#math cw

[mother-sauces]

Mathematical dual? Aint that the same shit that got Galois?

[fictionkinfessions]

BILL CIPHER MATH ANON — STARTED STUDYING THE FOURTH DIMENSION! ALWAYS FOUND IT INTERESTING, BUT I NEVER REALLY LOOKED INTO IT TOO HARD. I’M FIXING THAT MISTAKE! IT FEELS NICE. EXCITING. NOSTALGIC. I MIGHT NOT BE ABLE TO ACTUALLY SEE THE FOURTH DIMENSION, BUT THIS IS AS CLOSE AS I CAN GET (UNLESS I SOMEHOW FIGURE OUT HOW TO BREAK THE LAWS OF REALITY, WHICH, HEY! WHO KNOWS!). I’M STILL AN N-DIMENSIONAL CREATURE TRYING TO FIGURE OUT THE N+1-DIMENSION!

x

[handageddon]

it’s called math in the US and not maths because we only learn one mathematic

[lenniershairlessredacted]

the other day I saw an (apparently old) joke that I keep chuckling about at random:

do you know what the b. in benoit b. mandelbrot stands for?

benoit b. mandelbrot

[20dollarlolita]

A-line Lolita Petticoat VS Party Petticoat VS Square Dance Petticoat

CW: underwear (stock photos of models in translucent petticoats, underwear is visible.)

Let’s go over it one more time for anyone new to the party: if you wear lolita, you need a petticoat. Your petticoat is what makes your dress look right, or, depending on the petticoat, wrong.

A lot of lolita (sweet, classic, and gothic) uses the bell/cupcake style petticoat:

(The y=-x^2 petticoat)

And a lot of classical and gothic lolita also uses the a-line petticoat.

(The y=-|x| petticoat)

If you wear the wrong type of petticoat for the wrong situation, you can end up looking rumpled or just a little off.

If you’re new to lolita and don’t know what kind of petticoat to get, you’ll probably want to get a bell/cupcake petticoat, because that opens you up to a wide range of styles across all lolita subtypes.

So you spend whoever much you spent on your cupcake petticoat, say $60 with shipping and shopping service fees, because the major foundation garment of your whole style’s worth saving up and getting a really good one. But then what if you find a dream dress that just can’t handle a cupcake petticoat? You need an a-line petticoat and you’ll only wear it with one kind of dress.

There’s cheaper petticoats out there on Amazon, Ebay, Spirit Halloween, The Party Store. Can any of them work for lolita? What would you have to do to make them functional petticoats?

Let’s take a look at petticoat construction, so that we can all learn the rules of what makes petticoats shaped like they are. Once you understand the theory, you can look at a garment for yourself and judge if it will work or not. Once you have that power, you can find deals and discounts wherever you go.

Onward we go!

First thing first, let’s get one thing out and over with:

y equals the negative absolute value of X (a-line graph, above) is not the same as (x^2/a^2)-(y^2/b^2)=1,  or this:

You can see how the bottom of that graph starts by going straight down, and then swoops out wider? That’s not the shape we’re going for.

And that’s the shape you get with this:

We’re going to call this shape a “flared” petticoat, since it gets exponentially wider as it goes down.

Here we have a Party City tulle petticoat, an actual square dance petticoat, and some weird stuff someone’s selling on ebay under “lolita petticoat”.

There’s one huge, major thing these petticoats share in common, that makes them unusable for lolita: they are all designed so that the under layers are visible when you’re wearing a skirt. That $17.37 petticoat even has different colored frills, to catch attention at them hem and show off the petticoat.

In lolita, we consider our petticoat to be underwear, and much in the same way that wearing high-rise underpants and low-rise jeans together can be a legitimate fashion statement, but you wouldn’t wear it to work, wearing petticoats designed to hang out can be a fashion statement, but probably not one you want to shoot for in lolita.

And let’s have some terminology here:

Tiers are the strips of fabric that are sewn together to make a longer skirt-part of the petticoat. The individual skirt-things are called layers. This specific image has 6 tiers (including the top fabric one) and either two or three layers. It’s harder to count the number of layers than it is to count the number of tiers.

