sniffnoy: (Golden Apple)
So, last summer, I learned from HK about the horeshoe crabs in New York City. She had -- I think, I forget the exact story -- been out exploring the distant parts of the city, and had come across Broad Channel, and she saw the horseshoe crabs on the beach there, and she gathered some people to go see them again. So she and I and some other friends went out to see them.

See, around May and June, it's horsehsoe crab mating season. I had no idea that horseshoe crabs even lived in the waters around here, but they do. And during the mating season they come up into the shallows to mate. But, you can't see them at just any time. My understanding is, they come out mostly when the tide is highest. You want to go at high tide, but not just at high tide, you want to go at high tide around a spring tide.

Anyway, last year's trip out to Broad Channel and we saw lots of horseshoe crabs. Although, at first, we made the mistake of heading north to the wildlife refuge, rather than south to the beach. The wildlife refuge did not have horseshoe crabs. What did have was lots and lots of mosquitoes. We all got really bitten before deciding to turn back and head to the beach instead.

I wanted to try the same thing again this year -- but maybe we didn't need to go all the way out to Broad Channel? The city has a webpage listing where horseshoe crabs can be seen. None of these locations are all that convenient, but Kaiser Park, in Coney Island, seemed like the most overall accessible.

So a few friends and I tried it out. Unfortunately, we made a bit of an error in timing. Yes, we went near a spring tide... but I hadn't checked the detailed tide charts, and it was actually near low tide when we went. Oops. We ended up mostly only seeing dead horseshoe crabs -- although we did find one live one, buried in the sand. Once the tide came in, it came out, and we got to see it swimming around! Still, just one is a bit disappointing. We did see a really cool bird which I have since learned was a yellow-crowned night heron!

We (well, not the same group) tried again two weeks later, same location. In this case we knew it would be around low tide, but, well, Kate insisted it be then (and then didn't show up...). The result was basically the same. A bunch of dead horseshoe crabs, only one living one. Well -- I spotted one living one buried in the sand before the others arrived, but once they showed up and the tide had come in a bit, I couldn't find it anymore. I guess it swam away? We did not see any herons this time, which was a bit disappointing, as Kippy (a birder) had come hoping to see one.

We did see lots of fiddler crabs! They're difficult to see, because in addition to being small, you can only see them from some distance -- get anywhere near them and they'll scuttle back into their holes (their holes covered that section of the beach). But move away and they'll come out and wave their claws up and down!

Anyway, I decided to make one final attempt this past Sunday. This time, the tides should be right. (And for whatever reason, this spring tide was projected to be especially high, higher than the other recent ones. I wonder why?) HK suggested we try Calvert Vaux Park, across the creek from Coney Island rather than on it, although, like Kate, she then didn't show up! But this time it worked -- we did find horseshoe crabs! And also an egret, and a fiddler crab that wasn't so good at hiding, and also lots of cats? By the entrance near 27th Ave I encountered 3 cats. Huh.

The park seems to consist of 3 peninsulas (do all 3 count as the same park? I think the south one is separate), but the three seem pretty split off from one another -- I think they all have separate entrances. We didn't try the northern peninsula because we just kind of missed it, but it looks like there's a beach there we could have tried. We did try the cove that lies between the northern and central peninsulas, since there was a beach there, but it was a dump, and also had no horseshoe crabs. (It also required taking some paths through the brush that were not the easiest.)

Ultimately the place we found horseshoe crabs was one of the beaches on the south side of the central peninsula. This is basically where we found all the animals, except the cats! There weren't as many as at Broad Channel last year -- possibly still an issue of timing, we were a bit after high tide and a few days before the highest tides -- but still a good number. One of them was tagged! So, yes, I reported it afterward.

We didn't try the southern peninsula (which I think is actually not part of the park). We were kind of tired from the heat at that point, and I'm not sure that the southern peninsula has any beaches, or how to get to them if so.

June is ending now so I think that'll be it for this year. But I'll go look for them again next year, I figure, and that time I'll be more considerate of just when high tide is...

(Meanwhile, the goats arrive on July 12th!)
sniffnoy: (Chu-Chu Zig)
Huh, I never posted about this, did I? So Kappa finally moved in back in January, and with her, many boxes of books. This month she finally unpacked most of those boxes. (There's still two remaining, to be handled at some undetermined future date.) There had been preparations earlier -- she and Geoff had gotten new bookshelves to replace Geoff's old bookshelves in the main room, that had been kind of falling apart for years. One of the old bookshelves was thrown out, the other I ended up taking into my room.

But now finally it was happening, the unpacking of the books. The idea was that Kappa and Geoff's book collections would be merged, with the new books going onto what had been Geoff's bookshelves -- whether the ones in the main room or the ones in their room. But -- as was anticipated -- in merging the collections, many duplicates were found. Where do you think these went?

That's right. They went on the giveaway shelves.

Now, which shelves those are had to change. The sheer number of books unpacked was such that Geoff and Kappa ended up taking over all the shelves along the west wall -- my books got kicked out to the bookshelf that was now in my room. So did my board games on the bookshelf in the corner. Now the west wall is all Geoff and Kappa's books, and the east wall is all books being given away. (Due to the change, I decided to finally label the shelves of free books.)

It wasn't just duplicates from between Geoff and Kappa that got put on the giveaway shelves, though. On unpacking her books, Kappa found various ones that she actually just didn't want anymore, so those ended up on the giveaway shelves too. Meanwhile, in preparation for the unpacking, Geoff had had to actually organize his book collection... and in doing so, turned up some duplicates purely within it! (And some books he decided he didn't want, either.) So that's more for the free shelves!

Of course, putting books on the free shelves doesn't actually immediately save any space -- it only does so once someone takes them. Maybe once enough books are taken, I'll be able to move my own books back out into the main room. But that'll take a while.

Prior to the deduplication, the number of books on the giveaway shelves sat at 137. After, it sat at... I don't know, I didn't even try counting. See, among the books being given away due to inter-spouse duplication was a nearly-complete set of Discworld books. These got placed on the top shelf of the tall bookcase behind the chair, double-stacked. I didn't want to deal with the inconvenience of counting that, so I didn't.

Now of course, these new books are going rather faster than the old ones. After all, the books still on the shelves before this happened were subject to adverse selection; of the books we were giving away, these were the 137 that people wanted the least. By contrast, the bulk of the ones being newly added were because they were books that two distinct people had bought! (Also, it's a lot of fantasy and science fiction, whereas the 137 that were there prior were heavy on abstruse philosophy.)

