Russia: Everybody is enjoying Hetalia: Math and Science Edition, da? Well I was thinking we needed a little more RUSSIA!

So I went and took over, and now Hetalia: M & S is one with Russia! If anyone have any questions, leave it to me to be answering!

Now I was looking over old calculus textbooks other night and I came across something that I am thinking would be real fun for me and also real educational for you. OH BALTICS!

(Lithuania, Estonia and Latvia come out) Y-Yes, Mister Russia?

Russia: It is time to be doing lesson on quadratic surfaces!

Latvia: T-that sounds hard, Mister Russia...

Estonia: It's not that hard, especially if you've got a good graphing calculator handy...

Lithuania: B-but I think he wants the fans to have a better understanding of how they work, r-r-right, M-Mister Russia?

Russia: Indeed! Now, you, Lithuania, get to be x-axis. Lie down on floor. (presses Lithuania to floor)

And you, Estonia, get to be y-axis. Lie down on floor perpendicular to Lithuania. (presses Estonia to floor)

And you, Latvia, get to be z-axis. Stand perpendicular to Lithuania and Estonia. (picks up Latvia and places him to stand at Lithuania's and Estonia's feet)

Now let's say their heads are in positive direction. (Flips through calculus book) OK, so let's say we've got x squared over a squared plus y squared over b squared plus z squared over c squared equals one. Ooh, looks like we are needing sticky notes now!

(Slaps a squared symbol to each Baltic's head, a squared to Lithuania's feet, b squared to Estonia's feet, and c squared to Latvia's feet)

Now we are having an ellipsoid, which is like 3D ellipse, shaped sort of like Easter egg. Now the a and b and c don't matter so much for shape, they just tell you how big the egg is, which direction it's bigger in, and basically they depend on whichever numbers divide to make one on other side. But you can multiply and have bigger number instead of one if you like, but it's easier this way. Now I will make Baltics into ellipsoid.

(Pours bucket of water on Baltics, then blows a bubble that encases them in a spherical shape.)

Ah, so here we are having a sphere. This happens when a squared, b squared and c squared are all equal! Now, you know the bigger the number underneath, the longer the ellipsoid along that axis is! So suppose we decide to make c squared bigger, then Latvia here has more room to grow inside bubble! Now we say c squared because it goes for both directions, so negative numbers work too, for they are same as positive when squared. BUT if we decide to make c squared smaller...

(Shrinks bubble down over Latvia's head until it pops)

Well... you get picture. Now suppose one of these variables is not positive but NEGATIVE... say, instead of plus z squared over c squared we have minus z squared over c squared? It can really be x squared or y squared that is negative too, but we're just going to pick on z for entire time.

Latvia: Why does he always pick on me?...

Russia: Well, that would make sort of, um, hourglass shape. Anybody have hourglass?

Prussia: Kesese I have this! (throws corset at Russia)

Russia: What is this? Oh, is corset! Perfect!

Prussia: Kesese got it out of Specs's closet! I didn't know he used this to keep his girlish figure!

Austria: That was HUNGARY'S!

Prussia: Sure, sure.

Russia: Anyway, graph will be in hourglass shape, and it will be open along whichever axis is negative. Which in this case...

(Shoves corset over Latvia's body)

Latvia: Mister Russia... this is bone-crushingly tight, even for my skinny body...

Russia: Good, then it is working! This shape is what you call one-sheet hyperboloid. It is shaped like corset worn by whichever axis is negative. Now, when x squared over a squared plus y squared over b squared minus z squared over c squared equals zero, you got elliptic cone, and it looks kind of like this:

(Pulls corset strings tight, pinching Latvia's waist and making him scream)

Russia: Well, basically it's supposed to be tiny and have zero width at the origin, and open up without curving like ice cream cone on top and bottom. But it appears that Latvia is too fat to be squeezed to be that skinny...

Now, to make hyperboloid of two sheets, we got to have x squared over a squared plus y squared over b squared minus z squared over c squared equals NEGATIVE one. Multiply it all by negative one and you get negative x squared minus y squared plus z squared equals positive one. Which means that x and y axes must share corset.

Latvia: What a relief, not me for once...

(Russia stretches corset over Lithuania and Estonia until it rips)

Russia: Oops, well you see, they do not call it two sheet hyperboloid for nothing. What it looks like would be if I had two bowl shapes, one on top of Latvia's head and one below his feet...

Prussia: How about cup shapes? (tosses bra at Russia) ALSO from Austria's closet!

Austria (flushes) ALSO Hungary's!

Russia: Right, right. (Removes cups from bra) Now if I take cups out, and balance one on Latvia's head so that it opens upward, and make him stand on the other one so that it opens downward, then we've got a two sheet hyperboloid.

Latvia: Well, this isn't so bad...

Russia: Now we going to change things, get rid of some of squaring.

(Removes the "2" superscript from the Latvia's head, and erases superscripts from the a, b, and c post-it notes.)

And we multiplying, not dividing.

(Removes a, b, and c post it notes and puts a and b on Lithuania's and Estonia's foreheads, stops and stares at c)

Well, actually, we can get rid of c. (Tosses c paper)

So what we got is z equals a times x squared plus b times y squared. Now this looks kind of like a parabola when you slice it on x and y axes, right?

(Baltics shiver in fear at the mention of slicing)

Haha, we not going to actually slice. We just going to make elliptic paraboloid, which actually looks like we just got rid of bra cup from under Latvia's feet.

(Yanks bra cup from under Latvia's feet, causing him to fall down. Russia props him back up and puts bra cup back on his head.)

Now, if a times x squared and b times y squared were both negative, he'd be standing on bra cup instead. But what if, say, a times x squared were negative and b times y squared were positive, or vice versa? Then it curves up on one axis and down on the other, and we got what looks like a horsey saddle. Anybody got horsey saddle?

Poland: Ooh! Like, I do, Mister Russia! (Tosses saddle for Russia to catch)

Russia: OK, so when a times x squared is the Baltic - erm, axis that is negative, it will be Estonia, I mean, y axis that is wearing saddle. (Puts saddle over Estonia, makes Latvia stand on it)

Lithuania: This saddle... is digging into my neck...

Russia: It's OK, soon it will be your turn to wear it! Now Lithuania wears it when Estonia instead is negative! (Switches saddle to Lithuania)

Estonia: This saddle is poking me in a very uncomfortable place...

Russia: Well... that's it for today's lesson! Please be sending in more questions for Russia to be answering! I will be answering a whole bunch next time, for everything is one with Russia now and so Hetalia M & S! And remember, in Soviet Russia, math learns YOU!

Estonia: What's-that-supposed-to mean? (gags and groans) Get Latvia off of me now since we're done...