So, petticoat theory for today, part 1:

If there are multiple layers, and each layer is the same length, the inner layers will hang below the outer ones.

You can see how the outer edges of these petticoats fall higher than the inside layers. With these petticoats, even if your skirt is a few inches longer than the outer layer, the inner layers would still hang below the skirt and be visible.

This a-line lolita petticoat also has a curved shape on the bottom, but the inner layers are not visible. They are invisible because those layers are slightly shorter than the top layer.

Found petticoat advice #1: If you are buying a non-lolita-specific petticoat, you should have a plan for how to shorten the inner layers. You do not want the inner layers hanging outside the outer layers, or you will not have the ability to conceal them under your skirt.

Okay. Tiny line to move onto point 2

The second thing that many non-lolita petticoats lack, that a-line lolita petticoats have: Support at the waist.

Most petticoats, party tutus, and the like, have a smooth, thin fabric as the top tier of the petticoat. This is designed so that you can wear the petticoat at your waist, and the flare it provides starts at your hips. (You usually wear petticoats at your natural waist because petticoats are heavy and your waist holds things better than your hips). 

Found Petticoat Tip #2: If you’re determined to use one of these petticoats for lolita, you need to be ready to either remove this thick waistband, or put a new layer on top of it to add support where it’s needed.

Today’s petticoat theory, part 2:

A flared petticoat often has zero support at the waist of the garment. An a-line lolita petticoat has support at the waist.

An a-line lolita petticoat provides equal support all the way down the skirt. A flared petticoat has more support at the hem than it does at the waist.

Tip #3: If your found petticoat has too much support and volume at the hem, some of that has got to go. There’s a good chance that, if you shortened your inside layers by just chopping them off (inelegant, but effective), you also cut out some volume at the hem. However, be very critical of your petticoat. If the little hem frills are too much volume, or too eye-catching, they might need to go. Once they’re gone, you need to make sure that your petticoat still has enough poof to work as a lolita petticoat.

Some petticoats are too flared and attention-grabbing with their bottom frill tier intact, but are too flat with it removed. There’s not always a possibility for a sweet spot.

Tiny line:

Moving right along:

Why are petticoats shaped like petticoats?

And, more importantly, how do I make a petticoat shaped like the petticoat I want.

So there’s a lot of petticoat tutorials that throw around ratios like 1:2:4:8 or something, but never explain how they got to that number.

So there’s two more pieces of petticoat theory:

A petticoat’s shape is contained within its outermost layer. If your inner layer is bigger than your outer layer, it won’t make the petticoat any bigger than the outer layer (not counting if the outer layer rides up or breaks)

A petticoat’s shape is based more on the shape one tier makes connecting to the tier above it than it does on the overall length. This one’s kind of hard to explain in one sentence so hang with me.

First part of that: If your outer layer is not as large as you want your final petticoat to be, you can shove the whole thing full of balloons and expanding foam and it still won’t ever reach the size you want.

This means that, when choosing or constructing a petticoat, you need to make sure your largest layer is at least as big as you want the final size of your petticoat to be. Tip 4: If your found petticoat doesn’t have enough fabric at the waist or on the outer layer, no modifications except adding a new outer layer will be able to save it.

The other one’s harder to explain, but here goes:

When you’re planning and patterning a petticoat, once you’ve established your general shape, you need to figure out how full/wide/long your individual tiers should be. When you’re at this step, stop thinking about the petticoat as a whole, and just look at each individual tier.

Here’s my a-line shape, and I’ve drawn lines on it to represent all the tiers in my petticoat. When I look at the place where a purple line and a red line connect, I can see that those lines intersect at constant angles. This means that each tier needs to be an equal expansion from the last tier. If my tier 2 was 3x the fullness of tier 1, I know that tier 3 should be 3x the fullness of tier 2.

Each tier is getting fuller, but they’re not getting proportionally larger.

If I take my cupcake shape, I can see that each tier slants down more than the tier before it did. This means that at each tier, the next tier needs to have proportionally less in it than the one above it.

If my tier 2 was 4x the fullness of tier 1, my tier 3 might be 2-3x the fullness of tier 2, and my tier 4 might be 1.5x the fullness of tier 3.

