#questionable math

[todays-xkcd]

Phil Plait once pointed out that you can calculate the total angular area of the sky this way. If the sky is a sphere with radius 57.3 degrees, then its area is 4*pi*r^2=41,253 square degrees. This makes dimensional analysts SO mad, but you can't argue with results.

Radians Are Cursed [Explained]

[redowlkitchen]

Today I read One Plastic Bag by Miranda Paul to one of my younger classes. This is a true story about a woman in Gambia who takes the plastic bags that are littered around her town and turns them into something new that can be sold to help her family. (She cuts the (washed) bags into strips and then crochets them into purses.)

After we read it, one student asked when the story took place. We didn’t have time to read the information at the end of the story, but I told them the author visited Gambia in 2003 and heard about the woman’s work then, so sometime a little before that.

Student: Wow. That was a long time ago. She must be dead.

Me: Are you sure? It’s 2022 now. Let’s say the story took place in 2002 just to make things easier. How long ago was it?

Another student: 20 years.

Student: *completely seriously* Yeah, that was ages ago. She’s definitely dead.

Third student: ....what?

According to this kid’s logic, I must be a ghost.

[context-is--irrelevant]

Okay, but 4 - 19 = 7.

Kiara, mid-argument with a 7 y/o

[zentromple]

Questionable Math: Pacific Rim

Long ago I watched Pacific Rim and thoroughly enjoyed it.  However sometimes I get distracted when movies (or books, or whatever media) give me concrete details that don’t make any god damn sense.  Its like they’re inviting me to consider what they’ve done.  I mean, the did do it; they went and put it in the movie and showed it to me.

The image that I have in mind is where 8 helicopters were carrying the jaeger out to the ocean battle.  4 helicopters strapped to each shoulder.  Lets not even get into the weird physics of how they were all connected together like a bunch of balloons.  For this exercise I’ll ignore that bit and treat them like they can all lift strait up very effectively.

 The helicopters make me ask: How heavy is a jaeger?

Lets find out.

The heaviest lift helicopter I was able to find was one with a lifting capacity of 20,000 kg

That gives an upper bound on the weight of the jaeger of 160,000 kg (20,000 kg * 8).  But is that a lot for a giant battle robot?

Lets look at density to figure out what the jaeger could maybe be made of.

A quick google tells me that a jaeger is 76 m tall and that an average person height and weight of 1.8 m and 80 kg would not be an unreasonable approximation.  The jaeger is approximately 42 times taller than the person (76/1.8) so using the cube rule a jaeger sized person would be 74,088 times heavier (42^3).  The inverse is true and if we scale down a jaeger to person size it would weigh 2.16 kg (160,000 kg/74,088).  This is clearly an unhealthy weight for a person sized object.

Some more googleing to find the volume of a person gave a pretty wide range (unsurprisingly) and so I’ve settled on calling the average person that I’ve constructed at 78 L as the density of a human body is 985 kg/m^3.  The density of the jaeger is 0.0278 kg/L (2.16kg/78L) which converts to 27.8 kg/m^3

Density list:

Steel 7850 kg/m^3

Chocolate 1325 kg/m^3

Balsa Wood 160  kg/m^3

Jaeger 27.8  kg/m^3

Aerogell 3  kg/m^3

So it can’t be solid balsa wood and is only 10 times denser than the least dense solid ever made.

But maybe I’m being unfair, it isn’t a solid chocolate jaeger after all, it does have air spaces in it.  So how does it compare to real sturdy military vehicles?  An M1 Abrams tank has a length, width, and height, of 7.93 m, 3.66 m, and 2.44 m respectively and a weight of 54,000 kg which gives it a minimum density of about 763 kg/m^3.  Its a minimum density because I calculated a maximum volume given its length, width, and height so it was actually smaller its density would increase.  The tank density of 763 kg/m^3 is about 10% of steel, which I assume is what a lot of a tank is made of.  With that ballpark density modifier I’m happy to say that the jaeger could be arguably made from a material of up to 278 kg/m3 and very happily indeed I have discovered that the density of aluminum is 270 kg/m3 so there we have it!

The jaegers as depicted being hoisted by the 8 powerful helicopters could plausibly be constructed from aluminum; plausibly with respect to weight alone, and only if they were 90% void spaces internally.

[wibblywobblywanderlust]

north american magic + math = ERROR

[Word of Warning: most of my info about population comes from wikipedia, so grains of salt and all that] 

as of the 2015-2016 estimate there were roughly 579 000 000 people in north america

supposedly, 1/10 is magical = 57 900 000 magical people 

approximately 27% of the pop. is children = 15 633 000 magical children

according to JK there is only one school in north america, but it’s a commonly accepted headcanon there is another school (salem academy, area 51, etc)

judging by magical america’s loyalty to ilvermorny, let’s say only 1/3 go to Other School = 10 422 000 magical children at ilvermorny 

well, let’s be (more than) fair and say 1/5 are squibs = 8 337 600 children at ilvermorny

divided into 4 houses = 2 084 400 kids per house

spread across seven years = roughly 297 771 kids per year

europe is pretty big, coming in just above north america at the 3rd biggest continent by pop. but europe also has three schools, cutting their individual pop down significantly. hogwarts in north west europe, beauxbatons in south, and durmstrang in the east, divides it pretty evenly 

the 2nd biggest by pop is africa, which has a magical population of about            121 613 000, and 32 835 510 kids, also with 1 school

then at 1#, you probably guessed it, asia . . . 

i don’t know what you want me to say, it has like 60% fucking percent of the WORLD’S population and 1 school. mahoutokoro better be fucking gigantic to fit   119 778 048 children

tl;dr JK Rowling needs to fix her fucking math

[lintamande]

Problems with Years

I am having problems with the timeline of the Silmarillion. Specifically, since the Silmarillion doesn't actually have a timeline, I am having problems with the timeline in the Annals of Aman.