In particular, those Discworld books lasted on the shelves less than a week before Zvi and Laura took the collection in its entirety. After that, counting became practical. The new total -- noting that when Zvi and Laura took the Discworld books, that same night some other people took other books, so this shouldn't be taken as "the grand total minus Discworld" -- was 222.

Two weeks later we're down to 212. And two of the books taken have been ones that were already on the shelves prior to this -- perhaps having better books to give away encourages people to explore the other ones more thoroughly? We'll see how low it goes!
sniffnoy: (Kirby)
So, here's something that had bugged me for a long time. Jacobsthal multiplication is associative, but the proof of this is unsatisfying. Normally the way you prove an operation is associative is that you show that a multiary version of it makes sense; you don't just literally show that a(bc)=(ab)c, you show that abc makes sense on its own as a ternary thing (or with even more operands) and that a(bc) and (ab)c are both equal to this.

But for Jacobsthal multiplication, the only proof I knew of its associativity was to show that a×(b×c) was equal to (a×b)×c; I wasn't able to find an interpretation of a×b×c on its own.

Now, some years ago, Paolo Lipparini found an order-theoretic interpretation of the Jacobsthal product, but it wasn't one that led to an immediate proof of associativity. But based on an observation of Isa Vialard, I think I finally have one! It's a little awkward, but of course it is; it's Jacobsthal multiplication, after all.

Let's say A and B are WPOs, and let's consider the lexicographic product A·B (note: B here is the coordinate that's compared first). Then o(A)o(B) ≤ o(A·B) ≤ o(A)×o(B).

But, actually, we can say more. Suppose B has k maximal elements, so that o(B) has finite part greater than or equal to k. Then, Vialard noted, we have o(A·B) = o(A)·(o(B)-k) + o(B)×k.

In particular, if o(B) has finite part equal to its number of maximal elements, then o(A·B)=o(A)×o(B). (In fact, as long as A has multiple distinct powers of ω in its Cantor normal form, the converse also holds.)

Btw, the finite part of o(B) is equal to the number of elements of B that have only finitely many elements above them. We could call such elements "almost maximal". So B satisfies this condition iff every almost maximal element is maximal.

Anyway, let's call a WPO that satisfies this condition "flat-topped". Note that flat-topped WPOs are closed under disjoint union, under Cartesian product, and, yes, under lexicographic product! So we can interpret α×β as the type of A·B for any flat-topped A,B with o(A)=α and o(B)=β; and we can do similarly with α×β×γ, etc. So this does it! We've interpreted multiary × and proved associativity nicely. Hooray!

EDIT 5/5: Actually, on writing to Lipparini about this, he pointed out to me that another observation of Vialard provides an even easier answer, namely, without needing the flat-topped condition, one has w(A·B)=w(A)×w(B), where w is width. I hadn't noticed this because, well, I've never really paid a lot of attention to widths, tbh.
sniffnoy: (SMPTE)
I'm trying to just write down more old thoughts, and you know what, I haven't written this one down.

The Python programming language has an interesting construct I haven't seen elsewhere: while/else. A while block with an else block following. Now this is terribly misnamed; based on the name you'd think that, like, the else block executes if the while block ends up being skipped, like if/else, but that's not what it is. The way it actually works that the else block executes after the while block *unless* you broke out of the while block.

This is misnamed -- it should be called something like while/coda or while/then -- but I think this is a nice enough construct that it should be standard! Yes, obviously just like labeled breaks this can be replicated by setting a flag or whatever (and often is), but it would still be nicer to just have it directly.

In particular, it really bugs me that Rust in particular doesn't have it, and here's why. Rust has this neat feature where any block can return a value (indeed, *does* return a value -- if it doesn't seem to, it's returning a value of unit type), and in a loop block, when you break, you can specify a value, and then the loop returns a value! That's pretty neat!

But, while you can do this from a loop block, you can't do this (with a non-unit type) from a while block! Why not? Because if the while block concludes naturally, what value would it return? The only sensible thing is to say that the while block returns unit type -- since it must return a consistent type -- but that means it can't return a non-unit value, you can't break out of it with a value. But if you could add a coda to a while block, this wouldn't be a problem anymore!

(Obviously all of this also implies to other types of looping blocks that naturally terminate.)

This lack of consistency really bugs me, and it's something that people have repeatedly suggested adding, but it's never been added because they haven't come up with a syntax for it that is both clear and not too breaking. (Apparently breaking out of a loop with a value was a somewhat late addition to Rust, which is why this wasn't gotten right at the beginning.) Boo! Anyway yeah it bugs me.
sniffnoy: (Golden Apple)
Why didn't they call it the Super Switch??
sniffnoy: (Golden Apple)
This is a problem that has come up several times recently, where I have mentioned the Freemasons to someone, and it turned out they'd never heard of them, and I had to try to explain it, only to find that it's hard to actually explain! But I think I've got it: The Freemasons are an organization about pretending they're about something. :P

...OK, sure, they're not exactly an organization per se, but still...
sniffnoy: (Chu-Chu Zig)
There's a question that I occasionally see asked on the internet -- what's the big deal about Eliezer Yudkowsky? Oh, he suggested that artificial intelligence might be dangerous? That's not original! People have been talking about robot uprisings for as long as they've been talking about robots. Why is this Yudkowsky guy getting credit for it?

And I think this question is interesting for what it reveals about how much things have changed since, oh, 20 years ago, let's say. Yes, people have been talking about robot uprisings for as long as they've been talking about robots. But the question you have to ask is, who was saying artificial intelligence might be dangerous? Because back then, the general assumption among... technical people, science fans, transhumanists, etc -- the general assumption among that crowd was that of course artificial intelligence would be a good thing, just like technology generally is a good thing; and that if you thought otherwise, if you thought actually it might be dangerous, you were a backwards luddite. (This is roughly the view expressed now by the "e/acc" crowd.) Moreover, the arguments for it being dangerous were largely based on anthropomorphization. Yes, there were people talking about the dangers of artificial intelligence -- but the sort of person who might go work on artificial intelligence, or become any sort of expert on it, would never listen to that sort of person, and with good reason; they generally weren't worth listening to. They really were often anti-technology luddites, and their arguments were generally pretty bad.

(Frankly, I'm not even the best person to be relating this -- someone older, who had an adult's view of all this, would be better. I was prompted to finally write this post due to a conversation with an acquaintance today who occasionally attends OBNYC, but who also attends New York skeptics' meetings, and mentioned how the latter is a very different, substantially older crowd. I like to point out sometimes that a lot of Yudkowsky's writing is fairly continuous with older writings on rationality by people like Feynman, Sagan, etc; but this is only somewhat true, as evidenced by this split!)

