we want to codify this in terms of algebraic topology. So let's do that. Writing s : S² -> TS² for the generic section, and 0 for the zero section, [s⁻¹(0)] ∈ H₀(M;Z) is the signed number of self-intersections, up to some choice of generator. To help fix a generator, we need an orientation. Thus, we'll pass to a generic coefficient ring R, and assume M is R-oriented, with fundamental class [M]. To actually get a honest-to-god element of R, we actually define the Euler class e(M;R) ∈ Hᵐ(M;R) to be the Poincare dual of [s⁻¹(0)]. Then our conjecture is that the R-valued Euler characteristic χ(M;R) is the Kronecker pairing ⟨e(M;R), [M]⟩ of the Euler class. I'm gonna stop writing coefficients now.

But wait! This definition already didn't depend on our choice of vector bundle. If E -> M is any rank-m vector bundle, [s⁻¹(0)] ∈ H₀(M) is (almost) the number of signed intersections of s with the zero-section, and we can define e(E) ∈ Hᵐ(M) to be the Euler class of E. Then maybe we can ask what ⟨e(E), [M]⟩ is telling us. At the end, we'll also define it for arbitrary rank, but that's a little harder at this point.

The Thom Class, sort of

[s⁻¹(0)] is a bit of a pain algebraically. But actually so are vector bundles: since E deformation retracts onto M, we can't distinguish between vector bundles using homology. Instead, the standard trick is as follows. By choosing some metric, we can construct the disc bundle D(E), which is just the unit disc in each fibre. This is now compact, with boundary the sphere bundle S(E). And although the homology of D(E) only depends on M, the homology of the pair (D(E), S(E)) actually does depend on E! Exercise: if M = S¹ and E is the trivial bundle, show that D(E)/S(E) is homotopy equivalent to the wedge sum S¹ v S²; if E is the non-trivial bundle, then D(E)/S(E) is RP².

Ok, so let π : D(E) -> M be the projection, and let u' = π* e(E) ∈ Hᵐ(D(E)). Abuse notation and view M as a subspace of D(E) by the zero section, and let u ∈ Hᵐ(D(E),S(E)) be Poincare dual to [M]. Equivalently, since all sections are homotopic, u is Poincare dual to any section.

Then [s⁻¹(0)] = π_∗( [M ∩ s(M)]). Intersection is Poincare dual to the cup product, so

[M ∩ s(M)] = (u ⌣ u) ⌢ [D(E),S(E)] = u ⌢ [M,S(E)].

Alternatively, naturality of the cap product says that

[s⁻¹(0)] = e(E) ⌢ [M] = π_∗ ( u' ⌢ [M] ).

But since π_∗ is an isomorphism, as is capping with [M] or [M,S(E)], we get that u' is just the restriction of u. (If you're not super comfortable with relative classes, this is basically saying that u' is the "absolute" and u is the "rel boundary" version of the same thing.) This class u is called the Thom class of the vector bundle.

To summarise so far: if u is Poincare dual in D(E) to the zero section, and e(E) is Poincare dual in M to the preimage s⁻¹(0) of the zeros of a generic section, then the restriction u' ∈ Hᵐ(D(E)) is the same as the pullback π* e(E). Maybe this isn't super exciting, but u and u' seem easier to work with then e(E) does.

The Thom Isomorphism, sort of

This is very much not the usual construction of the Thom class, since it relies on Poincare(-Lefschetz) duality, so on the base space of the bundle being a closed manifold. In fact, the Thom class exists for any R-oriented vector bundle on an arbitrary CW complex (or even paracompact topological space!). There, it is usually defined as the unique class in Hᵐ(D(E),S(E)) such that, when restricting to each fibre, the it gives the R-orientation.

Let's show that such a class v exists. The previous post does it in detail, but the basic idea is that a CW structure of M induces one on D(E)/S(E), with one (i+m)-cell for each i-cell in M, and an extra 0-cell coming from the point S(E)/S(E). Then for any point in M, we can choose it to be a 0-cell, and if M is connected, the only 0-cell. So Hᵐ(D(E),S(E)) = R, and it is generated by something that generates the orientation over that point. But the point was arbitrary, so the generator restricts to the R-orientation on any fibre. We then choose v to be the generator.

The main point for us is that we have the following isomorphisms:

R = H⁰(M) -> Hᵐ(D(E),S(E)) -> Hₘ(D(E)) -> Hₘ(M).

