JUNE 18TH, 2017
S: Today is, as all Beatlemaniacs know, Sir Paul McCartney’s 75th birthday. So, some of this lesson must be about that.
J: It’s all about the birthday hats for his birthday celebration.
S: Is it a previous birthday so that the other three are involved?
J: Sure! Pick a birthday.
S: Okay… Hmm… Maybe his 16th birthday? John and Geo would probably be there… or his 21st; that was a nice one… or his 22nd; that was in my favorite year… You choose out of those three!
J: The 22nd.
S: Actually, I just got a really good idea for the 16th.
J: What is it?
S: Aunt Mimi. She crashes the party because John did something or forgot to do something.
J: John forgot to do his conic sections math homework.
S: He never did ANY homework, though. Aunt Mimi just about gave up. But it’s supposed to be a story, so she comes in later. Any ideas for how the story problem will go?
J: Hmm… Do they have any fans yet?
S: No! This is 1958! By late ’59, they probably did, but I don’t think they did in ’58… Wait – what kind of fans do you mean? Rabid fans, or friends who just said, “Hey, I like your music!”?
J: Fans who want parts of their hats.
S: Oh! I know! Fans going back in time! And if they’re really going back in time, it must be at least year 2300. Okay, begin!
J: So, we’ve gone back in time-
S: Since when are YOU a Beatles fan?
J: The FANS have gone back in time to bring back samples of the Beatles’ birthday hats, but without disturbing the timeline.
S: Oooh. That’ll be hard!
J: You’d think so. But we have the ability to mathematically specify specific parts of these conical hats – hopefully by taking infinitely thin conic sections of the hats – and there will be no disturbance of the Beatles’ timeline.
S: Even though one of them is not there.
J: Here is the picture we’re basing this off of:
S: Okay, so, taking sections of cones that produce different shapes!
J: Suitable for bracelets or tiaras or other Beatlemaniac accessories.
S: But how would the fans wear them? They are 2D shapes, not strings, and they are infinitely thin!
J: They’re easy enough to put on – but ONLY YOU will know that you wear part of a Beatles’ birthday hat.
S: The poor Ringo fans… Don’t you feel sorry for them?
J: Oh, well, they’ll just have to go back in time to a different birthday for themselves.
S: Okay, start the math component. We’ve already seen the picture.
J: We’re taking a circle from Paulie’s hat.
S: Thanks for calling him Paulie!
J: So, we’re taking these sections by intersecting infinite planes with the cones, or party hats.
S: Are “we” using lasers?
J: No. We’re using the time machine computer. We give it the equation for the conic section, and it automagically extracts the conic section from the physical object in question, using an infinite plane to intersect it.
S: But doesn’t that mean that, since the section, no matter how thin, goes all the way through the hat, the top of the hat would fall off into Paul’s cake and disrupt time?
J: Well, we can rely on physics to hold it together. So, first, the circle. Do you see it?
J: It’s where the infinite plane is parallel to the bottom of the party hat (the so-called “cutting plane”).
S: Of laserness.
J: The equation for the circle is x2 + y2 = a2. We should also note that, parenthetically, conic sections have other useful, descriptive features, including eccentricity.
S: Here is the table with those descriptive features:
Could you please tell me what all those columns actually mean? Like, what is the eccentricity of a given shape?
J: The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular. So, if you look at the table, we’ve got an eccentricity e… So, you can see the eccentricities for different shapes with a fixed focus f and directrix-
S: Well, that doesn’t make sense because not all shapes are ellipses, so how could they all have eccentricities!?
J: So, if the eccentricity is between 0 and 1, there’ll be an ellipse.
S: Oh, I see! Because the only eccentricity in the table between 0 and 1 is for the ellipse. Okay, let’s move on to the next shape. What’s the difference between a parabola and a hyperbola?
J: There are many different ways of looking at it, but the hyperbola intersects the double cone in two places, whereas the parabola intersects in one.
S: Oh! That makes sense. Here’s another image to show that:
J: That plane that creates the hyperbola doesn’t have to be vertical… But the plane for the parabola – that’s not going to hit the mirror image party hat. The hyperbola is going to hit it in two places…
S: I can see that. So, is there any sort of problem to solve?
J: Well, I thought we’d just describe the conic sections.
S: Okay, so, we already described the circle… next, the ellipse! The ellipse as a conic section… is… at an angle – how do I explain it?!
J: It is an eccentricity between 0 and 1.
S: Yes, that is the crazy math term. What is it in human terms?
