Those crazy islands are the result of water filling in Noctis Labyrinthus (the Labyrinth of the Night), a bizarre set of canyons whose origin is still a mystery.
The oceans on Mars wouldn’t last. There might be some transient greenhouse warming, but in the end, Mars is just too cold. Eventually, the oceans will freeze over, become covered with dust, and gradually migrate to the permafrost at the poles.
However, it would take a long time, and until it did, Mars would be a much more interesting place.
When you consider that there’s a ready-made portal system to allow transit between the two planets, the consequences are inevitable:
Twitter
Q. How many unique English tweets are possible? How long would it take for the population of the world to read them all out loud?
—Eric H, Hopatcong, NJ
High up in the North in the land called Svithjod, there stands a rock. It is a hundred miles high and a hundred miles wide. Once every thousand years a little bird comes to this rock to sharpen its beak. When the rock has thus been worn away, then a single day of eternity will have gone by.
—Hendrik Willem Van Loon
A. Tweets are 140 characters long. There are 26 letters in English—27 if you include spaces. Using that alphabet, there are 27140 ≈ 10200 possible strings.
But Twitter doesn’t limit you to those characters. You have all of Unicode to play with, which has room for over a million different characters. The way Twitter counts Unicode characters is complicated, but the number of possible strings could be as high as 10800.
Of course, almost all of them would be meaningless jumbles of characters from a dozen different languages. Even if you’re limited to the 26 English letters, the strings would be full of meaningless jumbles like “ptikobj.” Eric’s question was about tweets that actually say something in English. How many of those are possible?
This is a tough question. Your first impulse might be to allow only English words. Then you could further restrict it to grammatically valid sentences.
But it gets tricky. For example, “Hi, I’m Mxyztplk” is a grammatically valid sentence if your name happens to be Mxyztplk. (Come to think of it, it’s just as grammatically valid if you’re lying.) Clearly, it doesn’t make sense to count every string that starts with “Hi, I’m . . . ” as a separate sentence. To a normal English speaker, “Hi, I’m Mxyztplk” is basically indistinguishable from “Hi, I’m Mxzkqklt,” and shouldn’t both count. But “Hi, I’m xPoKeFaNx” is definitely recognizably different from the first two, even though “xPoKeFaNx” isn’t an English word by any stretch of the imagination.
Our way of measuring distinctiveness seems to be falling apart. Fortunately, there’s a better approach.
Let’s imagine a language that has only two valid sentences, and every tweet must be one of the two sentences. They are:
“There’s a horse in aisle five.”
“My house is full of traps.”
Twitter would look like this:
The messages are relatively long, but there’s not a lot of information in each one—all they tell you is whether the person decided to send the trap message or the horse message. It’s effectively a 1 or a 0. Although there are a lot of letters, for a reader who knows the pattern of the language, each tweet carries only one bit of information per sentence.
This example hints at a very deep idea, which is that information is fundamentally tied to the recipient’s uncertainty about the message’s content and his or her ability to predict it in advance.1
Claude Shannon—who almost singlehandedly invented modern information theory—had a clever method for measuring the information content of a language. He showed groups of people samples of typical written English that were cut off at a random point, then asked them to guess which letter came next.
It’s threatening to flood our town with information!
Based on the rates of correct guesses—and rigorous mathematical analysis—Shannon determined that the information content of typical written English was 1.0 to 1.2 bits per letter. This means that a good compression algorithm should be able to compress ASCII English text—which is 8 bits per letter—to about ⅛th of its original size. Indeed, if you use a good file compressor on a .txt ebook, that’s about what you’ll find.
If a piece of text contains n bits of information, in a sense it means that there are 2n different messages it can convey. There’s a bit of mathematical juggling here (involving, among other things, the length of the message and something called “unicity distance”), but the bottom line is that it suggests there are on the order of about 2140 × 1.1 ≈ 2 × 1046 meaningfully different English tweets, rather than 10200 or 10800.
Now, how long would it take the world to read them all out?
Reading 2 × 1046 tweets would take a person nearly 1047 seconds. It’s such a staggeringly large number of tweets that it hardly matters whether it’s one person reading or a billion—they won’t be able to make a meaningful dent in the list in the lifetime of the Earth.
Instead, let’s think back to that bird sharpening its beak on the mountaintop. Suppose that the bird scrapes off a tiny bit of rock from the mountain when it visits every thousand years, and it carries away those few dozen dust particles when it leaves. (A normal bird would probably deposit more beak material on the mountaintop than it would wear away, but virtually nothing else about this scenario is normal either, so we’ll just go with it.)
Let’s say you read tweets aloud for 16 hours a day, every day. And behind you, every thousand years, the bird arrives and scrapes off a few invisible specks of dust from the top of the hundred-mile mountain with its beak.