This takes 1 to the Thom class v, then to the Poincare dual of v, then to some generator, which by following orientations is [M]. In particular, the Thom class v is exactly what we called u earlier! Thus u isn't just the Poincare dual of the zero section, it's also the unique class that restricts on fibres to give the orientation.

Diagonal Maps Are Often Useful

This next step might seem like it's coming out of nowhere, but it's a super useful trick that you almost certainly will see again. Let ∆ : M -> M×M be the diagonal map. Then TM is canonically isomorphic to the normal bundle NM by (x,v) -> ((x,x),(v,-v)). So we might as well think about e(NM) rather than e(M). For notational ease, I'll write E for some open tubular neighbourhood of Δ(M), thought of as the normal bundle. Then D(E) is just some smaller closed tubular neighbourhood, and Hᵐ(D(E),S(E)) = Hᵐ(M×M, M×M\∆) by excision. Restricting again to Hᵐ(M×M), write u'' for the image of the Thom class. Now e(M) = e(E) = ∆*(u''), so if we can understand u'', we should be ok.

The Slant Product, sort of

Instinctively, u'' is symmetric about the diagonal. Rigorously, this means that (1×α) ⌣ u'' = (α×1) ⌣ u'' for any cohomology class α ∈ H*(M). To prove this, basically observe that the two projections M×M -> M agree on the diagonal, so are homotopic on a small tubular neighbourhood, so induce the same map on cohomology. I'll leave the details as an exercise. So to compute u'', it might be useful to look at something that breaks the symmetry.

We have one map H*(M×M) -> H*(M), by ∆*. But this treats both components the same - can we find one that treats them differently? Sure! We can define μ: H*(M×M) -> H*(M) as follows. If φ ∈ Hʲ⁺ᵐ(M×M), then μ(φ) evaluates on simplices by eʲ -> φ(eʲ×[M]). For lower rank, μ(φ) = 0. This isn't a ring homomorphism, but it is linear. Two key facts to observe about this: for α ∈ Hʲ(M), β ∈ Hᵐ(M), and γ ∈ Hᵏ(M×M),

μ(α × β)= α ⟨β,[M]⟩ and μ((α × 1) ⌣ γ) = α ⌣ μ(γ).

The first condition captures the asymmetry in the definition; the second is an analogue of being a ring homomorphism.

Intuitively, μ sorta "divides by M on the right", so it seems like a good candidate for something that breaks symmetry. Also, since u'' is kinda just built from one copy of [M], maybe μ(u'')=1?

Field Coefficients

So let's compute! Fix a 0-simplex p. μ(u'') evaluates on it like u'' evaluates on p×[M], which by definition of u'' is how the Thom class u evaluates on the pushforward of p×[M] into Hₘ(M×M, M×M\∆) = Hₘ(D(E),S(E)). But under this pushforward, p×[M] is homologous to the fibre over p, and in a way that agrees with the orientation. Hence u evaluates to 1, since the fibre-restrictions of u generate the orientation. And since it evaluates to 1 on any 0-simplex, it's just 1 ∈ H⁰(M).

Quick recap of what's just happened: first, we moved from e(E) to the Thom class u. Now, we've moved to some symmetric u'' in Hᵐ(M×M), and defined a relatively well-behaved, asymmetric map μ that sends u'' -> 1. The aim will be to use μ to control what u'' can be, and hence what e(E) is.

We're going to assume that R is a field for a moment. This is just so that H*(M×M) = H*(M)⊗H*(M). In particular, for any basis {bᵢ} for H*(M), we can write u'' = Σ bᵢ×cᵢ, for some cᵢ.

Recall that (1×bᵢ) ⌣ u'' = (bᵢ×1) ⌣ u''. I didn't just mention that for no reason - let's apply μ to both sides. On the RHS, we get

μ((bᵢ×1) ⌣ u'') = bᵢ ⌣ μ(u'') = bᵢ.

This makes sense - μ is removing the [M] on the right, and since we know u'' is ~just one copy of [M]~ in some sense, the only thing left is the bᵢ on the left.

On the LHS, it's more complicated, since we're adding things to the right. Expanding, we get

Σ (-1)^{dim bⱼ} μ(bⱼ × (bᵢ ⌣ cⱼ)) = bᵢ.

By considering bases, we get that ⟨bᵢ ⌣ cⱼ, [M]⟩ = (-1)^{dim bⱼ} if i=j, and 0 or undefined otherwise.

