Billion Story Building

My daughter—age 4.5—maintains she wants a billion-story building. It turns out not only is that hard to help her appreciate this size, I am not at all able to explain all of the other difficulties you'd have to overcome.

Keira, via Steve Brodovicz, Media, PA

Keira,

If you make a building too big, the top part is heavy and it squishes the bottom part.

Have you ever tried to make a tower of peanut butter? It's easy to make a little tiny one, like a blobby castle on a cracker. It will be strong enough to stay standing. But if you try to build a really big castle, the whole thing smushes flat like a pancake.

The same thing happens with buildings. The buildings we make are strong, but we couldn't make one that went all the way up to space, or the top part would squish the bottom part.

We can make buildings pretty tall. The tallest buildings are almost 1 kilometer tall, and we could probably make buildings 2 or even 3 kilometers tall if we wanted, and they would still be able to stand up under their own weight. Higher than that might be tricky.

But there would be other problems with a tall building besides weight.

One issue would be wind. The wind up high is very strong, and buildings have to be very strong to stand up against the wind.

Another big problem would be, surprisingly, elevators. Tall buildings need elevators, since no one wants to climb hundreds of flights of stairs. If your building has lots of floors, you need lots of different elevators, since there would be so many people trying to come and go the same time. If you make a building too tall, the whole thing gets taken up by elevators and there's no space for regular rooms.

Maybe you can think of a way to get people to their floors without having too many elevators. Maybe you could make a giant elevator that takes up 10 floors. Or you could make fast elevators that work like roller coasters. Or you could fly people up to their rooms with hot air balloons. Or you could launch them with catapults.

Elevators and wind are big problems, but the biggest problem would be money.

To make a building really tall, someone has to spend a lot of money, and no one wants a really tall building enough to pay for it. A building many miles tall would cost billions of dollars. A billion dollars is a lot of money! If you had a billion dollars, you could rent a giant spaceship, save all the world's endangered lemurs, give a dollar to everyone in the US, and still have some left over. Most people don't think giant towers a few miles tall are important enough to spend a lot of money on.

If you got really rich, so you could pay for a tower to space yourself, and solved all those engineering problems, you'd still have problems making a tower a billion stories tall. A billion stories is just too many.

A big skyscraper might have about 100 floors, which means it's as tall as 100 little houses.

If you stacked 100 skyscrapers on each other to make a mega-skyscraper, it would reach halfway to space:

This skyscraper would still only have 10,000 floors, which is way less than your billion floors! Each of those 100 skyscrapers would have 100 floors, so the whole mega-skyscraper would have 100 times 100 is 10,000 floors.

But you said you wanted a skyscraper with 1,000,000,000 floors. Let's stack 100 mega-skyscrapers to make a mega-mega-skyscraper:

The mega-mega-skyscraper would stick out so far from the Earth that spaceships would crash into it. If the space station were heading toward the tower, they could use its rockets to steer away from it.[1] The bad news is that space is full of broken spaceships and satellites and pieces of junk, all flying around at random. If you build a mega-mega-skyscraper, spaceship parts will eventually smash into it.

Anyway, a mega-mega-skyscraper is only 100 times 10,000 = 1,000,000 floors. That's still a lot smaller than the 1,000,000,000 that you want!

Let's make a new skyscraper by stacking up 100 mega-mega-skyscrapers, to make a mega-mega-MEGA-skyscraper:

The mega-mega-MEGA-skyscraper would be so tall that the top would just barely brush against the Moon.

But it would only be 100,000,000 floors! To get to 1,000,000,000 floors, we have to stack 10 mega-mega-MEGA-skyscrapers on top of each other, to make one Keira-skyscraper:

The Keira-skyscraper would be pretty close to impossible to build. You would have to keep it from crashing into the Moon, being pulled apart by the Earth's gravity, or falling over and smashing into the planet like the giant meteor that killed the dinosaurs.

But some engineers have an idea sort of like your tower—it's called a space elevator. It's not quite as tall as yours (the space elevator would only reach partway to the Moon), but it's close!

