# Orders of Magnitude

[Mathematics] [Computer science] [Programs] [Linux] [Literature]

## The size of things

### Size of mankind

(Inspired by Le Petit Prince by Antoine de Saint-Exupéry.)

What is the total volume of humanity? What is the radius of the smallest ball that would contain all of humanity, densely packed? (Try to give a rough estimate before reading what follows.)

Now that's an easy one. The volume of a typical human being is around 70 liters. Six billion human beings make therefore a volume of 400 million cubic meters. This is held in a ball of radius just over 450 meters.

Incidentally, if humanity were spread in surface rather than in volume, say at one person per square meter (this is not too uncomfortable), in a circle, that circle would have a radius of 44 kilometers: all of mankind would fit in a rather small island.

### Size of mankind, II

What if humanity were spread evenly over the surface of the Earth: how much would each person have to himself?

The surface of the Earth is 5.089e14 square meters. Divided by 6 billion, that's 85000 square meters per individual (a square of side 290 meters).

### A volcano spewing out air

At the end of the movie Total Recall (I think it is), we see a volcano massively sending air in the martian atmosphere so as to make it breathable. Let us examine whether this plausible.

The radius of mars is about 3400km (it is roughly half that of Earth). This means its surface is around 150 million square kilometers. We want the air to be present up to a characteristic altitude of, say, 1km (this is very generous, considering the height of some volcanos on Mars!), so we must fill a volume of around 150 million cubic kilometers. Assuming the volcano crater is one square kilometer in area (one million square meters: a fairly large one), and the air is coming out of it at a speed of hundred meters per second (this is a considerable velocity: 360km/h, you don't want to be hit by a wind like that), it still takes one and a half billion seconds to fill the atmosphere. That's over 45 years!

## Information

### The Bible and a phonebook

(Inspired by Douglas R. Hofstadter, in Metamagical Themas.)

How do you think the relative sizes of the Bible and a phonebook compare? It has been claimed that some people have memorized an entire phonebook: does this seem likely?

The number of characters in the Bible is known to a fairly good precision: I counted 4017010 characters in the Authorized version. Say four million. Now consider the phonebook: I took a phonebook of Paris (because that's one I happen to have with me), and I estimated around 30 to 40 characters per line, 150 lines and 4 columns per page, thus, around 20000 characters per page. And there are two volumes of 1500 pages, making a grand total of something like 60 million characters. This is fifteen times the size of the Bible.

Actually, it is probably much more difficult to memorize than fifteen texts of the same size as the bible, because the Bible has much more continuity, has so many repetitions that make memorization much easier, and does not contain the (essentially random) phone numbers which make memorization so much more difficult. A better comparison is probably to say that the Bible is 750000 words long, that each line in the phonebook is about as difficult to remember as twelve words (8 digits for the phone number — since the first two don't change — plus the name of the person, their first name, the name of the street, and the number on the street). This gives a ratio of 30 this time.

It may be possible that some people have memorized the entire Bible; but I do not thing it possible that anyone should have memorized an entire phonebook (at least, not one with a size comparable to that of Paris).

As a side note, the size of the Linux kernel is around 50 million characters, wich is comparable with the phonebook, though a lot less random.

### Libraries and disks

Which contains more: a library or a hard drive? (We are assuming that books contain only text with nothing like images.)

Of course, this question is essentially meaningless: which library and which hard drive? A very large library (such as the Library of Congress) can contain several million books. As for the average size of a book, I estimate it around a third to a half of a million characters. This means that the content of a very large library is measured in terabytes: this is more than hard disks contain (I do not think any single hard disk exists yet that has a capacity above a terabyte). However, an individual's library (one that is found in a house) might contain two or three thousand books: this corresponds to a mere gigabyte, a rather small size for a hard drive.

To summarize, the two are more or less comparable.

### Information in the Universe

(Inspired by a paragraph in Bruce Schneier's book on cryptography.)

This is a very wild computation: the total amount information stored in the Universe. I have performed several calculations which are off by a factor of more than one million. (Not counting the fact that the question is meaningless to start with!) But let us proceed anyway.

The Universe's age is about 15 billion years, or 5e17 seconds. So its radius can be estimated by the distance light travels in that time (this is wrong, of course), that is 1.5e26 meters. That is 5000 megaparsecs. So its volume is the cube of this (this is also wrong, of course): 1e11 cubic megaparsecs. Now the density parameter of the Universe is estimated at 2e-26 kilograms per cubic meter: this is an energy (ee equals em see squared!) of 2e-9 joules per cubic meter. Now the mean temperature of the Universe is well known: 2.73 kelvin; using Stephan Boltzman's constant, we see that a napier of information (that's 1.44 logons, but considering the approximations we are doing, we don't need to distinguish between napiers and logons) corresponds to an energy of 3.8e-23 joules. Dividing the one by the other, we find a completely ad hoc value of 4e13 napiers per cubic meter (that's 7 terabytes per cubic meter of the near-void the Universe is made of). For a volume of 3e78 cubic meters, we find a total information of 1e91 napiers. The log base 2 of this number of logons is around 306.

Thus, the Universe can be described by ten-to-the-ninety-one bits of information, and 307 bits suffice to index any particular bit of its informational contents.

## The Universe

### Its density

How many hydrogen atoms are there per cubic meter in the Universe (on the average)?

### Its emptiness

The Universe is mostly empty. If it were not, how big would it be?

What is meant is: if the mass of the Universe were as densely packed as an atomic nucleus (or a neutron star), how large would that be?

Actually, you don't need to know much to answer this one. From the previous question, each hydrogen atom in the Universe occupies alone a cube of size about 50 centimeters. But in atomic matter, this size is about one femtometer (one “fermi”). So if the Universe were full, its radius would be smaller than its present one by a factor of the ratio of these two quantities. A rough estimate is 1e12 meters, or 5 astronomical units: the Solar system up to the orbit of Jupiter.