And even if you've never picked up the Annals of Aman, you should read this, because it involves literally billions of Elves and an argument for why Feanor probably should have had nuclear weapons. Bear with me. 

In Y.T. 1133, the Noldor and Vanyar finally make it to Valinor, where (to make a long story short) they'll chill for 362 Y.T. until the Darkening. Tolkien went back and forth on how many Years of the Sun equated to a Valian year; estimates range from 9.58 to 144. That means the Elves spent a minimum of 3600 Years of the Sun in Valinor before the Darkening. Why is this a problem? Well, LACE tells us that elves marry soon after coming of age (age 50) and at this time in history, had lots of children. (The one family we know about in detail certainly bears that out). If we assume that your average elf-couple has 4 kids by age 200 and then stops, we get a population that grows by a factor of 10 every 500 years. If we started with two hundred Noldor, by the time of the Darkening we'd have 2,000,000,000. 

Yeah, immortality + high birth rates + 3600 years gets you billions of elves. (Humanity pulled it off even without the immortality). There's absolutely no way that a force of this size would lose to Morgoth. 

(This problem is slightly less pronounced if elf families only average two children apiece. We'd have about 50 million Noldor, which is still too many to plausibly lose the war.) 

And 3600 years is using the lower-end estimate for how long a Year of the Trees is. If we use the 144 years estimate, then the Elves have 52,000 years in Valinor.

... you don't want to know how many Elves that is, but let's just say they could march ten thousand abreast past Angband until Arda is remade and never run out of Elves because their population is expanding more quickly than they can march. This version also has Feanor as 45,000 years old when the Darkening comes; I have a feeling that had Feanor gotten 45,000 years to invent stuff before Morgoth screwed things up, the Silmarillion would have ended with Feanor flattening Angband with intercontinental ballistic missiles and then flying his private jet over to Beleriand to retrieve his Silmarils from the rubbish. (Can someone write this fic? Please?) EDIT: Wrote this fic. 

But aside from the demographic issues, there are other reasons to doubt that there was time for a hundred generations of Elves in Valinor before the Darkening. Specifically, the only Elf family we know about, the House of Finwe, has precisely four generations - and the only great-grandkids in the picture, Celebrimbor and Idril, are both very young at the time of the Darkening.

And it gets worse: there are 16 Valian years between Feanor's birth and Fingolfin's, so at least 160 Y.S. (Annals of Aman). We're told that Feanor married young; if marrying shortly after coming of age was standard (LACE), then he must have married on his fiftieth birthday, or perhaps even younger, to be remarkably early. That means he and Nerdanel would already have been married over a hundred years when Fingolfin is born. It seems really unlikely that they wouldn't have started on their record-setting childbearing spree by then. But we know that Maedhros isn't older than Fingolfin: he cites Fingolfin's age specifically when he hands over the crown! 

Oh, and there's one more problem. Tolkien went back and forth on by how much the Sindar in Beleriand outnumbered the Noldor, but he was quite clear (Letters) that they did outnumber them. But the Sindar lived outside the bliss of Valinor, under far more difficult conditions; additionally, their starting population should have been smaller, since lots of the Teleri never left Cuivienen, many of them wandered off along the way, and half of them ended up going to Valinor. And no one died in Valinor; Beleriand was far more hazardous. If the Noldor had large families, the Sindar birthrate would have to be outrageous to end up with an even larger population.

There's no way of equating Valian years and Years of the Sun which solves these puzzles for us. That leaves one possibility: While in Valinor, Elves mature in Valian years.

 This solution is tremendously elegant, by which I mean that it answers all of our objections at once. We know that time seems to pass differently in Elven realms; even Lothlorien seemed surreal to the Fellowship. I don't think time literally passes differently in Valinor (a la Narnia), but I think the pace of life is inarguably far, far slower. It makes sense that Elves would be slower to grow. Elvish pregnancies would last one Valian year. There's only been time for about five generations in Valinor; if we started out with a couple thousand Noldor, we'd have around 100,000 to 150,000 by the time of the Darkening. By contrast, the Sindar probably have a lower birthrate and a high death rate, but they've had time for two dozen generations while the Noldor have only had five; their population can easily be much higher. And Feanor was still quite young when his father remarried and his half-siblings were born; Maedhros is significantly younger than Fingolfin, and we still comply with the timeline in Annals of Aman. 

And all of our beloved Noldor were a few hundred years old (some even younger) at the time of the rebellion, which perhaps excuses some of their more stupid decisions. 

Was this completely obvious to everyone else?