For a movie with the tagline “Live, Die, Repeat,” it should therefore not be too surprising that the movie Edge of Tomorrow feels a little… familiar.
For a movie with the tagline “Live, Die, Repeat,” it should therefore not be too surprising that the movie Edge of Tomorrow feels a little… familiar.
[ We open on a devastated, post-apocalyptic world, struggling to recover from a decade-long zombie war. Life on Earth is nearly extinct, and humanity clusters in a few countries around the planet -- a thriving Cuba, a democratic China, a struggling USA -- while hordes of zombies still roam the ocean floors or remain frozen near the poles--]
Hollywood executive: Cut! Cut! Cut! What the hell is this? The zombie war is already over? The movie just started!
[ Well... yeah. The book is called World War Z: An Oral History of the Zombie War. The word "history" kind of implies these events happened in the past. ]
Hollywood executive: Yeah, yeah, TL;DR. That’ll never sell. We need to start at the beginning. We need action. Wait, did you say America was struggling?
[ Yeah. In the book, America is nearly destroyed by a combination of the arrogance of its military leaders and the pseudo-scientific gullibility of its citizens. ]
Hollywood executive: No, no, no, nix that. This is America. We win wars, dammit! USA! USA! USA!
[ We open on the modern world, where everything is honkey-dorey... but we show a montage of news-reel footage to suggest the upcoming zombie apocalypse. ]
Hollywood executive: Great. Now who’s the hero in this movie.
Max Brooks: That would be me, I guess. I’m a United Nations agent compiling a post-war report for a special commiss–
Hollywood executive: Boooooooooring. No, you’re a gritty, no-nonsense UN investigator. Who’s good with guns! And can fly planes! Also, you’re Brad Pitt!
Brad Pitt: Kick. Ass.
[ Brad, his wife Mireille, and his two daughters are driving through Philadelphia... when the zombie apocalypse happens. Hundreds of slow, lumbering undead shamble aimlessly through the streets-- ]
Hollywood executive: No, no, no, cut, cut, cut! What the hell is this? Slow zombies? That’s been done. We need something new: fast, aggressive zombies. Something that’s never been done before!
Hollywood executive: Didn’t I fire you?
[ Brad, his wife Mireille, and his two daughters are driving through Philadelphia... when the zombie apocalypse happens. Hundreds of fast, agile zombies sprint through the streets. They leap onto unsuspecting public, ripping into their skin, tearing our their flesh, gorging themselves on brains. Blood flows-- ]
Hollywood executive: No, no, no, cut, cut, cut. We need a summer blockbuster here, people. That means we need kids and moms to come see this movie too, alright? Strictly PG-13, right?
[ Brad, his wife Mireille, and his two daughters are driving through Philadelphia... when the zombie apocalypse happens. Hundreds of fast, agile zombies sprint through the streets. They leap onto unsuspecting public, and bite them, once and bloodlessly, on their arms, and then move on, kind of like that creepy purple-plague episode of the Smurfs. ]
Hollywood executive: That’s better. That’ll bring in the kids…
[ Brad and family commandeer an RV, drive to New Jersey, and meet a recently orphaned Hispanic kid. Brad and
Angelina Mireille adopt him. ]
Hollywood executive: … and that’ll bring in the moms.
[ Brad and his family are helicopter-rescued by is UN buddy Fana Mokoena and brought to an aircraft carrier at sea. ]
Fana: Brad, I need you lead a team around the world to find the origin of the zombie virus.
Brad: No, I can’t leave my family.
Fana: Alright, but they’re got getting any more speaking parts ion this movie.
Brad: Fine. I’m on my way.
[ Brad assembles a crack team of Navy seals and expert virologist Elyes Gabel, and fly to South Korea. ]
Elyes: I’m an expert on plagues and viruses and am humanity’s last hope for finding a cure for the zombie pandemic. My mental prowess cannot be overstated. We shall first need to locate the virus’ origin, deduce the mode of transmission from this, and reverse engineer a—
Hollywood executive: No, no, no, cut, cut, cut. The plan is to have some egg-head save the day with science? What the hell is this, Fantastic Four? No, that crap tanked. Lose the nerd.