So the significance of Eliezer Yudkowsky -- I mean, certainly not the only significance, likely not the main signifiance, but the significance for the purposes of this question -- isn't that he proposed that artificial intelligence could be dangerous; it's that he A. convinced of the risks of artificial intelligence the sort of people who were inclined to dismiss such, and B. did so by pointing out that the arguments for AI being safe are in fact based on anthropomorphization and rebutting these arguments in detail.

But it seems that people just coming to this conversation now often don't realize all this! They don't think there's anything usual about technical people, transhumanists, etc., considering AI to be dangerous -- they think of it as continuous with earlier arguments about robot uprisings, rather than considering those earlier arguments as something such people would have dismissed.

Somewhat similarly -- and this is something Yudkowsky himself has often remarked upon -- you get people who've never bothered to actually read Yudkowsky calling him an anti-progress luddite, which is of course not remotely correct. Indeed, the fact that he was not a luddite, and had credibility as a transhumanist, lent credibility to his eventual turn against artificial intelligence, made it influential rather than something reflexively dismissed! The thing (well, one of the things, but the relevant thing) that was unusual about Yudkowsky (but, thanks to him, is not unusual anymore) was his stance of being pro-technology and pro-progress but against the development of artificial intelligence specifically. But some people who comment on things based only on impressions assume that it still must be the case that anyone opposed to artificial intelligence must be a luddite. This is basically the opposite way one might miss Yudkowsky's significance; instead of failing to realize that the past was not like the present, one could fail to realize that the present is no longer like the past. (Well, OK, that's not quite right, because these people aren't so much assuming that the present is like the past, as they are assuming that the present is like the obvious thing you expect. It isn't!)

(Things are also different from 15 years ago in that it seems like transhumanism as it used to exist seems to be much reduced, because discussion of artificial intelligence has largely subsumed it all! Not entirely, but to a pretty good extent.)

Anyway, yeah, some context for those confused about such things...

-Harry
sniffnoy: (Golden Apple)
Have you all read about the geology on Venus? How there's no real cracks in the crust, no distinct tectonic plates, just one big ("stagnant") lid? How this doesn't allow heat to escape slowly like it does on Earth? How it's hypothesized that every half a billion years or so, the trapped heat finally breaks out in the form of giant volcanic eruptions all over the entire planet, where possibly the planetary entire surface just kind of melts, erasing all record of what came before?

Yeah. You know what they call this kind of worldwide catastrophe?

A global resurfacing event.

That really needs a better name.

ADDENDUM: OK, I'll kick things off and suggest "omnivolcano apocalypse". Or, if we want to to tone things down just a little, perhaps "omnivolcanic catastrophe"?
sniffnoy: (Chu-Chu Zig)
Yes solutions have been up for a while but I've been busy! Anyway, Mystery Hunt this year!

I once again went up to Boston for Mystery Hunt this year. Still on Plant.

The Hunt had an interesting structure this year. The first round had five different metapuzzles -- four ordinary metas and one supermeta, although I believe it was designed to be solvable without solving all of the other metas first -- and solving each one of the five unlocked a different subsequent round. Each of these later rounds had their own metapuzzle or metapuzzles, and solving the metapuzzles for all rounds unlocked the runaround. We made it to all five of the later rounds, but didn't solve any of their metas.

The other way the structure of the hunt was unusual was that, rather than solves unlocking specific puzzles, instead they separately "discovered" puzzles and gave you keys. You see, when you discovered a puzzle, it would still be locked -- they'd give you a name and a minimal description that they considered spoiler-free -- and you had to spend a key to unlock it. The idea was that this would allow teams to focus their efforts on the sort of puzzle that they prefer. I think this was a positive effect, but there was also a negative one. You know how clues and free solves require strong team leadership to make sure they get used? I think keys have a similar sort of thing going on. Without strong leadership a team might just not spend their keys and not unlock enough puzzles!

Also, one of the subsequent rounds this year was a fish round, and, I have mixed feelings about that. I mean, it's nice to have some easy puzzles, but they're so long, and it's kind of exhausting to solve so many puzzles and still not unlock the meta. If you want to have a fish round, maybe break it up into several rounds, rather than making it one long one?

The other reason I'm maybe a bit cool on fish rounds is that if there's always easy puzzles available, it becomes less appealing to sit down with the harder ones and stare at them and try lots of things. It's distracting! At least it's not like last year where, AIUI, the fish round barely contributed to progression... here at least it was a necessary part of making it to the runaround. But, yeah, if you want to have easier puzzles, maybe just mix them in with the other puzzles as an occasional "here have an easy one" rather than sticking them all in one round to make sure one's always available?

I'm actually a little unsure I helped all that much this year! I mean I definitely contributed to the team but I feel like I was often so distracted... well, I guess I'll go over the individual puzzles and see!

(Also: Li-Mei was on the writing team this year! Hi Li-Mei! :) )
Cut for spoilers )
OK and I think that's all I have to say about this year!

Next year I think I might go to the JMM because it'll be in Washington DC, so I'll probably stay home for Mystery Hunt and solve remotely. But we'll see...

-Harry
sniffnoy: (SMPTE)
So today I learned that the entire time I've been playing Space Alert, I've been getting a rule wrong. It's something that doesn't affect all that much, I think, and we're likely going to just keep playing the same way? I dunno.

It's just funny because like -- I'm like the guy who knows all the rules and interactions in Space Alert (well, in the base game; the expansion is way too complicated for that, also I don't even have it), but, uh, this one rule I had wrong because there was never any reason to look it up!

And that is: Rockets. When Oren taught me the game, he said, the rockets are smart rockets, so if there's nothing within range for them to hit they don't explode, they just circle back and return to the bomb bay. Only if there's something to target are they used up.

Turns out that's not what the rules say! The rules say that when you shoot a rocket you do in fact use up that rocket, no matter what; so you can only shoot rockets 3 times per game, that's it.

...hm, I guess that does maybe affect that a bit, because it means you can't to the same extent spam rockets as a vacuum action. But then, 3 is still quite a bit -- many games rockets don't get used at all -- so uh actually you still likely can? Are you really going to fire more than 3 rockets all that often?

So yeah I dunno. Probably not going to change it, everyone I know uses the "smart rockets" rule -- usually because I taught them. :P I dunno where Oren got the rule from! But yeah.
sniffnoy: (Golden Apple)
Let's start with the less surprising one: A formula for primes (thanks to Jeff Lagarias for bringing this to my attention).