J: Oh, well, it’s from a plane that’s NOT parallel to-
S: -the bottom of the cone, OR to Paul’s head! Or to George’s or John’s, or the other people there. But I don’t know who the other people there are. Ooh! I bet Ivan’s there! I’m SURE Ivan’s there. They have the same birth- wait. They have the same birthday. It’s probably not a joint party, but maybe it is! Or maybe they just never go to each other’s birthday parties because they’re always on the same day… I don’t know. So, Ivan MIGHT be there, or he might not, and if he is, then it’s also his birthday, or maybe they had birthday parties at different times on the same day, or maybe, in 1958, June 18th was a weekday, so they would have had birthday parties on separate weekends, or maybe it wasn’t… What do you think?
J: *sweeps hand over head* Sheeeeeeew!
S: Okay, so, we were talking about ellipses, right?
J: Right. Here’s a diagram of the parts of an ellipse:
S: So, the point (if we’re just talking about conic sections) is that ellipses are not parallel to the top of the head. But no! Human heads are not flat! So this is ALL wrong!!! DUN, DUN, DUUUUUUUUN!!!
J: The party hats may be just fine.
S: Well, they’re on heads that are not flat-
J: Well, perhaps they are very stiff, or else have been placed on the heads very gingerly, and so are not distorted.
S: But the point is that the heads aren’t flat, so the bottom of the cone isn’t flat, so you can’t measure this stuff! Oh, whatever.
J: So, the equation for the ellipse?
S: *looks at table shown previously* x2/a2 + y2/b2 = 1
S: What ARE x, a, y, and b?
J: Well, we’ve got a and b, and this diagram- where is it? *takes a long time looking for the diagram*
S: *is waiting*
S: *gets bored*
S: Oh, and that other guy… whose name was Ian, I think? Once upon a time, I got him and Ivan confused all the time, and by the way, they called Ivan “Ivy”.
J: *makes uninterested sound*
J: *still looking for diagram*
S: “Jive with Ive the Ace on Bass” was his catchphrase…
J: What now?
S: “Jive with Ive the Ace on Bass” was Ivan Vaughan’s catchphrase when he played tea-chest bass in the Quarrymen!
S: Okay, so, GET TO THE MATH…
J: *doesn’t get to the math*
S: What IS a tea-chest? Is it a chest with tea in it?
J: I think so.
S: Jive with Ive the Ace on Bass, Jive with Ive the Ace on Bass, Jive-
J: So, you know what a radius is, right?
S: Of COURSE I know what a RADIUS is!!!
J: But an ellipse doesn’t have one radius, right? Instead, we’ve got a and b.
S: Oh! Okay.
J: a is in the x direction; kind of the like the x-radius, if you will. And you can think of b as the radius in the y direction.
S: So, y is left and x is right! I mean… That reminded me of that time in A Hard Day’s Night when Paul says, “I’m his grandfather! I mean…” Anyway, y is vertical and x is horizontal.
J: So, you could figure out what the effective radius would be, depending on the angle as we went around the ellipse from 0 to 360.
J: With this equation-
S: -the equation in the bottom left of the previous image-
J: r = b2 / a – c cos θ
S: Yes. r is the radius, by the way.
J: c is the distance from the center of the ellipse to the intersection with a-
S: Cosine? Does it say cosine?
J: Yes – cosine theta. Next, the parabola!
S: Yes, we know ALL about parabolas… ParaboliCAT *cough cough*
J: Well, we’re not cutting through, say, both sides of a cone in the way that we would get a circle or an ellipse; we’re just cutting through one side of it. And the equation for the parabola is…?
S: y2 = 4ax
J: And the eccentricity is 1.
S: So… that means… that it is an ellipse? Or something? Or does it mean that it’s a circle?
S: Well, then-
J: It means that it’s a parabola.
S: Oh… But I thought you said- OH! Because any eccentricity between 0 and 1 is a measure of how circle-like an ellipse is, right?
S: And a parabola is not a circle or an ellipse, so it doesn’t- I said “circur” instead of “circle”!
J: For the eccentricity of an ellipse to be 1, b2/a2 would have to be 0.
S: Oh. Do you know what we should never do? Talk in Manchester accents.
J: Oh, I couldn’t pull anything off there.
S: *in a Manchester accent* MANCHESTA!!! *goes back to Liverpudlian/London/Cockney/Yorkshire/other British stuff accent that she was using before* No! I’m so sorry, Beatles, forgive me! Because I can’t remember if it was just John or if it was all of Liverpool who hated Manchester people and were their mortal enemies. I just can’t remember if it was John or the entire city!!!
J: Oh no.