Each tier is still getting fuller, but the amount that they’re getting fuller by is decreasing.

So how do you know how much fuller to get every tier? There’s math for that! Disclaimer: I don’t actually do this math when I make a petticoat. I like winging it. If you don’t like winging it and you like being correct, well, I can give you the validation that you’re doing it The Right Way. This is completely optional, though! Once you derive these equations, you can keep them on a piece of paper and use them with any waist measurement or length.

Get a piece of paper, folks. We’re about to get mathy. We’re going to do the math for an a-line here. Let me know if you want to see the cupcake math.

Actually, get three pieces of paper, a pen, and a roll of tape.

The first thing you want to do is to cut a donut shape out of a piece of paper.

Cut a line through that donut, and then adjust it so that it’s the shape of an extremely full a-line skirt. Mark the ends of the donut, and then cut out the part that overlapped. Now you’ll have two pieces of your donut. In mine, I cut out about 1/3rd of my donut. This means that my ideal a-line skirt shape I’m shooting for is a 2/3rds-of-a-circle skirt.

Now, we do the math to figure out the maximum lemgth of out tiers. This is standard circle skirt math:

Your waist measurement divided by pi is 2x your inner radius.

Your inner radius plus your skirt length is your total length.

I draw a picture and draw where my tiers are. Each tier should be 1/3rd of your total skirt length.

Now you have to calculate the circumference of your hem and the bottom later of each tier. Calculate on the full circle. We’ll subtract for a 2/3rds circle skirt later.

(Radius+tier length)x2 times pi is the circumference for the bottom of tier 1.

(radius+tier 1 length+tier 2 length)x2 times pi is the bottom of tier 2.

(radius+tier1+tier2+tier3)x2 times pi is the bottom of tier 3. In my skirt, tier 3 is also my hemline.

Now multiply all those bottom-of-tier measurements by however big your circle on your skirt is. My skirt’s 2/3rds of a circle, so I make my numbers smaller by multiplying them by 2/3rds.

Now you cut each tier of your petticoat to something longer than that measurement. If you’re using chiffon, you might need to cut them 3x the length, and if you’re using crinoline you might need 1.5x the length.

So, what’s all that math doing?

Basically, a skirt cannot be larger than the amount of fabric in it allows. If your hem of your petticoat is only 30″, no matter how much poof you shove under there, your skirt will never be poofier than 30 inches.

So what we just did was calculated how long lines on a circle would be at specific points. We’re going to cut the tier strips of fabric to (based on) that measurement. This means that, on those lines on your circle, you know your fabric will be that long. We just cut it that long, so it has to be that long. Then, you take the other edge of the fabric, and shorten it by gathering it. You gather it to the length of the line above it, and sew them together. If you then laid your petticoat over a 2/3rds circle form or whatever you picked, it’d fit that shape perfectly.

That’s confusing and I think I need to add some pictures, but I have to run to work in like 2 minutes, so let me know if that makes zero sense without pictures.

If you don’t like math, I’ll actually make a worksheet for the a-line petticoat and post that so you dont’ need to derive anything. G’night, y’all.

[adjoint-law]

Me, in summary

[magpiescholar]

Do we happen to have any idea how many Noldor came to Beleriand, either by crossing the Helcaraxe or by the stolen ships? asking for Reasons.

This essay attempts to find some reasonable numbers for how many elves there are at a given time, figuring based on “some guestimate of the largest possible population of Noldor in Aman which might not require the wholesale slaughter of Orome’s hunting stock just to feed the masses.”  I love that quote.

He comes to about 70,000 crossing the Helcaraxe and about 25,000 taking the ships, yielding less than 100,000 arriving in Beleriand.

The numbers Michael Martinez comes up with feel really big, which is an objection he anticipates -- and his logic seems perfectly sound to me. However, I have time on my hands and a burning fever so I’m going to work backwards from the ten thousand troops Turgon shows up to the Nirnaeth Arnoediad with (because I always prefer to work backwards from Turgon).