Everybody know there's no "formula for the prime numbers", right? Most mathematicians would I think view the question of a "formula for primes" as a bit silly -- the obsession with "closed-form formulas" is for high-schoolers, and what constitutes a "closed-form formula" isn't well-defined anyway, outside of very specific contexts. Most mathematical functions don't have formulas; high-schoolers always ask about a formula for this, a formula for that; you have to just tell them, look, stop worrying about a "formula" for it, it is what it's defined to be and that's it, OK? Math isn't about formulas.

Well, uh, it turns out that one of the premises above is wrong. And in fact this information isn't new -- the article I linked cites a result of Mazzanti from 2002 (!) that any Kalmár elementary function can be expressed in terms of addition, multiplication, exponentiation, monus (minus, but saturating if you try to go below 0), and floored division.

That's a huge class of functions -- implying that, contrary to the point of view expressed above, most number theoretic functions like divisor count, sum of divisors, etc, do have "formulas"! And yes, that includes the n'th prime function. The paper I linked isn't proving the existence of such a formula, which follows from the Mazzanti result; rather it's just trying to optimize it a bit.

Now all this assumes you're OK with monus and floored division. But:
1. I think very little of the power here is coming from monus; I think pretty much every instance of monus used to construct pn (and lots of these other number-theoretic functions) could just be minus instead, I don't think they ever really perform a subtraction without knowing that the difference is nonnegative.
2. If you're not OK with monus, but you are OK with max or min, well, then monus is just max{n-m,0}
3. If you're not OK with that, well, we're allowing exponentiation, so you can write |n-m| as √((n-m)²) and then say it's (|n-m|+(n-m))/2 (this is a little questionable in my book but hey it would satisfy a high schooler :P )
4. As for floored division, unlike monus it really is quite essential to making this work -- evidently floored division is way more powerful than I ever realized! But like, would a high schooler be OK with floors in their formula? I'm pretty sure the answer is yes. :P

So, wow, that sure overturned some things I thought! But wait, here's something crazier:
"Any function I can actually write down is measurable, right?"


Like, every mathematician knows, that if you can explicitly write down a function, you don't really have to worry about whether it's measurable right? Not only do all the things one might normally do to make a function preserve measurability, but it's actually impossible to constructively prove the existence of a non-measurable set (and therefore function). So if you can write down a function explicitly, you can't prove it's non-measurable.

Somehow it never occurred to me before there's a loophole there: No, it can't be provably non-measurable... but maybe it's not provably measurable. And that's exactly what this paper claims to do -- it writes down an explicit function whose measurability is independent of ZFC. I haven't yet taken a good look at this one to understand how it does it, but I should really sit down sometime and do that.

But, even having looked inside a bit, I was still confused -- how could it be the case that it wouldn't be provably measurable, given that the definition wasn't anything too crazy, and all the common ways of making functions preserve measurability? I think the answer here is that while the infimum or supremum of a countable collection of measurable functions is again measurable, there isn't any principle that says that if f:X×Y→R is measurable, then so is x↦infy∈Y f(x,y). Huh! I would never have guessed that that operation need not (provably) preserve measurability. Crazy!

Sometimes, the right formula can do a lot!
sniffnoy: (SMPTE)
So, some time ago I noticed -- I don't recall how, it's very odd this ever came up -- that my computer supported some nonstandard flag emoji; specifically, US state flags. Today I decided to explore this some more to see just what it supports.

Now I'm sure somewhere on the system is a file listing exactly what it has, or that this is documented somewhere online, but I don't know where I'd find either of those, so I just did some experimentation. I'm assuming that it's the OS (on some level) that's responsible for handling this? If it's the desktop environment, well, I'm running MATE 1.26.0; maybe that's what's doing it rather than Linux Mint per se. Regardless, I'm assuming it's some sort of lower-level feature of the system.

Now, likely not everyone reading this is familiar with how flags in Unicode work, so let me summarize that briefly. Unicode, for obvious reasons, didn't want to add national flags as characters, so instead there are "regional indicator symbols" corresponding to the letters A-Z. If you put two of these together, you get a country's flag as per its two-letter country code (which comes from ISO 3166). So if I put the U regional indicator followed by the S regional indicator, I get 🇺🇸. (Hopefully that appears for you!)

This works well enough for national flags, but then people wanted flags for England, Scotland, and Wales. So the Unicode consortium devised a separate hacky system to handle flags for subnational entities, and I'm just going to skip going into that, but suffice it to say it involves spelling out the ISO 3166 code for the subnational entity with certain other special characters (that had once had a different special use which has since been deprecated, although don't worry they added this new use in a compatible way just in case).

The thing to note here is that officially, the only supported flags for subnational entities are England, Scotland, and Wales. If a system -- such as my computer -- supports anything further, like US states, that is decidedly unofficial; it is, as they say, not recommended for general interchange.

So, with all that out of the way, what -- beyond the officially supported flags -- does my system support?

Well, the first thing to note is that actually the national flags don't even seem to work in all applications; they don't work in the terminal emulator, specifically. They seem to work everywhere else, though! Like whether or not they work in Vim depends on whether it's terminal Vim or graphical Vim. Huh! And yet, the subnational flags England, Scotland, and Wales work everywhere, including on the command line. Some real inconsistency there!

But what about beyond the standard? For national flags, it appears to only support the standard ones; no SU for Soviet Union, alas. For subnational flags though, it has a little bit more.

As already mentioned, it supports US states. Not all of these are up to date, I should note; of course it's no surprise that it wouldn't have the new flags for Utah and Minnesota, which only just changed, but it still has the old flag for Mississippi! Now I got this computer back in 2017 but I update the OS whenever it tells me to, so these days it's running Mint 21.3. I guess probably it's MATE that's responsible for the flags, then, since like I said I'm running MATE 1.26, and that came out back in 2021; it's entirely believable that they wouldn't have gotten the flag updated, even if they did have months to do it. Apparently MATE 1.28 is now out, hopefully that got it up to date, although it still came out before the Utah and Minnesota changes so I guess those would still be out of date.

In addition to the US states, Washington DC also works, but the US territories don't, not even Puerto Rico. Interestingly it has this question-mark-flag that it uses for any flags it doesn't know; this applies to both national and subnational flags.

But it does support more than just England and the US -- Canadian provinces work, and so do Canadian territories! Canada gets its territories but not the US, huh? Completing North America, the flags of Mexican states are also supported, as well as the federal district of Mexico City.

