Work Text:
Lucifer finally hauls his pruny butt out of the bathtub - he was up at 4 in the morning playing Battleship and Marco Polo with those rubber ducks in there! - and arrives at the breakfast table in his pink houserobe and rubber duck slippers. I can't wait to see how he reacts to his breakfast talking back at him!
I've already finished my bowl of Jambalaya Crunch and I'm now lounging on the couch watching this silly little pageant called the "Overlord Games - Sponsored by VoxTek". It seems that for this year's theme they couldn't make up their mind on whether to do "kids' science fair projects" or "historical disaster reenactments" and right now they're sending off the bottle rockets slash Challenger disaster tributes.
I find VoxTek's rocket launches to be HILARIOUS. Oh, that TV-headed fellow wants to reach Heaven so bad... sadly, his test rockets always fail before they reach low ring orbit! Vincent always tries to save face by claiming that he meant to do that, he meant for it to crash-land, he knew they weren't ready to launch anything heavier than a satellite into orbit, not to mention all that malarkey about how it's necessary to fail a countless number of times in order to achieve success - as somebody who has failed countless times before succeeding in my attempts to make radio contact with Hell while still an Earthly human, I believe I am qualified to comment on this: there's a difference between failure that is calculated for the purpose of teaching a specific something, such as what configurations are unlikely to be fruitful to test in future endeavors; and sloppily planned failures that are nothing but wasteful, for the purpose of shock-and-awe to grab audiences' attention. And VoxTek's failures, for all that Vincent wants people to believe that they are well-planned calculated failures, more often than not prove under close scrutiny to be a better fit for the latter category of failure.
But what really gets me is the kettle logic of Vincent pushing the "fail faster" manifesto in one breath, and blaming signal interference in the satellites' GPS system (gee, I wonder who could be behind that?) for his failures in the next.
So this whole sponsored Overlord Games nonsense is nothing more than VoxTek's desperate attempt at headhunting - to find fresh blood to polish the company's tarnished reputation. The most impressive candidate so far, at least in my opinion, is the one-man team of "Baxter", a bioluminescent, fishy-looking sinner from 1910s Germany, for the sheer number of fields that he has to be competent in to be doing all of that on his own and still being able to compete with the multi-person "teams" who are registered for this competition. The second most impressive in my opinion, as much as it pains me to admit it, is the one-man team (if you don't count his little egg minions) of "Sir Pentious" - yes, that same snakelike fellow who is so pathetically desperate to be a cereal mascot and constantly trying to "do battle" with me - who, while underperforming relative to several of the multi-person registrants, at least has the self-respect to be doing all of it on his own.
Lucifer seats Charlie after Lilith has finished bathing her in the other bathroom and brings down the three new impulse-buys - ahem, cereals that he discovered the other day; he HAD to have them, because nothing entices him more than eating the same old food that he always eats, but with an all-new homology introduced as an optional format for consumption. The spherical puff of genus 0 is the simplest and cheapest homological format of course, and if a company wants to cut costs without resorting to such pitiful lows as shrinkflation then any features that make the cereal shape non-homotopic to a sphere will be the first thing to go. But of all these creative new ways they're finding to rebrand the basic sphere homology - "sherbet scoops", "bloopers", and so on - this "donut holes" rebranding, complete with a mathematical justification, definitely takes the cake.
It's really too bad that the math is incorrect - off by a factor of two!
I decide that I'm still hungry after eating my breakfast and head to the kitchen to pour myself another bowl of cereal. Wait, what's this - Pizza Blasted Goldfish? I think I'll try that! I pour it into my cereal bowl and sit myself down next to Lucifer's seat while he's pouring the chocolate flavored Krave donut holes to Charlie at her request.
Lucifer side eyes me as he begins pouring his own bowl of cereal. "What are you grinning at?" He turns his eyes to look at the box and immediately frowns.
"Al! You didn't have to insult the cereal box like that! What did the cereal ever do to you?"
I am still laughing at his reaction. I eat one of the Goldfish, pleased to see that the cereal piece is smiling back at me. "Oh, Lucifer... how is it that you literally wrote the book on Hell's geometry but you can't recognize an incorrect surface area equation for computing the surface area of the annular planetoids that you LITERALLY CREATED?"
"What? It doesn't look incorrect to me..."
"Are you confusing the surface area equation with the volume equation now? And even then it would be incorrect for volume."
"It's been a while since I created Hell's rings... what's the big deal anyway?"