S: But either way, I can’t talk in a Manchester accent, because *switches to dramatic Italian-ish accent* then I would be betraying them!
J: And, of course, for the parabola, the semi-latus rectum and focal parameter are both 2a. On to the hyperbola!
S: *switches back to British accent from Italian accent* Yes.
J: So, we have the hyperbola…
S: Which is an infinite plane that the computer shoots through two… party hats that don’t exist in that oppositely stacked position (please see third diagram from top of post). Okay, here’s how the arrangement for the hyperbola does exist: George has an idea that he thinks won’t work because he’s tried it before, but he stacks two of the hats – John’s and his own – on the table, but he stacks them so that the tips are touching (again, please see third diagram from top if you haven’t already). And everyone is so amazed that he actually succeeds and the hats don’t fall over!
J: And the bottoms of the hats are parallel.
S: Yes, they are. But no one knows that it’s just the time travel computer sending messages to his brain that make his fingers set the hats up perfectly, and making sure the air current around the hats is favorable.
J: But this is without disturbing the Beatles’ timeline.
S: Yes, but it still works, somehow.
J: Very good. Good computer.
S: No, wait a second – that would disrupt time because when George originally tried it in real life, they fell over, and now they’re staying up.
J: Perhaps for that moment when they were in perfect balance before they fell over.
S: Ah, yes.
J: That would be useful.
S: The computer is lightning fast! It doesn’t need the hats to stay balanced for more than a decisecond!
J: So, we could have a section of an infinite plane that was going through both party hats, and that would give us a hyperbola.
S: Yeah. Because both of the hyperbolas are the same, right? Are they congruent?
J: They’re part of the same curve, but the two different parts don’t touch.
S: I know! I was asking if those “two different parts” were the same shape!
J: It really depends.
S: On the angle of the curve, or does it have to be straight and parallel with the center line of the two hats to be the same on both sides?
J: It depends.
S: FOR THE DEFINITION OF A HYPERBOLA, DOES THE PLANE HAVE TO POSITIONED PERFECTLY PARALLEL TO THE CENTER LINE OF THE TWO HATS?
J: Well, the rectangular hyperbola – that’s a special form, there. But just a general hyperbola, the top and bottom may look different.
S: Oh! So the plane could be at any angle as long as it’s still intersecting both hats, but the rectangular hyperbola is different.
J: Yes. So, the equation for the, uh…
S: I just winked at myself in my computer screen’s reflection like Paul does during “You’ve Got To Hide Your Love Away” in Help!
S: But, obviously, there weren’t computers then and he didn’t wink at himself; I just copied his winking style, that’s all.
S: He moves his jaw and opens his mouth sideways sometimes when he winks. It looks weird when I try to do that.
J: -similar to-
S: But it somehow doesn’t look weird when he does it. Maybe that’s why everyone loves his wink – because no one else in the world winks using their mouth.
J: -that of the ellipse. So, we’ve got x2/a2 – y2/b2 = 1.
S: And the difference between that and the equation for the ellipse-
J: -is that where this uses subtraction, the ellipse equation uses addition.
S: *starts humming “Santa Claus Is Coming To Town”* Why am I humming “Santa Claus Is Coming To Town”?
J: And those are the conic sections that we take from party hats.
S: The fans are so happy!
J: “WAIT!” says Aunt Mimi, bursting into the room. “I know you didn’t do your homework, Johnny, but there are the degenerate conic sections.”
S: Would Aunt Mimi really know about degenerate conic sections? Maybe she would.
J: Well, some authors do not consider them conics at all!
S: Ugh! How DARE they!!!
J: So, the degenerate conic sections are… a point-
S: A point! One dimension. 1D. Oh no! 1D! One Direction! HOW DARE YOU MENTION THEM, MATHEMATICIANS!!!
J: You can imagine a plane that was between those two party hats that George stacked on the table.
S: A plane that’s between… OH! A plane that is between them, and is perpendicular to the bottoms of the hats; therefore, it doesn’t touch the hats at all, except for the point you were talking about, which is the only place where the two hats touch each other!
S: And that’s just one little dot – it’s not even considered a conic section! But I think it is; I think it’s like a circle, because a dot is a circle. And there is no space between the two hats! But wait – there must be space between the particles that make up the hats, or they would… meld together as one… a statue of the party hats of two Beatles… preserved forever! Actually, probably thrown in the recycling bin, as the Beatles weren’t famous yet… Did they even have recycling bins in 1958? Or did they just put everything in the garbage? Yes, I think they just put everything in the garbage. Recycling bins are a fairly recent development.