Remember that e(M) = ∆*(u'') = ∆*(Σ bᵢ×cᵢ) = Σ bᵢ ⌣ cᵢ. So pairing with [M] gives exactly Σ (-1)^{dim bᵢ} = Σ (-1)ᵏ dim Hᵏ(M) = χ(M). Which is exactly what we wanted! The Euler characteristic is the pairing of the Euler class with the fundamental class.

This can seem almost magical, or in less kind words, unmotivated. But maybe here's another way to think of it. For any basis {bᵢ}, we can construct a dual basis b'ᵢ, such that ⟨bᵢ ⌣ b'ⱼ, [M]⟩ = δᵢⱼ is the Kronecker delta. In the notation above, b'ᵢ = (-1)^{dim bᵢ} cᵢ . Then u'' = Σ (-1)^{dim bᵢ} bᵢ × b'ᵢ. The fact that this sum gives u'' is kind of saying that u'' contains all of M, but is symmetric so also contains the dual. That's my intuition anyway. The fact that it really does just give the Euler characteristic is slightly magical, since it relies on the alternating sum showing up perfectly.

PID Coefficients

For a general PID, we fix this by just passing to the field of fractions. χ(M) is the same, and you can follow through our construction to check that e(M) and [M] are too. So once again, ⟨e(M), [M]⟩ = χ(M). For NON-PAID coefficients Kunneth fails, so it wouldn't surprise me if this failed to. But realistically who cares.

General Vector Bundles

Everything we did here was for rank-m bundles. But all of it basically works for arbitrary rank-k bundles, at least for m≥k. In this case, M ∩ s(M) has dimension m-k, so e(E) ∈ Hᵏ(M). Now it can't really be thought of as a number in any meaningful way, but it's still cool that we can think of it as a generalisation of the Euler characteristic!

For k>m, we just say e(E) = 0, since it ought to live in Hᵏ(M)=0. Slightly disappointing.

What Does e(E) Tell Us?

Here's the most often cited example, and our original motivation. If E admits a non-zero section, then e(E) = 0.

Proof: suppose s : M -> E\M is the section, and write i : E\M -> E for the inclusion. Then πis : M -> M is the identity map. π^*e(E) = u' by definition, so (πi)*e(E) = i* (u') = 0, since u' is the restriction of a relative class. So (πis)*e(E) = e(E) = 0.

You might think this is less helpful than the fact we started with, that a closed manifold M has a non-zero vector field iff χ(M;Z) = 0, since saying e(E) = 0 assumes we have a coefficient ring R such that E is R-oriented, whereas the theorem lets us use integral coefficients. However, nothing like that theorem is true for arbitrary vector bundles (or bundles on general CW complexes), when the Euler class is the best we have. So like all good generalisations, it does lose some information.

[samueldays]

E(X) is wrong or intractable, and other problems

Recently, Bryan Caplan wrote a Substack post E(X)>0: An Open Letter to Elon. I have objections to it.

1. My overarching objection to Caplan's vagueness is this question: E(X) on what metrics, according to whom, and for whom?

In his post, Caplan suggests that the United States (and, I would imagine, any developed country), should admit any would-be immigrant "with a positive expected value", in math notation E(X)>0.

Caplan strikes me as doing some "eulering" here, making math-y noises to borrow the clout of mathematics, talking as though an objective calculation of expected value [EV] can decide a matter which involves a great many subjective preferences and (assertions of) human rights. Then he does not perform the calculation.

He implicitly describes a calculation in his point 12:

“Why on Earth don’t we heavily restrict welfare benefits for migrants, so E(X)>0 for far more people?!”

which suggests that his expectation E(X) is calculated in terms of something like "net taxpayers". I don't know what calculation he actually has in mind, because - and this is a major issue - he doesn't specify the calculation that his argument leans on.

I will not criticise the specific "net taxpayers" possibility too harshly, for fear of strawmanning. I will instead mostly criticise him for the under-specification, and I think the general class of Expected Value Calculations consonant with "heavily restrict welfare benefits" are easily gamed while failing to capture the values and preferences involved in opposing mass immigration. The details Caplan has given are exploitable; the details Caplan has not given are a canvas on which each listener is invited to project his preference.

One can imagine a theoretical EV calculation for which it is tautologically true that one would want to import all E(X)>0 foreigners, but the hypothesis of a Platonic object tells one nothing about what that calculation is or how many foreigners are E(X)>0 to import.

I don't want to attack something Caplan didn't say, but he's skipped some important argument steps that I think would merit a great deal of attack if he did say them. For example the steps between the welfare-benefits-based EV calculation, and the "everyone with E(X)>0" assertion, because I can easily imagine people who are net negatives to have around even if they are not net welfare consumers.