Elyes: Pardon me a moment while I clean this loaded gun.
[ Elyes shoots himself in head. ]
Brad: I guess it’s up to me now.
Hollywood executive: Better.
Brad: Do any of you soldiers stationed here know where the virus started?
Brad: Well, poo. Where should be go next?
Soldier: [ Sneaks a quick peak at a copy of World War Z ] In the book, there’s a bit about Isreal…
Brad: Sold. Let’s go to Jerusalem.
Soldier: Just remember… the zombies are attracted to sound. The only way to get you back to your plane is to go… really…. quietly….
Brad: Sorry, I was setting by phone’s ringtone volume to 11. Did you say something?
[ Brad's phone ring. Zombies kill most of the soldiers and seals... but, you know, bloodlessly. Brad flies off to Jerusalem. ]
Ludi Boeken: I’m a high-ranking official of the Mossad, the Isreali intelligence service. I correctly predicted the zombie outbreak, and instructed the country of Isreal to secretly big a ginormous wall around the country to keep the zombies out.
Brad: Wait… you had enough time to build a 500 foot tall wall around the entire country of Isreal…. but you never thought to tell the rest of the world about the oncoming plague? And how the hell did you keep it a secret? Did you put a big tarp over the whole country? None of this makes any sense.
Ludi: Well, actually, the book explains this in great detail—
Hollywood executive: Details are boooooooooooring. Pick it up.
Ludi: Oh, look. Zombies.
[ Zombies form a 500-foot-tall undead human bridge to reach the top of the wall and attack the city. One zombie bites the hand of Daniella Kertesz, Brad's newly acquired Isreali soldier. ]
Brad: Don’t worry. I saw this once on The Walking Dead.
[ He effortlessly chops off her hand to stop the spread of the infection. ]
Brad: Wow. That was significantly easier and less bloody than The Walking Dead led me to believe.
[ Brad and Daniella hop on board a passing 747. Zombies form a 30,000-foot-tall undead human bridge to reach the plane and attack the passengers. Unfortunately, the plane crashes in Wales. Fortunately, it does so across the street from a World Health Organization -- or W.H.O. -- research facility, staffed by Peter Capaldi, Pierfracesco Favino, and the flower-dress lady from Agents of Shield. ]
Peter Capaldi: Hello. I’m the Doctor.
Brad: Doctor? Doctor W.H.O.?
Peter Capaldi: Just the Doctor.
Brad: I have a theory. The zombies have a weakness. Their weakness is weakness. Specifically our weakness. Our weakness is their weakness. If we’re weak, we’ll make them weak in a week.
Peter: Good idea. This being a cutting-edge research facility, we should easily be able to synthesize a pathogen that would trick the zombies into thinking your terminal…
Hollywood executive: What did I say about eggheads and science?
Peter: I meant to say, rats. All of our deadly disease stuff is stored in a single room, housed in the deepest, darkest part of the building, overrun with zombies.
Brad: Right. Here’s a crowbar, a bat, and an axe. Let’s go bash some zombie heads.
Hollywood executive: PG-13.
Brad: …metaphorically speaking.
[ They go into "B Wing," and are chased throughout the building by zombies, Benny Hill style. ]
[ Eventually, Brad makes into into the deadly disease room, but gets trapped inside when he bumps into the chattering Cenobite from Hellraiser. He injects himself with a random syringe of disease, and walks right past all the zombies. ]
Peter Capaldi: The good news, Brad, is that your plan works! We’ll be able to synthesize a camoflage virus in no time.
Brad: And the bad news?
Peter: You’ve got radioactive syphilis. Next time read the label on the bottle, okay?
[ Brad is reunited with his family, and everyone lives happily ever aft--- ]
Hollywood executive: Hold it, hold, hold it, cut. Two words, everybody: Se. Quel.