Outside of North America, while I didn't try everything, I wasn't able to find anything that works except the United Kingdom. (No Australia, even!) And the flags of its subdivisions -- England, Scotland, and Wales -- are actually standard, not an extension.

But wait... what if I tried Northern Ireland? Northern Ireland doesn't have an official flag, because it's a contentious political issue. But if you try to make my computer display one, by spelling out "gbnir" in the appropriate special characters... yup, it the Ulster Banner! Interesting.

Anyway those are the subnational flags my computer supports. Quite North-America-centric! But hey -- none of this is standard, so they're not under any particular obligation to do more!
sniffnoy: (Kirby)
So here's a question Shaked asked a bit ago. He said, say you've got a finite poset P; let's associate a number to P that is the smallest number of pairs x<y you need to specify for this to generate the order via transitive closure. Given a poset on n elements, what's the largest this number can be?

My initial thought was that (at least when n was a power of 2) maybe the cube would maximize it. This turned out to be pretty far from correct. Like, very approximately, this gets you n(log n)². What I didn't notice at the time is that Shaked's number actually has a much simpler description, because the minimum set of pairs you need to specify is precisely the order's covering relation! So the question is just, what poset on n elements maximizes the size of the covering relation, or equivalently the number of edges in the Hasse diagram?

Well, you can get ⌊n²/4⌋ by just making two tiers, divided up as evenly as possible, and declaring everything in the top tier to be greater than everything in the bottom tier (with different elements in the same tier incomparable). Is this the best possible? Yes! A Hasse diagram has to be triangle-free (can't have transitive triangles because it's a covering relation, can't have cyclic triangles because that wouldn't yield a partial order), and Turán's theorem or Mantel's theorem says this is the (unique) triangle-free graph on n vertices with the maximum number of edges. There you go! The uniqueness of the undirected graph doesn't quite translate to uniqueness of the poset, though, because you could instead have three tiers, with the size of the middle tier equalling (or almost equalling) the combined sizes of the other two -- that's a different poset, but it gets you the same undirected graph. It's not too hard to see that this is the only way to vary things, though.

When Shaked initially posed this problem -- before we realized what the invariant he defined actually was -- I said that it reminded me vaguely of order dimension. In reality I don't think it's all that similar, but that was what I thought at first. This raises a question: What's the highest order dimension you can make with a poset on n points? This question has been studied, it turns out; the answer is ⌊n/2⌋ for n≥4. (This is violated for n=2 and n=3, since dimension 2 is obviously possible for both -- if the maximum were 1, every order would be total!) When n is even, this bound is attained by putting n/2 points in a top tier, n/2 corresponding points in a bottom tier, and having everything in the bottom tier be less than everything in the top tier except its corresponding point above. So, kind of similar to the maximizer to Shaked's problem! Moreover, for even n≥8 this is also unique, and in the case n=6 it's pretty close to unique (there's only two alternatives, or only one if we don't count duals as separate). (The n=4 case isn't at all unique, as anything non-total will do.) For the odd case you can just do this for n-1 and then toss in an additional point that does whatever you want (so this case won't be at all unique).

I don't know, not a lot to say about this but I thought it was neat.
sniffnoy: (Kirby)
So I haven't posted in a while but woooo I beat Slay the Spire on Ascension 20 (as Watcher, no heart). :D

I, uh, did not expect I would be able to do that (without lots of going and looking up strategy, I mean, as opposed to figuring things out myself). I was going to content myself with just getting all my characters to Ascension 20. But now I guess I've got to beat Ascension 20 with all of them! That still seems daunting, though... (currently, I have Ironclad at 18, Defect at 19, and Silent at 20).

When I beat Ascension 19 as Watcher I had very little health left and was like how will I ever beat 20?? Well uh part of the answer was having Fossilized Helix to tank some big hits near the end! (And then having enough block to not waste it on small hits, due to Kunai + Duality.)

(Beating Ascension 19 as Silent, uh, that was an Apotheosis run, so. :P )

If I do beat Ascension 20 as every character I think I'm just stopping for now -- I'm not going for heart ascensions, no thanks. That's too much. Maybe after another yearlong break from the game. :P

(Slay the Spire 2? Yeah that looks cool but uh I've still got the first one to play... also who knows how long it will take to emerge from early access...)

Meanwhile other things going on! I'm finally looking for work again. Well -- I've been procrastinating on this, because it's a pain, but it's a thing I need to do!

Andreas Weiermann wanted me to come out to Belgium again sometime in 2025 but I don't think it's going to work out, a new job is hardly about to let me take a month off like Truffle did...

Speaking of math, a guy named John Campbell wrote to me recently to suggest a new variant of complexity: What's the smallest number of 1's you need to make a *multiple* of n? I don't know that there's much to do with this (and usually it will equal the ordinary complexity, though not always; 1499 is a counterexample), but it's kind of neat. I guess it satisfies f(mn)≤f(m)+f(n). And it's computable, although I certainly don't know of any good way to compute it... maybe somebody will find one?

Uh, I had a large belated birthday party a few weeks ago! I announced it well in advance in the hopes that some of the always-busy people would show up... it was partly successful at this. Linda showed up so I finally got to show her that look I still have your drawings! But Liz Goetz did not so I did not get to show her that look I still have your sign. Oh well.

We're down to just 163 books to give away, though...
sniffnoy: (Chu-Chu Zig)
We're down to just 178 books remaining now. :P
sniffnoy: (Chu-Chu Zig)
So I wrote a while back about going on that unplanned trip with Yanan and some of her friends to go see the solar eclipse.

Since then, more things have happened in the sky around here, some of which Yanan alerted me to! Unfortunately, I've missed most of it...

The big aurora a few months ago I missed because I was down in North Carolina at the time, visiting Hunter and Lee and Colin and Aviva. The more recent aurora I missed because various things kept me inside that day -- I ended up going out to try and see it, heading down to Battery Park, at like 11 PM, but by that point it was too late. (Although I did there run into some other friends of mine who made the same mistake...)

I did catch the recent partial lunar eclipse, although wow that was a lame one -- but I've certainly seen (total) lunar eclipses before, even here in New York. But the big one recently was Comet A3.

Nic, Hunter (who was visiting), and I tried to see it from Nic's roof on Wednesday, but we weren't able to see anything. The next day Yanan and I tried to see it from Pier 76, but we didn't manage to see anything either. I do have to say, not being much of a stargazer, that wow, I did not realize Venus was so bright! But, uh, that's always there I guess. :P The two of us did run into another pair of stargazers who had a fancy telescopic camera; it, like, just barely managed to register the comet. So oh well. Guess it wasn't bright enough to see from the city. I mean, I guess that's not a big surprise, but I was hoping...