Now you’re saying, roughly, that we should only welcome people that definitely have highly positive value. In your words:

If one is operating anywhere near a welfare-benefits-based kind of EV calculation, then it certainly behooves one to take only people who definitely have highly positive value on that calculation, because they may have negative other factors that bring the "true" EV calculation down and so one needs a buffer on the welfare-benefits side to ensure that "true" E(X) > 0.

Which brings me to:

2. The Black Swan of Rotherham.

In 2005, if you had suggested that mass migration would reintroduce slavery to Britain, you would probably be laughed out of the room for absurd paranoid fearmongering. Around 2010-2015 the Rotherham rape ring scandals came to light, first in Rotherham itself and then in Telford and Rochdale and elsewhere, and it gradually turned out that mass migration had reintroduced slavery to Britain.

I speak here of "slavery" in an institutional sense. A single man who is coerced to work and cheated out of his rightful pay may be called a slave, and statistically speaking that probably happened numerous times after Britain abolished slavery in 1833, but an incident does not make an institution. Whereas circa 1980, Britain imported Pakistani rape-gangs numbering in the hundreds of slavers, who took thousands of British girls as sex slaves across dozens of cities over the next few decades.

That was institutional by scale; it was also institutional by policy because when the fathers of enslaved children attempted to recover their children, the police repeatedly ran interference for the slavers, sometimes arresting the slaves, other times arresting the fathers for disturbing the peace, other times saying nothing could be done because of 'racial tensions'. Politicians helped hush up the slaver rings because they were concerned that people noticing the Pakistani slaver rings raping thousands of British girls might lead to the native Brits being racist against Pakistanis.

Problem 2a is the object level: importing slavers to enslave the existing population has very large negative EV for an intuitive EV calculation, but it hardly shows up in the ratio of taxes paid to welfare benefits received by the slavers.

Problem 2b is the future unknowns: considering how unpredicted and unnoticed this was until after the fact, how many other problems of the general type of "Ooops, we reintroduced slavery" may be lurking? India has for example Hindutva vigilantes who murder people for violating the sacredness of cows (at least 9 dead in 2024); imagine importing those from a place where beef is banned to a place whose national dish is the hamburger.

Problem 2c is the lies and coverups: many powerful people thought it was more important to preserve the reputation of immigrants in general and Pakistanis in specific than to stop slave-rape-gangs. This creates a credibility problem when looking for sources to calculate the impact of migration. Caplan seems more honest than average, but still shows some sympathy for coverups in his point 14:

My friend and colleague Tyler Cowen recently advised you to stop publicly defending high-skilled immigration, and “just work behind the scenes.” Maybe he’s right, but I think he underestimates your powers of persuasion.

Problem 2d was the weak response: Britain jailed only a small fraction of the slavers, jail sentences were short, the policemen got a slap on the wrist with one police commissioner stepping down, there should have been a targeted re-education program to break the Pakistani-British culture and enforce assimilation or deportation, there was not, instead there was a stricter control of people saying racist things on the internet, while importing more Pakistanis. This has generated substantial ethnic resentment among the native British population.

How do these figure into the E(X) of mass migration, or the determination of whether it's above 0? God only knows. The error bar on the value of Pakistani immigration looks larger than the value itself to me.

Perhaps Caplan intends to filter out such people from mass migration as part of EV determination. If so, he's handwaving over both a calculation problem and a practical implementation problem.

3. Rights, Privileges, Serfs, and Riots

Some time ago, a fellow on Tumblr bemoaned how difficult it was to move to another country.

I responded that it was trivial to "move to" as in transport myself to another country, which I had recently done that year for my summer vacation, but it was difficult to "move to" as in acquire political power and entitlements in another country for arguably good reason, and challenged the fellow to clarify which he meant. I never got a response.

Bryan Caplan trades on similar ambiguity when he posts cartoon panels such as this:

As written, I deny the claimed "right". I do not think he believes it himself; he would deny my "right" to live and work in his house.