[ --but the war rages on. ]
After what seemed like an eternity, winter is finally over, and summertime heat waves are upon us:
Sorry about that, Box Elder.
Apparently, the heat is getting to the folks at the Weather Channel, too. They’ve apparently forgotten how bar graphs work:
(On a related note, WTF TWC?)
The Butterfly is rather fascinated by the cemetery near our house. (More precisely, she’s fascinated by the fact that the cemetery is full of dead people.) As we were driving home the other day, road construction detoured us by the cemetery, and so, having nothing pressing to attend to, the Butterfly and I decided to visit it.
We walked up and down the rows of headstones, and she asked all manner of questions:
…and so on.
As we were wondering through the graveyard, the usual bank of summer afternoon storm clouds rolled in from the west. Suddenly there was a bright flash of lightning, followed by a deafening crack of thunder than made both of us jump.
“We should probably go,” I said. “We don’t want to be out here in the open, because we might get struck by lightning.”
“Alright,” said the Butterfly. “Although, if we do get killed by lightning, at least we’re already in the cemetery.”
You can tell a lot about a driver by the vehicle they drive:
The administration building on the Komplexify U campus is an impressive, pillared edifice whose top floor houses a free public museum of geology that includes both bones of dinosaurs, plesiosaurs, and icthyosaurs (which I recently found out are not all the same thing). Fittingly, therefore, the walls and floors of the administration building are made of polished stone, and (future) geologists and paleontologists (like the Ladybug) can admire the colors and shapes of minerals immersed in the stone.
Me, on the other hand…. all I ever see is this:
Don’t… blink…. Don’t… blink….
About a year ago, several folks forwarded me Facebook links to stories about David Laferriere — a graphic designer who draws pictures for his kids every morning on the plastic sandwich bags in their lunch, collected for posterity on his Sandwich Art flickr page — together with a follow-up along the lines of:
Now, while I appreciate the apparent iron-clad rigidity of the logic involved here, I politely told them that sandwich bag art was David Laferriere’s thing, and me doing sandwich bag art for my kid would be a cheap knock-off of a clearly superior and more original product.
So I started making napkin pictures for the Ladybug’s lunch instead.
Apples and oranges, people.
I didn’t do it everyday (since the Ladybug eats a “hot lunch” at school once a week) and I never bothered to take pictures of them, but it turns out the Ladybug had been keeping some of her favorites stashed in a box in her room. Here are some of the ones she kept:
Unfortunately, near the start of January 2014, I simply ran out of ideas for quick pictures in the morning… so I handed the napkin over to the Ladybug, and told her to draw a squiggle on it — a random curve or two — in the hopes it would inspire something.
For example, the first squiggle I got — a swooping “C” shape — became an awkward meeting between Mr. Peanut and a fan:
My second squiggle was a crazy zig-zag curve, which became the Doctor hanging out with a Dalek:
so, without further ado, here is my collection of squiggle napkins from the first half of 2014. What do you see in them, and then you can click on them to see what I eventually did with it.
Seething with anger and embarrassment, Old Man Kronecker could think of nothing any more past running the Hilbert Hotel to the ground. His original plan of adding a single extra room to the hotel failed to produce a hotel with more rooms, and neither did adding what effectively amounted to adding a complete, second hotel. Kronecker realized that even if he built a hundred copies of the hotel and appended them to his Kabins, the Hilbert Hotel will still be able to accommodate the guests: the guests from the first set of Kabins would go in Hilbert Rooms 1, 101, 201, 301,…; the guests from the second set of Kabins would take 2, 102, 202, 302,…; the guests from third set of Kabins would take 3, 103, 203, 303,…; and continuing on this way, until the guests from the hundredth set of Kabins filling in the empty rooms 100, 200, 300, 400, …
With a sigh, he tore down in Krocker Kabins, and went back to the drawing board.