"What's the big deal? WHAT'S THE BIG DEAL?!" I cackle and grab another Goldfish and almost eat it until I notice that it isn't smiling. It doesn't even have a face - what kind of vegan rubbish is this?! I won't eat any food WITHOUT a face! I hand it over to Charlie.
"Lu, tell me. If you were to ask yourself, which shape has the greater surface area to volume ratio, a sphere or a torus - and you answered according to your gut-feeling, your visual intuition, which would you answer?"
"Why the torus of course... I mean it just looks like it has more surface for its area... I can like feel it in my mind almost... but sometimes intuition can be deceiving, which is why the mathematics is all that matters."
"All right then, so let's see what the mathematics has to say!" I run off and return with a pencil and a notepad. "Now you know what the surface area of a sphere is, correct?"
"Yeah, that would be 4 pi r squared..."
"And do you know how to find the volume of a sphere from the surface area?"
"The equation for the volume is 4 over 3 pi r squared... I mean, cubed if I'm not mistaken?"
"Yes, it's the result of integrating with respect to r from the radius value of 0 to its full value on the sphere..."
Lucifer peers at my work. "Okay, you skipped a load of steps, but I follow your logic..."
"All right, so now, what is the surface area equation for a torus? If your feeble little mind needs a refresher, then how about I walk you through how to solve it right here and now... you take the circumference with respect to little r, that's 2 pi r around the poloidal axis, and integrate it with respect to the toroidal axis - I'm expecting that you can tell which angular coordinate is which from the context..."
"The angular coordinates are... interchangeable... both integrate over 2 pi... I see, oh. They're wrong. I'm guessing they integrated over only one pi!"
I laugh while taking a break from scribbling equations down to pick out all the faceless Goldfish to give to Charlie. "Which would be the correct equation if this were a sphere... but it's not! They're forgetting that angular fraction of surface area through the hole!"
Lucifer says, "Well you proved yourself right, Mister Smartypants, but the main point that they were trying to make is that the sphere homology maximizes the glaze to dough ratio - at least relative to the torus. Can you prove they are wrong about that now?"
"I'm way ahead of you..." I compute the volume of a torus on paper and show it to Lucifer.
"And now we divide surface area by volume for each, sphere and torus, to find out which has the greater glaze to dough ratio. What's the glaze to dough ratio for the donut hole, Lucifer?"
"That would be... 4 pi r squared divided by 4 over 3 pi r cubed... so that's 3 over r."
"Excellent. Now what's the glaze to dough ratio for the donut in all its toroidal glory?"
"Ok that's - 4 pi squared big R little r divided by 2 pi squared big R little r squared... that's 2 over little r... that's actually smaller... but it really depends on how the torus's little r compares to the sphere's r, they're not the same thing so what am I missing here..."
"Perhaps you should set a constraint on their volumes so as to make sure they are both equal to a constant?"
"Right, set 4 over 3 pi r cubed equal to 1... so we get r equal to the cube root of 3 divided by 4 pi... so that means that the value we get for the glaze to dough ratio is 3 times the cube root of 4 pi divided by 3... now for the torus, set 2 pi squared big R little r squared also equal to 1... solve for little r and we get little r equals the square root of 1 divided by 2 pi squared big R. So 2 over little r equals 2 times the square root of 2 pi squared big R. Wait, wait..."
I snicker as the Devil, stumped, clutches his forehead and sprouts horns and breathes fire in frustration over this.
"There isn't just one answer... even with the volume constraint, the big R is still a free parameter... there is a whole range of little r and big R coordinate pairs that will fit the volume constraint but they all probably have different surface area values!"
"Then pick one."
"Pick one? But how? Do I find the maximum or the minimum of the equation relating them to each other? I'm not sure that equation is even bounded..."
"Just pick a pair of values that fits the constraints. If you can show that even one such pair of values exists for a torus then you've already proved that the sphere is suboptimal by showing that a superior non-spherical configuration exists."
"That doesn't sound very rigorous..."
"Well if you want rigor then try proving the isoperimetric inequality. That's the proof that the sphere actually MINIMIZES the glaze to dough ratio! HAha, so it really doesn't matter which pair of radii values with the volume equals 1 constraint that you pick for your torus - it's guaranteed to have a greater surface area than any volume 1 solid of spherical homology! It's genius all right - a genius way to save money on glaze!"
"But some of those radii values will result in a torus with a lower glaze to dough ratio if you use the wrong torus surface area equation, that's smaller by a factor of two..." Lucifer remarks.