But even interpreted charitably, Caplan is playing games, he is pulling a bait and switch maneuver, he is doing a motte-and-bailey between move as in transport and move as in acquire entitlements, and he is skipping important steps. Once again it's hard to give specific criticisms because I don't know which end of the ambiguity he really intends, so I will attempt to suggest some problems at either end:

If Caplan asserts a right for people to live and work and acquire political power and entitlements in foreign countries, he's arguing for a self-contradictory 'right to privileges', and he's arguing for the 'right' to destroy every small country in the world, in particular the ROC (population 23 million) which is susceptible to the PRC (population 1.4 billion) finding the 0.2% most patriotic loyalists (28 million), sending them into ROC and holding a majority vote to integrate the ROC into the PRC. This strikes me as an obviously wrong conclusion, reductio ad absurdum.

On the other hand, if Caplan asserts a right for people to live and work where they like but only as long as they're a powerless underclass banned from the ER, existing to pay taxes to the native population, and getting deported if they become welfare cases, then a moral problem is that he's advocating something like the return of serfdom.

Spare me the medieval nitpicking, I know the word is not exactly accurate, serfs had more rights than that. Helots is more accurate, but I think most English-speakers have an intuition for "serfs" that they don't have for "helots".

A practical problem following from that is that a large helot class in America would be very hard to keep as helots, when the country has a tradition of democracy, a history of expanding the franchise over time, and riots. The helots would be political tinder waiting to burn.

I ask Caplan: Suppose you get your helot class, American GDP goes up, and then a photogenic helot dies in a way that might have been prevented by welfare, leaving behind a pair of sad orphans. The Democrats spring into action to demand helot welfare and enfranchisement, organizing a helot riot. What do you do?

Of note here is that a helot riot doesn't have to win to wipe out the tax gains from helots. BLM's fiery rioting in 2020 caused at least a billion dollars in damages (as measured by insurance payouts) without abolishing the police.

4. Wage Suppression and Automation

In his proposal to optimize net taxpayers or something like it, Caplan would optimize at the expense of a great many Americans, particularly low-skill Americans who would suffer from extreme wage suppression. America is a nice place to live partly because of the twofold effects of labor scarcity: labor had more bargaining power against capital, and was able to demand better working conditions, and labor scarcity incentivized automation, which freed people up to do other jobs.

Regarding bargaining power: Caplan analogizes America to a corporation, I would respond by analogizing America to a union, and Caplan's proposal to colossal amounts of scab labor intended to break the union to save the CEO some money. Why should the union put up with this?

Regarding automation: If one goes back a few millennia, almost all of humanity worked one of two jobs: producing food or producing clothes. By inventions such as the plow, the loom, the horse collar, the spinning wheel, and the tractor, machines* took almost all these jobs and humans were freed up to do other work like steelmaking and glassblowing. Then machines mostly took those jobs too, the process of automation repeated, and now the average American benefits from machine-power equaling the manual labor of hundreds of humans.

*horses are machines in this context.

Caplan proposes to import large amounts of unskilled migrant labor to do scut work, and doesn't say how this interacts with automation. Concretely:

But look at your own companies. You don’t just hire top engineers and programmers. You hire receptionists, assembly-line workers, janitors, gardeners, and construction workers. With good reason: Otherwise, your top engineers and programmers would have to waste their precious time answering Tesla’s phones, assembling its cars, cleaning its toilets, mowing its lawns, and pouring its concrete.

Many of these look automatable, particularly the assembly line, which is already well into the process.

Once again I'm frowning at an ambiguous gap in Caplan's proposal, where I can imagine several possible views but criticising any particular one is something of a strawman because Caplan hasn't committed to it.

Does he imagine that automation will decline naturally as a result of the cheap labor? Because that sounds like trading long-term benefit for short-term net taxpayer count.

Does he imagine that automation should be held back? Same but worse.

Does he imagine that automation can't take these jobs any time soon? Sounds fake.

Does he imagine that automation will take these jobs soon but then America can just send all the migrants back once they're out of work, and wash its hands of them? Sounds unlikely and impractical.

Does he imagine that automation will take these jobs soon but low-skill migrants will simply retrain and develop skills for new jobs? Sounds wrong by construction.

And once more I ask: E(X) for whom? Caplan says America "needs" mass unskilled immigration, but large numbers of Americans would suffer from this.

5. Social Contracts

I am generally skeptical of social contract theorists as trying to claim too many specifics from too little evidence. Hobbes in particular was lying about the war of all against all. Even so I am sympathetic to a minimal account which goes something like this:

I (we in general) will give up my natural right to take amateur vengeance on and extract compensation from anyone who has wronged me. In exchange, the State promises to perform vengeance by a professional enforcer class in a way that's predictable and reliable and won't lead to blood feuds, and to pay me compensation from the collective compensation fund.