Some people think we can build a space elevator, but other people think it's a crazy idea. We can't build one yet because there are some problems we don't know how to solve, like how to make the tower strong enough and how to send power up it to run the elevators. If you really want to build a gigantic tower, you can find out more about some of the problems they're working on, and eventually become one of the people coming up with ideas to solve them. Maybe, someday, you could build a giant tower to space.

I'm pretty sure it won't be made of peanut butter, though.

Faucet Power

I just moved into a new apartment. It includes hot water but I have to pay the electric bill. So being a person on a budget ... what's the best way to use my free faucet to generate electricity?

David Axel Kurtz

You could build a tiny hydroelectric dam in your tub.

It would generate power, though not very much of it. The formula for power is pressure times flow rate.[1] Since bathtubs are pretty shallow, the pressure at the bottom isn't very high, so this works out to around two watts of power, or about 25 cents per month.

You can get more power if you increase the pressure of the water passing through the generator. To do this, you could increasing the depth of the water. If you have two floors in your apartment, you could have the water column stretch from the second to the first floor, generating at least ten times the pressure and ten times the power.[2] In effect, the local authorities would be paying to pump the water up to your apartment, and you're getting some of that energy back when you let it flow back down.

Could you use the faucet to pump the water up arbitrarily high, and get more and more power out of it as it falls back down?

No. First, you couldn't pump the water arbitrarily high. Household faucets have a pressure of around 4 atmospheres.[3] You can lift water about 10 meters per atmosphere of pressure, which means that a household faucet can only pump water up by about 40 meters.

Second, as you can probably guess by looking at the above picture, pumping the water up 40 meters with water pressure and then back down doesn't accomplish anything—you can just hook the faucet up to your device, and let the water pressure drive the generator directly. In either case, for a bathtub faucet, this works out to almost 200 watts, or $25 per month.

You'd have to make sure your plumbing could handle the water. If your pipes get plugged up and stop draining, the faucet could fill your house in a matter of days. And either way, eventually someone from the city would probably show up to ask why you're using 40 tons of water every day.

And really, with California suffering through its worst drought in history, this system might earn you some dirty looks. Sure, if you live far away from California, it's not like your water would have gone to ease their drought, but wasting a gigantic amount of water (and investing a bunch of money) to save a few dollars on your electricity bill might come across as a little rude.

A bathtub's flow rate is five or six orders of magnitude less than that of a river, but it's still a lot of water. Could we put it to a less selfish use?

There's a common piece of advice that says you should drink 8 glasses of water a day. No one really knows where this advice came from; people claim you should drink anywhere from 2 to 12 glasses of water daily,[4] and none of them have any real evidence behind them. The only real solid advice I've heard is that if you're thirsty, you should drink some water.

If we stick to the "8 glasses of water" standard, then a bathtub faucet provides enough drinking water to sustain about 10,000 people indefinitely. In other words, the city of Manhattan could survive on the water from just 150 bathtubs.

But if your goal is to save money on your electric bill, there's a much more lucrative option.

Single servings of bottled water sells for a dollar or two per half-liter. A lot of bottled water comes from municipal sources—that is, it's tap water. Bottled water isn't necessarily about the water; often, people are paying for convenience or because there's an issue with their water supply. Whatever the reason, however, there's no reason to let Coca Cola keep all the profits.

If you bottled the water from your bathtub faucet and managed to sell each bottle for $1.50, you'd make $72 per minute—$38 million every year.

Then you won't have to worry about your power bill.

Great Tree Great Axe

If all the seas were one sea,
What a great sea that would be!
If all the trees were one tree,
What a great tree that would be!
If all the men were one man,
What a great man that would be!
If all the axes were one axe,
What a great axe that would be!
And if the great man took the great axe,
And cut down the great tree,
And let if fall into the great sea,
What a great splish-splash that would be!

... How great would all of these things be?

—John Eifert (quoting a Mother Goose rhyme)

If all the seas were combined into one sea, it would look pretty much like the Pacific Ocean, only a little bigger.