So let’s assume that Turgon’s total fighting force is somewhat less than half of his people.  I figure this because, while the vast majority of the Noldor who followed Fingolfin would have acquired at least shooting and seal-clubbing skills, it seems that Fingon was the family’s warrior and Turgon was responsible for preserving Noldorin culture in peace and secret, so he probably got the lion’s share of noncombatants regardless of how much you think Fingon knew about Gondolin.  Also, more children would have been born in Gondolin than elsewhere, given the Noldor’s customs about childbearing in time of war.  Here’s where I differ from Martinez: I’m figuring Turgon brought almost his full strength to an alliance of such import as the Nirnaeth -- it’s secrecy that protects Gondolin, not force.  But -- and here’s that almost -- given that he left his heir behind in case he fell, he would undoubtedly have left Maeglin an honor guard at the very least -- and Turgon is too practical and loves his city too much to leave the very least.

So if Turgon’s 10,000 is a bit less than half of the city’s population, let’s give him (for the sake of round numbers) 25,000 total.  Working backwards (to account for breeding), let’s give him 20,000 who left Nevrast for Gondolin -- but wait!  We know that not all of Gondolin’s lords were Noldor, and presumably not all of their people were!  Rog is widely thought to have been non-Noldor (mostly because of the name; his name is the sole reason he got left out of the published Silmarillion, actually) and Galdor’s description suggests the Laiquendi.  I, personally, really like the idea of Gondolin as a very cosmopolitan and multicultural city, with the rewards and tensions that implies, so let’s go with that.  So (farewell, round numbers) let’s give him 18,000 Noldor per se, 2,000 additional elves, who go to Gondolin.

The Noldor that Turgon took to Gondolin were a third of Fingolfin’s people, so let’s make the Nolofinwean group number 54,000.  Martinez suggests that Fingolfin had about twice as many in his company as Feanor did, while Finarfin’s sons took about as many people as Feanor with them, which I find reasonable.  So -- and we’re back to Martinez’ number.  About 108,000.

Again, though, that’s ... a lot of elves.  I think most of us tend to conceive of Beleriand as a smaller world.  

And wait!  We have linguistic evidence that big round numbers that sound nice tend to become part of household speech for elves -- Menegroth is called “the thousand caves” (or the 1758 caves? I don’t really grok base-12). Do we believe that it was exactly 1000 caves and no one was ever allowed to build on or remodel lest they have to change the city’s name?  I don’t.  Could Turgon’s force of ten thousand be a “forty days and forty nights” situation, where this is cultural shorthand for “a whole bunch”?  That would certainly match how Tolkien’s narrative voice usually handles numbers, in hosts and companies and manies.  Maybe that force of ten thousand is more like 7,859.  Maybe it’s more like four thousand.  Especially given that we may, depending on which canon you’re reading, be translating from base-12 numbers to base-10 numbers (again, understanding that is quite beyond me), with accompanying uncertainties in either direction.

Personally, my (rather extreme) preference is to lop a zero off any number Tolkien gives.  I enjoy the tension between exaggerated-for-effect-until-it-came-to-sound-like-truth tens of thousands and a smaller, more desperate reality.  I call it a magnifying series of roundings-up for ease and effect on the part of the literary agents for each book, which survive because they feel epic enough -- so that I get a smaller, more intimate population to work with in fic, casting this ten-times-larger shadow.  Handily, this lets me just do my calculations over without a calculator -- Turgon brings 1,000 to the Nirnaeth, Gondolin’s population is 2500 total, giving us roughly 6000 under Fingolfin, 3000 under Orodreth, 3000 under the Brothers Feanorian -- and 12,000 total Noldor arriving in Beleriand.  (For interest’s sake, this means that, say, Doriath can’t have a population of fewer than about 15,000 in order for the local elves to significantly outnumber the Noldor.)

So, depending on your Reasons, I think any number of Noldor arriving in Beleriand between 12,000 and 110,000 is perfectly reasonable and shouldn’t raise a whole lot of eyebrows.

[mother-sauces]

Given 3 finite sets, is it always possible to construct a Venn diagram of circles, such that the area of each circle is exactly proportional to the size of the set it represents, and the area of each overlapping section is exactly proportional to the number of elements shared between the sets?

What about for >3 sets?