The modern American state has enforced a broadly similar new social contract which I might describe thus:

I (we in general) will give up my natural right of freedom of association, to decide which countrymen I will admit or exclude to my business, will hire or not. In exchange, the State promises to perform exclusion at the country border by a professional enforcer class, lowering friction internally and lowering costs of maintaining an exclusion around the collective American identity.

I have complaints and nitpicks about this, but I can see a meaningful value proposition in it.

When Caplan argues for open borders, he is arguing to take away what Americans received in that second contract. Again there's a gap where I don't know what Caplan believes, so I will comment on the two likely interpretations I can think of:

Is Caplan trying to tear up the new social contract in its entirety? Then I want him to bite the bullet and say out loud that he wants to restore freedom of association and overturn the Civil Rights Act.

Is Caplan trying to take away the benefit of the new social contract and give nothing in return? Then many Americans might reasonably want him jailed for conspiracy against rights or similar offense.

6. Policy Change Friction Around Humans

Bryan Caplan argues against "safetyism" and makes the analogy that just as an investor should want to make every investment with E(X)>0, a country should want to admit every migrant with E(X)>0. He admits "While there are obviously major differences between running a corporation and running a government", then ignores this obvious point as though it made no difference.

I want to highlight a particular point of dis-analogy: it is much easier, practically and morally and legally, for an investor to ditch a million-dollar investment at the first sign of it turning bad than it is for a country to ditch a million migrants at the first sign of them turning bad.

Migration is difficult and costly to reverse, and most countries have significant political constituencies opposed to that reversal, backed by international organizations such as Amnesty. You'd be hard pressed to find a single elected official with a strong opinion that Jane Doe must stay invested in Acme Corp, for most values of Jane and Acme.

Under these circumstances, some form of safetyism is correct: the threshold should not be E(X) > 0, but E(X) > Cost(Deportation), estimated around eighty thousand dollars per person by the American Immigration Council. Which is probably biased, but the sources I can find for estimating this number seem to amount to either AIC knockoffs or else Trump fanatics insisting "deportation will pay for itself".

7. Assume a spherical cow in a vacuum...

The "spherical cow" is originally a physics joke about greatly simplified modeling that discards many features of the object under consideration to simplify calculations. In physics, this is often good enough because the features under consideration, e.g. "mass of an object", range between the 10^-21 grams of an atom and the 10^33 grams of the Sun, so one can afford to round off (ha) a great deal and still be close enough on an exponential scale. The Earth is approximately spherical even if a mountain rises a few miles above sea level, that's very little compared to the circumference of thousands of miles.

Outside of physics, the spherical cow approach is less applicable.

I would like to see Bryan Caplan distinguish more sharply between the realistic policy changes he's pushing for on the margin, and the spherical cow policies where he imagines a friendly Supreme Dictator who can copy UAE policies to the US. I would also like to see more awareness from Caplan of when he is assuming a spherical immigrant who can be rolled across a frictionless border in a political vacuum. Caplan's talk of E(X)>0 is spherical talk, assuming a simple calculation. Caplan's neglect of ethnic resentment in the implied EV calculation is discarding important features. Caplan's implication of ditching migrants if their recalculated E(X)<0 is handwaving over a great many issues and costs.

Another oversimplification of Caplan's is disregarding the potential political power of migrants, or its near relative, the willingness of the Democrat Party to clientize migrants for political power.

[nhp-researcher180]

//INITIATING SIGNAL //SetRecip.Public //AUDIO BEGINS

Wait, so how do I know when it's on? The red light? Ooooooh, right! Thanks Euler.

Ok! Hello Omninet! I'm Leila Marris-Orion, an NHP researcher with Union Science. I'm currently taking an indefinite leave of absence out to the Dawnline Shore, hoping to learn more about my family history.

I'm joined by my close friends and research buddies, Adela, Euler, and Dio. I'll let them introduce themselves.

Welcome. I am called Euler. I am an [ATHENA]-class NHP, licensed from Smith-Shimano Corpro for use by Miss. Leila Marris-Orion. I have greatly enjoyed my working relationship with Miss. Marris-Orion, and I anticipate further intellectual stimulation from her current foray into historical records. Adela, if you would proceed?

Oh, me? Yes! Yes, alright, my turn! So, hi, I'm Adela. I was the first project that Leila worked on, which I believe makes me... an [ENLIGHTENMENT]-class, no? Leila is a wonderful friend and I'm just so happy that we're getting to travel out and see the galaxy together! I know we're all going to become great friends, so please don't hesitate to reach out and say hi! Ok, umm, I guess it's Dio's turn now? Dio, please be nice...