The poem's tree, axe, and human are more interesting.

The tree

Real trees can't grow taller than around 130 meters, thanks to physical limits on their ability to transport water. If they found a way around those limits, they'd face issues of fundamental physical strength; a kilometers-tall tree would crush itself.

Let's set aside these limits, and imagine that we built a single standard tree out of all the material in all the world's trees.

For our "standard tree," we'll use the oak tree from the Sylva Foundation's OneOak project. The project extensively documentsevery detail of a single oak tree. As part of the project, the tree was cut down in 2010. Frankly, I'm not sure what to make of the whole thing, but it's as good a candidate as any for our model "standard tree".

The OneOak tree was 23.9 meters tall and weighed 14.385 tons.[1] By comparison, one paper estimates that the world's forests have an aboveground plant mass of about 470 billion tons.

If—ignoring the physical constraints—we combined this mass into a single tree, modeled on the OneOak, the trunk would be two kilometers in diameter. The upper branches would stretch about 75 kilometers above the surface—most of the way to space.

The human

If we used the same approach to combine every living human into a single body—again, ignoring the obvious physical constraints—that person would be close to 3 kilometers tall.

Proportionally, the person would have grown slightly less than the tree.

The axe

How many axes are there in the world?

Thanks to multiple meanings of the word "axes", this is a hard problem to Google.[2] Instead, let's try to get a reasonable guess through Fermi estimation.

Since there's no central clearinghouse of axe-related information, I thought I'd try asking friends how many axes they had.

But this might miss a lot of axes; for example, some axes—like fire axes—are owned by organizations. To get a slightly fairer sample, I asked a bunch of friends around the country to estimate the number of axes and number of humans in their general vicinity.

Some people were in houses with sheds, and had a 2:1 human-to-axe ratio. Other people were in large offices with hundreds of people and at best one or two fire axes. The average seemed to be around 50:1—much lower than I expected.[3]

Of course, surveying random Internet-connected people who I know is hardly representative; rural people probably have more axes, while the very poor might not have any. But it also wouldn't make sense for there to be substantially more axes than people, if for no other reason than that humans can only really use one axe at a time.

But absent any other data, I'm guessing the ratio of humans to axes is probably somewhere between 50:1 and 5:1.

This means that our combined axe would be a little small for our combined person. It would be only a little over half a kilometer long—barely more than a flimsy hatchet.

If an experienced axe user can chop down an eight-inch tree in 15 minutes, then chopping down our giant tree—if the rate is proportional to the axe size and square of the tree's diameter—would most likely take a few weeks of chopping.

The fall

The tree would weigh between 1% and 10% as much as the Chicxulub asteroid that killed the dinosaurs.

It would strike the ocean with much less speed than the Chicxulub asteroid, and the energy release would be much less substantial. However, it would still be moving at kilometers per second, and would be able to displace a gigantic amount of water.

The Chicxulub impact created a giant tsunami; the buried layer of jumbled sand mixed with a fossilized forest it left along the coast of the Gulf of Mexico was a crucial clue in discovering the location of the crater.[4]

There have been some beautiful simulations of the Chicxulub tsunami. The exact details of the tsunami depend on a lot of factors, but it seems safe to say that waves at least tens or hundreds of meters high would inundate every coastline and destroy virtually every coastal city and many farther inland.

In other words, the Mother Goose poem in John's question probably wouldn't wipe out the human race, but it would probably be the deadliest single disaster in our species's history. Even by the ghastly standards of old childrens' fairy tales, that's pretty bad.

Axe law

In closing, I offer my favorite piece of axe-related legal trivia:[5]

Lawyer Kevin Underhill, of the legal blog Lowering the Bar, hilighted the wonderful 1998 case People v. Foranyic. In this case, he writes, appeals court ruled "that there was probable cause for police to detain someone they see riding a bike at 3 a.m., carrying an axe."

So if you're three miles tall and heading toward the coast to cut down the world's only tree ...

... watch out for cops.