"But even then there are still plenty of radii values that result in a torus with the same volume but greater surface area even by those miscalculations! HAHA!"
"But wait... maybe they meant like how physically speaking it's hard to get enough glaze in the centers of donuts so the inside of the donut's hole often comes out under-glazed!" Lucifer shoots me a smug look, as though being a habitual sweet eater had taught him something that a sweet abstainer such as myself wouldn't have known.
I laugh and tell him, "Even if that were the case - let's assume the glaze is missing from the inner half of the poloidal circumference - that would still be less than half of the surface area, not enough to justify the absent factor of 2."
"Well how about the Banach-Tarski paradox huh?!" Lucifer raises his voice, sounding just a little unhinged as he contemplates the controversial implication of set theory's Axiom of Choice. "According to the Banach-Tarski paradox a sphere can be taken apart and reassembled into two completely identical new spheres, maybe that's what they meant! Y'know that's how Jesus fed 4,000 people with only 7 loaves?"
"Really."
"Uh huh... I... think? Hehe..."
"And you think a cereal company has those same miracle working capabilities - which even you, the most powerful magical being in Hell, have failed yet to demonstrate."
"I... dunno... I'm just playing devil's advocate, haha, I'm giving them the benefit of the doubt..."
"Lucifer... first of all the Banach-Tarski paradox doesn't just apply to spheres. It can also be used to duplicate a torus. And secondly, even if they had used it, just once, to duplicate the sphere and not the torus, the torus equation would STILL be incorrect - and at that point, the equation for the sphere's surface area would be incorrect too - would need to change that to 8 pi r squared to account for the Banach-Tarski duplicate - and there would be not just one but TWO incorrect equations on this cereal box!"
"But... but the ratio of the spheres' surface area to the torus's surface area would be correct then at least! They'd just need to divide it all by a factor of two! And the doubled glaze ratio of the duplicated donut hole-"
"Lu, are you forgetting that the addition of a Banach-Tarski doppelgänger to double the donut hole's surface area also doubles the donut hole's VOLUME? The glaze AND the dough are BOTH doubled... so the glaze to dough ratio stays the same."
"...dang it you got me... oh who am I kidding, there is no defending this... this is all my fault for letting myself go, I used to be Hell's premier geometer and topologist, but then I created the Ars Goetia and they kind of took over my math responsibilities from there... speaking of which, I hope the homework assignments you're submitting to Andrealphus look neater than this," Lucifer made a face at the smiley faces and simplified veve I had doodled all over my work.
"So, aren't you curious to find out who's responsible for such an egregious oversight?" I ask.
"Absolutely! Well, you see, there are three cereals that came out with glazed donut hole versions - Apple Jacks, Frosted Flakes, and Krave."
"And who are the mascots for those cereals?"
"Oh well for Apple Jacks there's Apple and Cinna Mon but I don't think it's their fault, I mean are they even on your hit list?"
"Apple is."
Lucifer gasps - I imagine Apple is one of his favorite mascots, so he's not pleased to hear that he's on the hit list. "Oh, well, um, I doubt he's involved with this, as much of a little prankster as he is, haha, it's most likely to be Tony the Tiger's fault. After all he's the one who's been lobbying for influence on Hell's school systems..."
"What about the mascot for Krave?" I ask - the mascots for all three cereals are depicted on the back of the Apple Jacks box, and the Krave mascot appears to be an anthropomorphic piece of the regular Krave cereal - a square filled with chocolate gunk.
"Oh that's not a real mascot - you see, when a cereal has a cartoon piece of cereal as its mascot, well that's just a placeholder mascot, the type of mascot that they have when they can't get a real mascot..."
I take a look at the Krave Donut Holes box, and my eyes widen. "Um, Lucifer..."
I show him the box and he's shocked too. "No way... you're kidding!"
I nod, smirking - it appears that Krave, or the Donut Holes edition at any rate, has a new mascot, that mascot being none other than Sir Pentious.
I take Lucifer by the hand and rush him over to the couch to watch the rocket launch on TV - I notice that Sir Pentious has the logo for Krave Donut Holes on his ship - and not only that, but I see one of his Eggboiz doing some last-minute spray painting on the ship, and the images that he spray paints include the sphere and the torus and their surface area equations as printed on the cereal box, including the incorrect "2 (pi squared) Rr" for the torus.
"My, my... I hope our little snake friend didn't use that particular equation when calculating the heat and cold tolerances of his o-rings..." I take my staff out and attempt to use it to hack into the launch station from afar so I can have a little chat with him. It's only fair to warn him...