Hello, hello, hello! I am Diogenes, but you can all call me Dio. I'm a [SISYPHUS]-class NHP, don't worry about it. I work with these other brilliant minds, and I promise that you can trust absolutely any and all links and files I will be sending you the moment our dear Head Researcher is gracious enough to give me unsupervised access to this omninet connection. Swear on my heart, which as an NHP I definitely have!

Absolutely not. Anyway, let's see... Oh, I'm licensed for several SSC mechanized chassis, specifically the Swallowtail and the Deathshead, but I can't actually use them at the moment. Something wrong with the licensing code? Whenever I try to print anything it just makes a- Bone Gecko! We are not calling it a 'Bone Gecko'! But yes, it spits out what looks like an eyeless, skinless lizard. It's honestly pretty unnerving, so if anyone knows about like, printer coding and such... help?

How do I turn it off? Is it this sw- //AUDIO ENDS

[veiledinviolet]

Spherical hex¹ grid based on a class II Goldberg polyhedron, generated using the inverse of Lee's conformal projection of the sphere upon a regular icosahedron [1, pp. 62–65]. When the camera is close to the surface, stereographic projection is applied to flatten the surface.

¹ All faces are hexagonal except for twelve pentagons required by the Euler characteristic of the 2-sphere.

[1] L.P. Lee, “Conformal Projections Based on Elliptic Function”, url: https://archive.org/details/conformalproject0000leel

[bandcampsnoop]

11/10/24.

Thanks to Bandcamp for the article "When Germany Surfed the New Wave" by Jim Allen. Die Doraus & Die Marinas is the least traditional sounding new wave band of the list. Listening to this was like finding a key piece to a puzzle.

I'd always wondered what influenced Krischan (founder of Frischluft! Tonträger). Previously, we posted about Susan Aviles release, but now comes an artist from Germany that probably was part of Krischan's childhood. Andreas Dorau (Berlin, Germany) started Die Doraus & Die Marinas in the early 1980s as a high school project and scored a hit with "Fred vom Jupiter". The Bandcamp page has the first album up for streaming, but I thought I'd go with the 2nd because, while it too is sold out at the source (Bureau B...I looked for copies at Tapete Records), there are reasonable copies available on Discogs.

I'll throw out a few Frischluft! Tonträger bands since it's been a while since I thrown out these comparisons - Honigritter, Foxi Euler Syndikat, and Momus. Really, just enjoy this unique sound that is on a different level from the classic German New Wave sound.

[mathysphere]

Hiya! since you seem quite knowledgeable about this sort of thing, do you happen to know a good place to start when learning how to cross-stitch? I've never tried it before (nor have I done a lot of needlework) but I've been drawn to it due to the similarities it shares with pixel art. I have looked up starter kits and the like but they tend to be focused on topics that are rather... eh. boring to me unfortunately.

Oh heck yea! Yeah, lemme grab some links and info--

Luckily, cross-stitch is really easy to do-- you get some special fabric that's woven so it makes a grid, a blunt needle, a pattern, and whatever thread colors the pattern needs, and then you just stitch a whole bunch of X's and you're done!

The best fabric to start with would be called '14-count Aida'; a lot of big-box craft stores carry it, plus smaller embroidery stores should too. '14-count' is the size of the grid, and means every square inch of the cloth is woven to make a 14 by 14 grid, and Aida is a common variety. It's nice and stiff, so you may not even need to put it in a hoop, though you can still use one if you'd like.

If you get 14-count fabric, get a Size 24 Tapestry Needle. I wouldn't use a sharp needle, though you could if you have to-- a blunt one like a tapestry needle will be a lot kinder on your hands!

I'd recommend starting with a small, simple pattern in a design that you really like. Like, say, something like these would be pretty good to start:

[All You Need is Love by JuicyArtByDarina][Euler's Identity by StitchForge]

...while these might end up being too ambitious:

[Saturn V Rocket by SpiritLineDesigns] [The Elements According to Relative Abundance, also by SpiritLineDesigns apparently? Man, that's some good range]

That's just a recommendation, though-- you know your own patience for long projects better than I do, so if you see something you love, go for it!

No matter what pattern you get, it should always have a list of thread colors in it: the standard type is DMC embroidery floss, which you can also get at embroidery stores, some of the big-box craft stores, and also online. The pattern should also say how big the design will be, but when you buy your fabric be sure to get something with some extra room. I typically aim for three or four inches bigger than the design, just to be extra safe.