The same would be true for an extra thousand hotels, or extra million hotels, or extra billion hotels, or any extra finite number of hotels. And so Kronecker did better than that: to the single story of the Kronecker Kabins, Old Man Kroncker stacked copy after copy of the Kabins, producing (at the expense of most of his fortune) a massive, infinitely tall high-rise. He unveiled his new and improved Kronecker Kondos, a building with an infinite number of floors (Floor 1, Floor 2, Floor 3, and so on), each floor with an infinite number of rooms (Room 1, Room 2, Room 3, et cetera). It was the greatest wonder along Historic Root 66…
…Or so Old Man Kronecker proclaimed. Kronecker flooded the airwaves with advertisements showcasing the vertically infinite and horizontal infinite nature of the new Kronecker Kondos, with rooms stretching to the horizon, above and beyond. This building, the adverts proclaimed, put the Hilbert Hotel to shame: it clearly had more rooms than the old Hotel, since one could find an infinite number of copies of the hotel in it.
Now, Mr. Gamow, the proprietor of the Hilbert Hotel, once again responded to the accusations against his venerable hotel, accusing Old Man Kronecker of slander. “Your Kondos are a hideous eyesore, sir,” snapped Mr. Gamow, “and completely unnecessary too, for the Hilbert Hotel still has room enough for all your guests.”
And so, urged in part by the television news networks (what better way to fill an uneventful 24-hour news cycle?), the two hoteliers struck a deal: a challenge to see whether the Hilbert Hotel had vacancies enough to house all of the guests of the Kronecker Kondos. In front of television cameras, Kronecker invited passersby to stay a spell at the Kondos, and soon the rooms were all occupied. “Now let me show you,” said Old Man Kronecker, “that the Hilbert Hotel is too small!” And with that, he asked each of his first-floor patrons to walk across the street to their corresponding room in the Hotel: Kondo 1 to Room 1, Kondo 2 to Room 2, and so on. They did this, and sure enough, none of the guests from any of the higher floors of the Kondos had anyplace to go.
The interstate positively surged with displaced patrons of the Kondos.
“Your Hotel is obsolete,” Old Man Kronecker spat. “Finally!”
“Hardly,” replied Mr. Gamow. “The problem is not the choice of hotel, but of hotelier.”
He then turned to the throng of displaced guests. “Our motto is Yes! Vacancies. Please give me a moment.” With that, he turned on the PA system, allowing him to speak to all the rooms in the Hotel at once. “Excuse me everyone,” he began. “Mr. Kronecker has placed you in the wrong room. If you would step outside a moment.”
When all the patrons of the Kondo were back outside, Mr. Gamow pulled out his bull horn and spoke to the masses. “Everyone, please take a look at your Kondo key please: it should have a floor number and a room number. By adding extra zeros to the front of either number, you can make them both have the same number of digits. To find your room in the Hilbert Hotel, just take your floor number and intersperse its digits with your room number. For example, if you were on Floor 132 in Kondo 456, then your two numbers are
1 2 3
4 5 6
and if we “thread” the digits together we get
1 4 2 5 3 6
and so you’ll be staying in Hilbert Hotel Room 142,536.
“Similarly, if you were on Floor 39 and Kondo 5067, then your two numbers are
0 0 3 9
5 0 6 7
and if we “thread” the digits together we get
0 5 0 0 3 6 9 7
and so you’ll be staying in Hilbert Hotel Room 5,003,697.
“If anyone needs help,” he added, “I’ll be in the office to assist you. Thank you.”
And so the Kondo guests looked at their room keys, added their extra zeros (as need be), and found their way to their rooms in the Hotel. And soon the interstate was empty.
“Voila,” said Gamow. “There is room enough in the Hilbert Hotel for all your guests.”
“I can see you got the guests off the street,” sputtered Kronecker, “but you must have doubled up some of them in a room or two or…”
“You figure we would have heard a complaint from them by now,” observed Gamow.
“You’ve got audience sympathy,” Kronecker sneered. “You’re cheating.”
“You think I’ve got two people in a room,” said Mr. Gamow, “but I assure you I do not. Pick a room, any room, and I shall tell you who’s in it.”