"Why, hello there!" I buzz in. The snake immediately flares his hood in alarm.
"What-what?! Who'sss there?"
"Ahaha haha, let's just say I'm an old friend... the first time we met, if you recall, I taught you a lesson about why it's a bad idea to design a manned vehicle which doubles as a floating maser cavity."
The snake hisses with rage as it hits him. "YOU! ALASSSTOR! We meet again! Remotely."
"Ahaha, but alas, that lesson had to be learned the hard way. Hopefully you will choose to learn today's lesson the easy way? Today's lesson being the importance of proofreading..."
"Proofreading? You mean my math? Oh I agree! I have proofread ALL of my mossst difficult calculationsss, and the lesssss difficult onesss I have left to my Eggboiz."
"Your - I beg your pardon - your Eggboiz? Those little egg-shaped cretins to which you assign much of your grunt work? My, my, that... explains a lot..."
"Lisssten, I know my Eggboiz aren't the brightessst bulbsss in the egg carton, but I'm pretty sssure they can handle a sssimple tasssk sssuch as proofreading my calculationsss for the sssurface area of my o-ringsss... it's only calculusss after all it's not rocket ssscience... whoopsss hehe I guessss it actually IS literal rocket ssscience but you sssee what I mean?"
"Yes, I see, and do you know what I see? I see that you underestimate how difficult calculus actually is for individuals whose intellect is considerably beneath yours. Now, are you sure you don't want to double-check your work on those o-ring calculations with your own eyes? I already did my own calculations and you know what I've found? That YOUR surface area equation for the torus is off by a factor of two! That means that you underestimate the rate of heat transfer, or how fast the o-ring heats up or cools down, since heat flux is a function of surface area, so I reckon that once the thrusters are on your o-ring will rise to the temperature at which it begins to malfunction twice as fast as expected-"
"SSSHUT UP!!!" the snake snaps. "You are jussst like Baxter! Baxter wouldn't ssshut up with the nagging me to check my work, pessstering me about o-ring thisss and sssurface area that! That'sss why we sssplit up inssstead of working as a team thisss year for the Overlord Gamesss as we normally do! Now, if you don't mind, I'm due for launch in t minusss one minute and ssseven sssecondsss... sssix sssecondsss..."
"Suit yourself... Personally, I would get the heck out of that thing and use one of the egg critters to man it as a crash-test dummy, just in case..."
The TV had already cut to a commercial break, to spare VoxTek the embarrassment of getting hacked and hijacked by a stranger, so nobody got to hear our little spat. I pick up the phone to call a friend who owns a casino and make a bet.
"Hello there Husker? You're watching the Overlord Games correct? I'm betting my entire fortune that the snake suffers o-ring failure before hitting the objective."
"Hey wait a minute who's Husker?" Lucifer asks.
"Oh, a gambling buddy."
"I didn't know you had gambling buddies... you better watch it, if it turns out you've been pawning off our family treasures to pay off gambling debts then Lilith's going to have your head mounted on the wall..."
"I don't gamble, Lu." I hang the phone up. "And this is not a gamble. That would be like saying that engineers gamble when they make engineering design choices. I would hope that they don't, at any rate..."
When the commercial break is over I watch Pentious's rocket launch. I can't believe it... the idiot didn't take my advice but instead chose to be his own crash-test dummy... after closely observing the rocket for the first few seconds of launch I pick up my staff to count down to Pentious.
"Three... two... one."
The rocket explodes. Sir Pentious is seen blasting off again in his emergency jet pack parachute - at least he thought to wear that - hissing through his last staticky transmission, "CURSSSE YOU ALASSS-" before being completely cut off.
"Well there you have it! Sir Pentious's rocket failed because his torus equation was missing a factor of two! But as luck would have it, I happened upon that missing factor of two, and I used it to multiply my own fortune! Let this be a lesson to all you homework-weary schoolchildren out there - if ever you should find yourself wondering why it's important to know the precise formula for the surface area of a torus... THIS is the reason why! O-ring failure! If you want to be an astronaut or a rocket scientist when you grow up, heed my warning - whether or not you use the correct torus equation can make the difference between life and death!"
I remain tuned in to see how Baxter's rocket fares - his biomimetic rocket design, inspired by the jet propulsion of flying squid, with a fuel produced through chemosynthesis by the symbiotic bacteria from the deep-sea tube worms he had been studying, but with custom genetic modifications - definitely a sinner worth keeping an eye on; his innovations will spark the beginning of a new era for VoxTek - for better and for worse.