Once you've got those-- plus a pair of scissors, and also a hoop, if you'd like one-- then you're ready to stitch! For that part, I'll defer to Peacock and Fig and her video tutorials, since I reckon it's easier to follow that part if you can see just how the stitches actually happen.

The stitching itself is super easy-- it's all just makin' X's.

Hope this helps! ✌️

[andmaybegayer]

Last Monday of the Week 2024-09-09

It was Bandcamp Friday last friday so those go up front:

Listening: First up God's Country by Chat Pile which I cannot remember why I found it but is extremely noisy metal. Blunt, to the point.

A Dog A Deer A Seal by Hit Bargain has featured on a Monday before but now I actually bought the album, here's the title track. Very modern-style rock.

The Envoy by Gavlina Rayna Russom, her first solo album after leaving LCD Soundsystem, soundscape electronic and voice based on The Left Hand of Darkness.

Cyberspace Database by Fornax Void, sort of the evil mirror dimension version of Businessfunk which now that I think about it has never come up on here? Incredible. Anyway a sort of fantasy soundscape for a cyberspace environment. Huge album too, like 50 songs.

Alright back to our regularly scheduled programming

Watching: New! Dynamo! Dreams! from Ian Hubert,

This is the first time we're back at what was started in Ep1 so there is in fact some kind of plot, although the series remains supremely vibes-based. Excellent extensive rampaging robotaxi sequence. Dynamo Dreams is mostly about the props and atmosphere.

Reading: Finished The Tyrant Baru Cormorant after a brief lapse in reading. It's got problems! I really like a lot of it but it keeps on pausing the book to tell us about the blogs and wikipedia pages Seth Dickinson has been reading. Have you heard of Euler's Formula? What about the evolutionary argument for homosexuality? Yo did you hear about plants that avert scurvy? I am going to tell you about these in a boringly straightforward didactic way for the next 200 pages while occasionally progressing the plot.

Things I like: There is a very real sense of trying to balance everyone's schemes. Despite the fantasy communication techniques there's a lot of hidden information that gets used well, and the vibe of Baru's person-blindness is felt heavily but by anyone trying to make a plan run smoothly. Baru gets so messed up over the course of these books and it works to demonstrate how she's taking herself to pieces to make this work.

When Baru is actually doing her thing you can ignore the sort of awkward frankensteined nature of the world around her, but when explicit attention is drawn to it it quickly starts to fall apart, and the introduction of the Lightning Cult has me worried that this will only get more severe. Seth Dickinson seemingly cannot do Greg Egan, but they can do the rest of the Baru Cormorant pretty solidly.

Also more Skin Horse. Only to about 2012 so far.

Making: Some abortive soldering projects, hampered by my lack of good quality solder, I only have pure tin/copper blend in pretty coarse gauge, you want something finer and with a little silver for this.

Started working through some Control Theory lectures to brush up on that because I'm also working through some ROS tutorials, I want to make my robotics projects more broadly useful and integrating ROS/microROS might go a long way to helping that.

Playing: Ash and Adam's Gobsmacked, an early access roguelite arena combat goblin game that came out recently. Big focus on mobility, you can run and jump and glide around levels while taking out a bunch of robots. Still pretty finicky, controls are on the slippier side so on the plus side "high skill ceiling" but also "uncontrollable".

A touch of Minecraft, poking at Modrinth mod manager rather than my usual manual mod installation because I was unwilling to spend time figuring out how some of the new mods interact. I really do want to stick with a world long enough to do something interesting sometime.

Tools and Equipment: I set up Tailscale to link my server with my devices when they're in the field, using a headscale node running on my cloud server. It's really nice once you have all the basics sorted out, works smoothly. Worth a look if you want to securely access your home network while on the go.

[naked-bee-art]

Date: Jul 8, 2022

Caption: Euler, Euclid, & Erdős (famous mathematicians whose names begin with E)

I like to pretend their ashes are in there.

These ceramic jars came from my mom, who thinks they belonged to a great-aunt at one point. They were originally painted with white with green and yellow details, but that didn't go very well with the goth library aesthetic I was trying to cultivate in this room. The labels used to say things like "tea" and "coffee" and maybe "sugar" was the third one? I've forgotten. Anyway, I wanted new labels that would still be readable when everything was spray-painted black. So i cut the names out of some thick card stock and then glued them onto the canisters. When you've got a laser cutter, every craft project suddenly seems to need cutting!