“Let’s see. That number has six digits: 3 7 8 9 2 0. If we take every other digit starting with the first we get 3 8 2, leaving 7 9 0. Therefore, that would be the guest from Floor 382, Kondo 790.”
“Let’s see. That number has seven digits, so let’s add an extra zero at the first to make it 8: 0 4 5 7 1 0 0 0. If we take every other digit, starting with the first, we get 0 5 1 0, leaving 4 7 0 0. Therefore, it must be the guest from Floor 510, Kondo 4700.”
They tried this for several more minutes, with Old Man Kronecker giving out room numbers, and Mr. Gamow correctly identifying the unique resident in it.
“Look,” Mr. Gamow said, “this is a very simple algorithm. Every 6-digit number in the Hotel corresponds to a unique address in the Kondo — the digits in the odd-numbered spots give the floor, and the digits in the even-numbered spots give the room. Every five-digit number in the Hotel does the same thing, except the odd-spot digits give the room and the even-spot digits give the floor. Every guest in your massive Kondo has a room set aside for them in my Hotel, and there’s only one resident per room.”
And with that, the Hilbert Hotel became the single greatest news story on the cable networks… well, except for Faux News, who instead ran with the story that Mr. Gamow, who had been caught on camera saying the Islamic-sounding “al-gor i’bn,” was therefore almost certainly a traitor or jihadist, and probably both.
Old Man Kronecker, on the other hand, quietly took the plans for his next building…
…and tore them up into little tiny pieces, and resigned himself to accepting that the Hilbert Hotel was there to stay.
…to be continued.
The Hilbert Hotel was originally conceived by George Gamow in his 1947 book One Two Three… Infinity. I’ve taken his iconic metaphor and transformed it from a gala-metropolitan high-rise it into a 1950s-era roadside motel, but have otherwise kept the spirit of place intact, which is to illustrate many of the counter-intuitive properties of infinite sets.
This third fable of the Hotel illustrates once more the paradoxical nature of the infinite — that adding an infinite number of copies of an infinite set need not change its cardinality.
If the first fable showed that the set of counting numbers has the same cardinality as the superset of natural numbers , and the second fable showed that it has the same cardinality as the complete set of integers , then what does this third fable show us?
Well, if the Hotel is a metaphor for the natural numbers , then the Kronecker Kondo is a metaphor for the set of ordered pairs of natural numbers ; hence, the set of all pairs of counting numbers has the same cardinality as the set of (singleton) counting numbers. Gamow’s room assignment is based on the fact that any natural integer can be expressed uniquely in the form
where each of the numbers are integers between 0 and 9, and all but finitely many of them are 0. This is the precise way of stating that is the digit in the ‘s digit of the integer . Gamow’s “interspersing” or “threading” of the room numbers is given by the function given by
Its corresponding inverse map is
A neat consequence of this the fact that the set of nonnegative rational numbers — that is, the set of numbers expressible as fractions, like 1/2 and 114/287 and 4 (which is 4/1) — has the same cardinality as the set of natural numbers. For it is clear that, on the one hand, the set of nonnegative rational numbers has at least the cardinality of the natural numbers, since the function
is a one-to-one function from to . On the other hand, the cardinality of the set of ordered pairs of natural numbers is at least as great as the cardinality of nonnegative rational numbers, since the function
is a one-to-one function from , provided the input fraction is expressed in reduced form. Since we’ve shown that the latter set has the same cardinality as , this is equivalent to saying that
These two statements mean that the natural numbers and the set of nonnegative rational numbers have exactly the same cardinality by virtue of a classic result called the Cantor-Bernstein-Schroeder Theorem (here’s a pair of nice proofs at Wikipedia).
In fact, a similar line of argument shows that
Hence, we’ve just show that, even though
counting numbers integers rational numbers,
it turns out that all three of these crucially important sets of numbers have the exact same cardinality!
Is there nothing the Hilbert Hotel cannot fit?