When the mountain is worn flat to the ground, that’s the first day of eternity.
The mountain reappears and the cycle starts again for another eternal day: 365 eternal days—each one 1032 years long—makes an eternal year.
A hundred eternal years, in which the bird grinds away 36,500 mountains, make an eternal century.
But a century isn’t enough. Nor a millennium.
Reading all the tweets takes you ten thousand eternal years.
That’s enough time to watch all of human history unfold, from the invention of writing to the present, with each day lasting as long as it takes for the bird to wear down a mountain.
While 140 characters may not seem like a lot, we will never run out of things to say.
1It also hints at a very shallow idea about there being a horse in aisle five.
Lego Bridge
Q. How many Lego bricks would it take to build a bridge capable of carrying traffic from London to New York? Have that many Lego bricks been manufactured?
—Jerry Petersen
A. Let’s start with a less ambitious goal.
Making the connection
There have certainly been enough Lego1 bricks to connect New York and London.In LEGO2 units, New York and London are 700 million studs apart. That means that if you arranged bricks like this . . .
. . . it would take 350 million of them to connect the two cities. The bridge wouldn’t be able to hold itself together or carry anything bigger than a LEGO®3 minifig, but it’s a start.
There have been over 400 billion Lego4 pieces produced over the years. But how many of those are bricks that would help with a bridge, and how many are little helmet visors that get lost in the carpet?
Let’s assume we’re building our bridge out of the most common LeGo5 piece—the 2x4 brick.
Using data provided by Dan Boger, Lego6 kit archivist and operator of the Peeron.com Lego data site, I’ve come up with the following rough estimate: 1 out of every 50 to 100 pieces is a 2x4 rectangular brick. This suggests there are about 5–10 billion 2x4 bricks in existence, which is more than enough for our one-block-wide bridge.
&nbs
p; Supporting cars
Of course, if we want to support actual traffic, we’ll need to make the bridge a little wider.
We probably want to make the bridge float. The Atlantic Ocean is deep,[citation needed] and we want to avoid building 3-mile-high pylons out of Lego bricks if we can.
Lego bricks don’t make a watertight seal when you connect them together,7 and the plastic used to make them is denser than water. That’s easy enough to solve; if we put a layer of sealant over the outer surface, the resulting block is substantially less dense than water.
For every cubic meter of water it displaces, the bridge can carry 400 kg. A typical passenger car weighs a little under 2000 kg, so our bridge will need a minimum of 10 cubic meters of Lego supporting each passenger car.
If we make the bridge a meter thick and 5 meters wide, then it should be able to stay afloat without any trouble—although it might ride low in the water—and be sturdy enough to drive on.
Legos8 are quite strong; according to a BBC investigation, you could stack a quarter of a million 2x2 bricks on top of each other before the bottom one collapsed.9
The first problem with this idea is that there aren’t nearly enough Lego blocks in the world to build this kind of bridge. Our second problem is the ocean.
Extreme forces
The North Atlantic is a stormy place. While our bridge would manage to avoid the fastest-moving parts of the Gulf Stream current, it would still be subjected to powerful wind and wave forces.
How strong could we make our bridge?
Thanks to a researcher at the University of Southern Queensland named Tristan Lostroh, we have some data on the tensile strength of certain Lego joints. Their conclusion, like the BBC’s, is that Lego bricks are surprisingly tough.
The optimal design would use long, thin plates overlapped with each other:
This design would be pretty strong—the tensile strength would be comparable to concrete—but not nearly strong enough. The wind, waves, and current would push the center of the bridge sideways, creating tremendous tension in the bridge.
The traditional way to deal with this situation would be to anchor the bridge to the ground so it can’t drift too far to one side. If we allow ourselves to use cables in addition to the Lego bricks,10 we could conceivably tether this massive contraption to the sea floor.11
But the problems don’t end there. A 5-meter bridge might be able to support a vehicle on a placid pond, but our bridge needs to be large enough to stay above water when waves are breaking over it. Typical wave heights on the open ocean could be several meters, so we need the deck of our bridge to be floating at least, say, 4 meters above the water.
We can make our structure more buoyant by adding air sacs and hollows, but we also need to make it wider—otherwise it will tip over. This means we have to add more anchors, with floats on those anchors to keep them from sinking. The floats create more drag, which puts more stress on the cables and pushes our structure downward, requiring more floats on the structure . . .
Sea floor
If we want to build our bridge down to the sea floor, we’ll have a few problems. We wouldn’t be able to keep the air sacs open under the pressure, so the structure would have to support its own weight. To handle the pressure from the ocean currents, we’d have to make it wider. In the end, we’d effectively be building a causeway.