-Harry
sniffnoy: (Kirby)
So here's something unusual that happened recently. I got an email from some guy who wanted to ask me about something that had been added to the ordinal arithmetic Wikipedia page. He says, hey, someone added to the article this alternate recursion for natural multiplication, but didn't include a citation for it... you seem to be a person who knows about ordinal arithmetic; is this alternate definition correct? If so, do you know a citation for it?

I took a look, and, huh, this alternate definition was one I'd never seen before! So, if it was correct, I certainly didn't know a cite for it!

But, was it correct? The claim was that α⊗β was the supremum of all (α'⊗β)⊕β for α'<α together with all (α⊗β')⊕α for β'<β.

This recursion actually works for (α,β)<(ω2,ω2), but it isn't correct; at (ω2,ω2), it fails. The natural product of ω2 and ω2 is of course ω²4, but applying this recursion instead yields ω²3. So I wrote back and said all this and the guy removed it from the article (I didn't feel like getting involved myself).

But, huh! What an odd thing to happen. I wonder who added it -- OK, I guess I can go check that myself if I really care. But I wonder where it came from? Was it just a bit of incorrect original research? Like this person came up with what they thought was an alternate definition, just stuck it on Wikipedia like you're not supposed to do, and then it turned out not to be correct? Probably! But it would have been quite interesting had it actually turned out to be copied from somewhere...

There's a few other notable things here. First off, I'm flattered that someone thought I was the person to ask about ordinal arithmetic! I mean, I can't disagree that I'm a good person to ask, I've spent quite a bit of time examining ordinal arithmetic, not just the ordinary operations and the natural operations and some intermediate operations (that's the paper that caught his attention), but also weirder ones... have I never discussed [the operation that, having invented, I have named] Nim-Jacobsthal multiplication here?? Huh, I'll have to write about sometime. (No idea when I'd ever actually write that up for publication... it's somewhere near the bottom of the pile...) Still, I'm not who I would expect someone to go to first for such a thing as the semi-outsider that I am, even if I am a good choice!

The other thing is that I didn't notice this change myself. This page is actually on my watchlist, you see; the thing is that I haven't actually regularly checked my Wikipedia watchlist in years. And, wow, it looks like I haven't written about this here either.

Basically, some years ago, I got into a stupid edit war over Allan Lichtman's "Keys to the White House". I say it was a "stupid" edit war because it wasn't over the validity of the system or anything (which is obviously terrible!) -- it was purely about phrasing. At some point I just got too worked up about it and couldn't continue further, and, well, it led to me just not really checking my watchlist anymore.

Recently, though, I went and finally removed that page from my watchlist (after reading an article about how the "The Keys to the White House" isn't just bad, it's actually dishonest -- Lichtman goes back and changes things after the fact so that he appears to be always right) so I can start checking it again without encountering that. So far, well, I haven't actually done that. But maybe I will!

(And maybe I'll write sometime here about Nim-Jacobsthal multiplication, but if I do, you'll see why it's near the bottom of the pile publication-wise. :P But I guess that means a blog post is more suited to it...)
sniffnoy: (Dead face)
So, a few weeks ago some crazy guy attacked me on the street, basically across the street from where I live. Ran at me out of nowhere, hit me, knocked me down. Thankfully he stopped at that. Anyway, I'm OK -- got some nasty bruises and scrapes but that's it. (Really annoyingly, the one other guy in view did absolutely nothing when I shouted for help -- just kept on walking!) Well -- when I went back later, the guy was still there, so me and Alex were able to get some pictures of him.

Anyway, when I reported this to the police, a thing the officer who took the report asked me was, why didn't you call 911?

I found this pretty baffling. Of course I didn't call 911, because at no point that I could have called 911 was there an active emergency! (Well, that's not true -- actually there was one such point, when I was on the ground but before the guy left. But I didn't think of it then.)

Ever since I was a kid, it's always been, remember, 911 is for emergencies only, you'll get in big trouble if you call 911 for non-emergencies! And "emergency" to my mind means, y'know, one where failure to act will plausibly result in death or permanent injury, right? Or something like that. Go read Wikipedia on emergencies.

But no, apparently according to the police here, 911 is just the number to call for any situation where you urgently need a police officer to show up, even if it's not an emergency as normally construed. So instead of Alex and I getting pictures of the guy, we should have called 911 to get a police officer to show up and arrest him -- even though by that point any active emergency was well over! I dunno, I feel like I'd be kind of embarrassed to call 911 in a situation like that.

This isn't the first time this has come up. Sometime maybe last year or so, there was some drunk-seeming guy outside being really loud and annoying late at night. Usually with people being too loud I don't bother doing anything and just wait them out; on occasion I'll go confront them myself; but sometimes, yeah, I call 311 and make a noise complaint. Calling 311 is such a pain -- there's such a long unskippable message before you even get to the menu. It's so bad!

Anyway, in this case, I tried to make a noise complaint, but the dispatcher said, so this isn't due to a party or a car but rather to a disorderly individual? That's a 911 call, let me transfer you. Um! Yeah ultimately I didn't go through with it and did nothing because y'know I hardly wanted to call 911 on this poor guy! I mean obviously this is in no way an emergency. I just wanted a police officer to come by and tell him to knock it off or he'd get arrested -- not to come in with gun drawn or anything, which is what I imagine when I think of 911! And yet that is what the city requires you to do??

I really don't like this. None of these situations are emergencies, 911 is supposed to be for emergencies, I would not feel comfortable calling 911 in these situations (although obviously I would now call 911 if some crazy guy attacked me again because I'd consider letting him get away with it to be worse). They should change 311 to handle these things, IMO -- and also, y'know, make 311 usable by not having that long fricking preamble and menu system!

(I guess they could introduce an intermediate -- "I need a police officer urgently but it's not an emergency" -- but I don't really expect most people to remember 3 different levels and phone numbers like that, so I don't think that's really viable.)

I want to complain about this, but I'm not sure to who. Is this just a New York thing or is this how things are divided more generally in the US? And if it is a city thing, I'm not sure whether it's city council I would talk to or what...
sniffnoy: (Chu-Chu Zig)
This was my entry in this year's ACX book review contest. Since it's not one of the finalists, I'll repost it here. Apologies for the uncharacteristic use of fancy Unicode characters -- I originally wrote this in a Google Doc as per the contest rules and I don't care to re-edit it. :P (Also, yeah, the "book review" format is why there's the "I recommend this book" section at the end as I try to wrap up... not something I'd normally include!)