J: Well, if you had someone who had a shop-
S: That was 58 years ago!
J: -they would probably use metal recycling in some form.
S: No, wait, that was 59 years ago. I was like, ’58 was 58 years ago! But then I remembered that it’s 2017, not 2016. I keep thinking it’s 2016! Can you say that again? I couldn’t hear you because I was yelling too loud.
J: Um, what was I saying?
S: I don’t know! I couldn’t hear you over my yelling! Never mind. Okay, so, is that the end of this unordinarily long lesson?
S: Ohhhhhhhhhhhhhhhh nnnnnnnnnnnnnnnnnnnnnnnnnoooooooooooooooooooooo. My life is dead. But life can’t die. Life is the opposite of death. That’s like saying, “My hot is cold, my cold is hot, my low is high, my high is low, my Beatles are One Direction, my One Direction is the Beatles.” In other words, I sound like Dr. Seuss.
J: So, another degenerate conic section would be one that intersects the cone-
S: I JUST DID IT!!!
J: -in a line, or-
S: I JUST DID THE PAUL McCARTNEY-STYLE WINK PROPERLY!!!
J: Do it again!
S: I can’t – I don’t know how I did it!!!
J: Or it could intersect it in a pair of intersecting lines. So, you could imagine a plane that chopped the party hats in half.
S: Oh, I see, okay. Wait – is that not considered a conic section?
J: That is a degenerate conic section. You end up with an X in space. You can picture it, right? The X where it cuts them?
J: And you can imagine a plane that just barely touched the top hat and the bottom hat.
S: But through the very middle? Or through just one side?
J: Um, it goes through the middle, but it’s not flat; it’s just tipped until it touches them.
S: OOOOOOOOOOOOOHHHHHHHHHHHHHHHHHHHHHHHHHH, I SEE, I SEE!!! It’s like you have two hats, and the plane goes like THIS:
J: If it’s tilted a little bit one way, then it intersects it in a point, and that’s the other degenerate case.
S: Quiz: Which Beatle has the most nicknames?
S: No. Geo. Because, I mean, he’s called George, Georgie, Geo, Jorj, Jorg, Jorge, and Joj. At least, those are all the names I’ve seen the fans call him in their blogs, fan art, fan comics, comments, memes, fan fictions, and the like. But I shouldn’t say “they”, since I’m one of them. But the sentence sounded confusing with “our”. Anyway, Paul just has Paul, Paulie, Pol, and Macca. And John has the least, because, really, the only thing you can do with John is Johnny. And then Ringo has Ringo, Ringsy, Rings, Ritch, and Ritchie. They probably all have more nicknames; those are just the mainstream ones.
J: So, for review, what are the seven conic sections?
S: Um… they are the circle… the ellipse… the parabola, the hyperbola… the, um… the dot…
J: The point, you mean.
S: Dot Rhone! Sorry – whenever I say or hear the word “dot”, I can’t help thinking of her.
J: The line and the intersecting lines.
S: The line and the intersecting lines.
J: Yes, and those last three are…?
S: The dot, the line, and the intersecting lines? THAT SOUNDS LIKE THE NAME OF A BOOK!!! “The Dot, the Line, and the Intersecting Lines.”
J: Some authors do not consider those last three to be proper conic sections at all.
S: Yes, it’s terrible!
J: So, they’re called…?
J: Okay, I think today’s lesson is over.
S: Yes. BBBBBBBBBBBBBYYYYYYYYYYYYYYYYYYYEEEEEEEEEEEEE!!! HAPPY BIRTHDAY, PAUL McCARTNEY!!! I LOVE YOU!!!
J: HAPPY BIRTHDAY, PAUL!!!
S: Your 16th and your 75th, and all the ones before and in between.
Above fan art: I think I found this on DeviantArt a long time ago, but I can’t remember where, and I just spent an hour searching the entire internet to try to credit it here. The only place I could I find it was on Zerochan, and I don’t think the person who posted it there drew it because there were other things they had posted that I knew were by PepperMoonFlakes or Wakonaga or WingsOverDA (all three on DeviantArt). So, to whoever drew this, I’m sorry that I can’t find you. If anyone reading this knows who drew this, please tell me so that I can credit it. The only reason I searched for an hour and didn’t use something else that I COULD credit is that this is my favorite Paul McCartney birthday fan art EVER.
Walrus in party hat painting link to original: Link
All conic sections diagrams from Wikipedia: Link
NOTE: Yes, I know that this was not posted on Paul’s actual birthday, but at least we did the lesson on his birthday.