As a side effect, our bridge would halt the North Atlantic Ocean circulation. According to climate scientists, this is “probably bad.”12
Furthermore, the bridge would cross the mid-Atlantic ridge. The Atlantic sea floor is spreading outward from a seam down the middle, at a rate—in Lego units—of one stud every 112 days. We would have to build in expansion joints, or drive out to the middle every so often and add a bunch of bricks.
Cost
Lego bricks are made of ABS plastic, which costs about a dollar per kilogram at the time of this writing. Even our simplest bridge design, the one with the kilometer-long steel tethers,13 would cost over $5 trillion.
But consider: The total value of the London real estate market is $2.1 trillion, and transatlantic shipping rates are about $30 per ton.
This means that for less than the cost of our bridge, we could buy all the property in London and ship it, piece by piece, to New York. Then we could re-assemble it on a new island in New York Harbor, and connect the two cities with a much simpler Lego bridge.
We might even have enough left over to buy that sweet Millennium Falcon kit.
1Although enthusiasts will point out it should be written “LEGO.”
2Actually, the LEGO Group® demands that it be styled “LEGO®.”
3On the other hand, writers have no legal obligation to include the trademark symbol. The Wikipedia style guide mandates that it be written “Lego.”
4The Wikipedia style is not without its critics. The talk page argument over this issue featured many pages of heated arguments, including several misguided legal threats. They also debate the italics.
5OK, nobody styles it this way.
6Fine.
7Citation: I made a Lego boat once and put it in the water and it sank :(
8I’m going to get some angry mail about this.
9Maybe it was a slow news day.
10And sealant.
11If we wanted to try to use Lego pieces, we could get kits that include little nylon ropes.
12They went on to say, “Wait, what did you say you were trying to build?” and “How did you get in here, anyway?”
13My favorite Friends episode.
Longest Sunset
Q. What is the longest possible sunset you can experience while driving, assuming we are obeying the speed limit and driving on paved roads?
—Michael Berg
A. To answer this, we have to be sure what we mean by “sunset.”
This is a sunset:
Sunset starts the instant the Sun touches the horizon, and ends when it disappears completely. If the Sun touches the horizon and then lifts back up, the sunset is disqualified.
For a sunset to count, the Sun has to set behind the idealized horizon, not just behind a nearby hill. This is not a sunset, even though it seems like one:
The reason it can’t count as a sunset is that if you could use arbitrary obstacles, you could cause a sunset at any time by hiding behind a rock.
We also have to consider refraction. The Earth’s atmosphere bends light, so when the Sun is at the horizon it appears about one Sun-width higher than it would otherwise. The standard practice seems to be to include the average effect of this in all calculations, which I’ve done here.
At the equator in March and September, sunset is a hair over two minutes long. Closer to the poles, in places like London, it can take between 200 and 300 seconds. It’s shortest in spring and fall (when the Sun is over the equator) and longest in the summer and winter.
If you stand still at the South Pole in early March, the Sun stays in the sky all day, making a full circle just above the horizon. Sometime around March 21, it touches the horizon for the only sunset of the year. This sunset takes 38–40 hours, which means it makes more than a full circuit around the horizon while setting.
But Michael’s question was very clever. He asked about the longest sunset you can experience on a paved road. There’s a road to the research station at the South Pole, but it’s not paved—it’s made of packed snow. There are no paved roads anywhere near either pole.
The closest road to either pole that really qualifies as paved is probably the main road in Longyearbyen, on the island of Svalbard, Norway. (The end of the airport runway in Longyearby
en gets you slightly closer to the pole, although driving a car there might get you in trouble.)
Longyearbyen is actually closer to the North Pole than McMurdo Station in Antarctica is to the South Pole. There are a handful of military, research, and fishing stations farther north, but none of them have much in the way of roads; just airstrips, which are usually gravel and snow.
If you putter around downtown Longyearbyen,1 the longest sunset you could experience would be a few minutes short of an hour. It doesn’t actually matter if you drive or not; the town is too small for your movement to make a difference.
But if you head over to the mainland, where the roads are longer, you can do even better.
If you start driving from the tropics and stay on paved roads, the farthest north you can get is the tip of European Route 69 in Norway. There are a number of roads crisscrossing northern Scandinavia, so that seems like a good place to start. But which road should we use?
Intuitively, it seems like we want to be as far north as possible. The closer we are to the pole, the easier it is to keep up with the Sun.
Unfortunately, it turns out keeping up with the Sun isn’t a good strategy. Even in those high Norwegian latitudes, the Sun is just too fast. At the tip of European Route 69—the farthest you can get from the equator while driving on paved roads—you’d still have to drive at about half the speed of sound to keep up with the Sun. (And E69 runs north-south, not east-west, so you’d drive into the Barents Sea anyway.)
What If? Page 16