The Globe: How the Earth Became Round, by James Hannam

We all know that the world is round; people have known this for thousands of years. Why, even the ancient Greeks knew the world was round; Eratosthenes famously computed its circumference. But – naïvely – it looks flat. Presumably, people didn’t always know the world was round, and at some point they figured it out. Who figured it out? How did the idea spread? Who knew, when? “The Globe: How the Earth Became Round” gives us an answer to this question, and it’s a disorienting one.

The Greeks didn’t always know the world was round; they had to figure it out. But they did figure it out. So, given how important astronomy was to so many ancient civilizations, you might reasonably guess that other advanced civilizations of the time – the Persians, the Indians – figured it out too.

James Hannam tells us that no, they didn’t. Sure, they found out – but they all learned it, directly or indirectly, from the Greeks. According to Hannam, the idea that the world is round was discovered precisely once. (OK, maybe not by one person, but only by one civilization.)

In the case of, say, Persia, this perhaps isn’t too surprising; they had a lot of contact with Greece, so it makes sense that they’d learn it by transmission before they ever solved the problem themselves. But what about China? They had plenty of time to figure it out for themselves before Greek learning on the matter was brought to them. Well, two millennia later, when the Europeans showed up to bring the news… even their best astronomers still had no idea.

The world is round?

Now, there’s three things we might mean when we say “the world is round” – the Earth is circular; the Earth is spherical; or the universe is spherical. (Or the universe is circular, but that doesn’t seem to have been ever much believed; or, I suppose, that the universe is a 3-sphere, but that way of thinking wasn’t available to the ancients. These days we’d say that the universe is overall flat!)

Obviously, it’s the middle one of these that we care about here, because it’s the one that’s true, and the one that took work to determine. The idea that the Earth is circular was commonplace among cultures that believed the Earth to be flat – and so was the idea that the universe is spherical. Sorting all this out in the sources can be a bit tricky – these can all be described as “the world is round”, and not only in English, so statements are frequently ambiguous!

But it is possible to sort out a history with enough reading, and this book attempts to present that history; a history of who thought it was round when, and of many of the ways in which people thought it was flat.

The ancient Greeks were the ones who first figured out the globe. There’s not really one discoverer among the Greeks, but Hannam gives the credit to Aristotle for properly putting the pieces together; saying not only that the Earth is round but also how we can know this, and, importantly, stating explicitly the idea that on a round Earth, “down” is not a fixed direction in space, but rather towards the Earth’s center, thereby explaining why the oceans don’t drain off and such (a point that would confuse flat-earthers throughout the ages).

From Greece the idea of the Globe spread to Rome, to India, to Persia; to the Christians, the Muslims, the Jews; to Europe; and eventually to China. The book has chapters (sometimes more than one) on all of these, as well as some others I left out.

The book talks about opposition to the idea of the globe as well as support for it. Like, prior to the ninth century, Jews were largely skeptical of the idea of a round Earth because of the idea’s Greek origin. And yes, the book does discuss modern flat-earthism at the end.

But if you want a full rundown, go read the book. I’m not here to recap how the idea from ancient Greece outward; I’m here to discuss what in that history is particularly surprising, and the implications.

The Chinese case gets worse

So – until the Jesuits started showing up in the 1500s, hoping to trade superior knowledge of astronomy for permission to preach freely, Chinese astronomers were not aware that the earth was round.

Now, I say “Chinese astronomers”, but I should be clearer about what I mean by that. Because back in the late 13th century, when Khublai Khan conquered China, he set up a second astronomy bureau in his capital, one that practiced Islamic astronomy, as needed for computing the Islamic calendar. The Muslims, of course, had long known that the Earth is round; they’d learned it from the Greeks, and also from the Indians who’d learned it from the Greeks.

Which means, yes, that there were people in China who knew that the Earth was round. But the Chinese astronomers and the Islamic astronomers – both set up in the Chinese capital – apparently didn’t talk to each other, because 300 years later the Chinese astronomers still held to a flat Earth!

An unsettling counterfactual

All this raises the question: what if the ancient Greeks hadn’t figured it out?

It’s easy to say that if they hadn’t learned it from the Greeks first, India and Persia would have figured it out themselves. Or the Muslims, later. They all had perfectly capable astronomers, after all. But… so did China. And they didn’t.

What about Europe? In the very worst case, Europe would have to have figured it out by the time of Kepler at the latest, surely? I mean, one might say that the idea of a round Earth is a prerequisite to the ideas of Copernicus, Brahe, and Kepler – and I imagine not having it would have slowed things down some. But given that the scientific community of Europe (and not just a few isolated figures) were discovering heliocentrism and related ideas at this time, this to me suggests that there was a general capability there, that certainly they would have figured out the globe as part of the astronomical revolution regardless.

But whether they actually would have is impossible to know. And of course – the astronomical revolution is still a pretty late time to figure all this out! That’s almost as late as the idea spreading to China. Would they have figured it out earlier? One would think so, but again, the example of China suggests perhaps not.

Perhaps Europeans would have figured it out from all the long-distance sailing they were doing? That’s before the Copernican revolution, but only by about a century or so. (And even if sailors had known, how long would the idea have taken to spread to astronomers?) But other people, including the Chinese, were already doing long-distance sailing by then, including the use of stellar navigation, and Zheng He’s voyages recorded plenty of evidence for the globe, and yet China didn’t figure it out; was European long-distance sailing different enough that it would have mattered?

I’ve gotten pretty used to the idea of multiple discovery being the norm, of ideas coming up when their time is ripe. New things are figured out because the groundwork has been laid, the measuring instruments have been improved, new evidence has been coming in, and so forth. The astronomical revolution in Europe is a fine example – invent the telescope and the rest seems inevitable.

The idea I’ve most often seen as an example of the opposite is general relativity; people often say that if it hadn't been for Einstein, it wouldn’t have been figured out for another decade or longer. (Which is kind of odd, actually, given that Einstein wasn't the only one at the time developing a geometric theory of gravity. I would have thought that the idea of a geometrical theory of gravity, rather than the specific field equation, would have been the hard part, but I suppose not?)

I'm used to considering the idea of the globe as so basic that of course it must have been multiply discovered; after all, the Greeks knew it! And indeed, within Greece – among that particular milieu – it does seem to have been something of a gradual (if not multiple) discovery. Certainly it’s nowhere near as non-obvious as General Relativity! But perhaps we should regard it to be pretty high on the scale of non-obviousness, perhaps approaching that of heliocentrism? That doesn’t sound right to me. And yet, the lack of an Aristotle delayed the Chinese discovery of the globe not by a mere decade, but rather two millenia. What was so special about ancient Greece, that the time was ripe there but not elsewhere?

More ancient Greek speculation

It really does seem that ancient Greece had more speculation than other places about non-flat shapes for the Earth. Everywhere had cosmological speculation, but in most places all of this elaborated on the idea of a flat Earth rather than replacing it.

For instance, the very first person to suggest the idea of a spherical Earth may have been the Greek philosopher Philolaus; his surviving work doesn’t explicitly state this but seems to suggest it. The notable thing here is that Philolaus is not a geocentrist; he has the Earth, and the other planets, orbiting the “universal fire” or “universal hearth”. So… he’s a heliocentrist two millenia early? Nope! The Sun is distinct from the universal fire, and also orbits it. Huh.

Meanwhile, about a hundred years before Philolaus, a Greek philosopher named Archelaus correctly reasoned that the Earth can’t be flat due to the difference in sunrise and sunset times as one travels East and West… but his conclusion was that the Earth was shaped like a concave bowl, which gets backwards how those times should vary. Seems he had the basic idea, but failed to think it through! Still, ultimately it was ancient Greece that was thinking about this and ancient Greece that got it.

Where does the Sun go at night?

There is, as I mentioned earlier, a fair amount of discussion of just what sorts of flat-earth cosmologies various ancient peoples believed in. For instance, lots of civilizations thought that the Sun goes underneath the Earth at night, but others said, how can it go under the Earth, it’s solid underneath there! The Talmud, for instance, says that at night the Sun travels from West to East not underneath the Earth, but rather around its northern edge. I would suggest reading the book for more of this nature.

But once again it’s the case of China that’s eyebrow-raising. Let’s consider – one of the classical pieces of evidence that the Earth is round is that as you travel North and South, you see different stars; some stars become hidden behind the Earth, which is only possible because of its curvature.

Chinese astronomers were aware of this phenomenon, but they had a different explanation. Their explanation, which they also used to explain where the Sun goes at night was – and I swear I am not making this up – that the universe had a draw distance. If you get too far from something, its light simply can’t reach you. The book doesn’t use the term “draw distance”, but that’s basically what it’s describing!

Notice, by the way, how that explanation requires the Sun and stars to be quite close! “The Gnomon of the Zhou”, from around the year 20 CE, gave the distance to the heavenly canopy as 43,000 kilometers, and the “draw distance” as 90,000 kilometers. Which doesn’t seem to make sense offhand, because 90,000 is more than twice 43,000, so that would mean the Sun would be always visible. Hm, well, “The Gnomon of the Zhou” isn’t a very consistent text in the first place, and may not be reflective of what later astronomers thought.

It’s possible that I’m portraying things uncharitably here, and that actually Chinese astronomers said that the light becomes continuously attenuated over distance, only eventually fading out entirely, but the book sure makes it sound like a draw distance. Certainly “The Gnomon of the Zhou” does, although, again, that may not be the best text to rely on.

Either way this explanation is baffling, because when the Sun sets it clearly vanishes a bit at a time, rather than uniformly fading out (with the attenuation explanation) or vanishing suddenly (with the draw distance explanation). Apparently the common people generally said that the Sun becomes hidden behind a mountain at night – a fairly common belief among flat-Earth-believing cultures (who often further identified this mountain with the mountain believed to exist at the world’s center), and to my mind a more sensible one – but Chinese astronomers said otherwise.

I wish the book had gone into more detail on the question of how well this explanation worked for explaining the vanishing stars upon North/South travel; how well does this explanation actually match the data? How big is the anomaly, and were people worried about it? Did Chinese scholars of optics have opinions on this mechanism? Unfortunately, the book answers none of these questions.

For that matter, which way is North?

“The Gnomon of the Zhou” contains more surprising assertions I wish the book had discussed further. I mentioned above that most civilizations that believed in a flat earth believed that the world was a circle; in China, however, it was believed to be a square. Now if you believe in a flat Earth this raises the question of, where is the center of the world, something many civilizations had opinions about. In China, the center was generally believed to be the capital city, although which capital city varied (earlier Dengfeng, later Kaifeng).

“The Gnomon of the Zhou”, however, has it different. It says that the center of the world is at the North pole. And that’s a statement that I really have to wonder what it means.

As mentioned, the Chinese believed that the Earth was a square; moreover, it seemed they considered it to be an “axis-aligned” square, with sides running North-South and sides running “East-West”.

If the “North pole” means the northernmost point, then already we’ve got a problem; such a square Earth has a northern edge, not a northernmost point. But even if we ignore that, the northernmost point sure as hell isn’t at the center!

I really have to wonder how whoever wrote that was interpreting the ideas of “North”, “South”, “East”, and “West”. My general understanding was that in China at that time these were understood as fixed directions in space, with the axis-aligned picture I mentioned earlier. For the North pole to be at the center, that would instead suggest a view where North is not a fixed direction, but rather “towards the center” – much as how on a round Earth, “down” is not a fixed direction but rather towards the center. But if the authors were thinking this way, it doesn’t seem to have led them to the idea of a round Earth. Which isn’t surprising; that’s quite a leap.

This point, of what “The Gnomon of the Zhou” meant by this claim, is something the book really needed to discuss more, I’d say. I don’t know how possible it is to answer this question, especially as that particular work just isn’t all that consistent in the first place; unfortunately “The Globe” never mentions the question at all.

I recommend this book

I’d recommend reading “The Globe”. I’ve covered what I consider to be the highlights, but the book discusses a lot more. It does have some notable omissions as mentioned above. Also, a bit of a warning about the prose – when it goes off its main topic, it sometimes reads like a high-schooler’s essay (although never in the endnotes, so I guess this was a deliberate stylistic choice?). This mostly goes away once you get past the first few chapters, though, and the main topic comes more into focus.

Of course really the main thing I’ve taken from this book is that unsettling counterfactual that it never explicitly asks – what if the ancient Greeks hadn’t discovered the Earth to be round? How long would it have taken India, Persia, Europe, to all get it? This question is for sure going to haunt me. I guess you don’t really need to go read the book for it to haunt you too! If you want your speculation on the matter to be a bit more informed, though, probably this is a good place to start.
sniffnoy: (Golden Apple)
Why are arms parallel and legs serial?

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