Powers of Ten: The Scale of the Universe in Factors of Ten

In 1977 Charles and Ray Eames released a nine-minute film that starts with a couple on a picnic blanket by the lake in Chicago. The camera rises. Every ten seconds the view is ten times wider: the blanket, the park, the city, the lake, the continent, the planet, the solar system, the galaxy, until the whole of the visible picture is clusters of galaxies. Then it plunges back down into the man's hand, through his skin and a cell and a molecule, into the nucleus of an atom.

The film is called Powers of Ten, and its idea is the same one that sits under the scoring of Scale of Things: size is best understood in factors, not in differences. This piece is about that idea, where it came from, and how to use it.

The Eames film was adapted from a book. In 1957 the Dutch educator Kees Boeke published Cosmic View: The Universe in 40 Jumps. It is a book of pictures. The first shows a girl sitting outside his school. Each picture after it shows the scene from ten times further away, out past the Earth, the Sun and the Milky Way, and a second sequence goes the other way, ten times closer each step, down into the structure of matter.

Boeke was a teacher, and his point was a teacher's: no one can picture the distance to a galaxy, but anyone can picture "ten times this", and a couple of dozen steps of ten times this will take you from a child to the edge of the known universe. The trick is not to imagine the big number. It is to take a step you can imagine, many times.

Charles and Ray Eames, the American designers, made a rough version of the film in 1968 and the finished one in 1977, with the full title Powers of Ten: A Film Dealing with the Relative Size of Things in the Universe and the Effect of Adding Another Zero. The physicist Philip Morrison narrated it. The camera starts one metre above the picnic, travels out to 10 to the power 24 metres, about a hundred million light years, and then down to 10 to the power minus 16 metres, inside a proton of a carbon atom in the man's hand.

Powers of Ten was added to the US National Film Registry in 1998, and it is still shown in classrooms, because nothing else makes the scale of things so plain in so little time. It is also the ancestor of every size comparison video since, the ones that line up a mouse, a whale, a skyscraper and a star.

An order of magnitude is a factor of ten. Two things are an order of magnitude apart when one is about ten times the other, two orders when it is about a hundred times, and so on. It is a way of counting how many zeros apart two numbers are, which is a far rougher and far more useful question than how far apart they are.

On a scale built that way, a logarithmic scale, the step from 1 metre to 10 metres is the same length as the step from 10 to 100, or from a million to ten million. Equal ratios are equal distances. That is the only kind of scale on which a beetle and a galaxy can share one picture.

Here is the range of things in Scale of Things, one rough step at a time:

• A few millimetres: a body louse, 3 mm.
• Centimetres: a honey bee, about 2.4 cm.
• Tens of centimetres: a banana, about 21 cm.
• Metres: a person, 1.7 m.
• Tens of metres: a blue whale, 27 m.
• Hundreds of metres: the Eiffel Tower, 324 m.
• About a kilometre: the asteroid Ryugu, 900 m across.
• Tens of kilometres: Mars's moon Phobos, about 22 km.
• Hundreds of kilometres: the asteroid Vesta, about 525 km.
• Thousands of kilometres: the Moon, about 3,475 km.
• Tens of thousands: Earth, 12,742 km.
• Hundreds of thousands: Jupiter, about 140,000 km.
• Millions: the Sun, about 1.4 million km.
• Tens of millions: the red giant Arcturus, about 35 million km.
• Hundreds of millions: the supergiant Rigel, about 110 million km.
• A billion: Betelgeuse, about 1.06 billion km.

From the louse to Betelgeuse is a factor of about 350 trillion, more than fourteen orders of magnitude. Each line on that list is one step of Boeke's ladder.

A guess in Scale of Things is scored on how many times too big or too small it was, which is the same as asking how far apart the guess and the answer are on a logarithmic scale. Twice too big and half too small are the same distance either side of the answer, and they score the same, 37 out of 100. That is what lets a whale and a beetle be fair questions in the same game: missing a whale by 5 metres is a small error, missing a beetle by 5 metres is absurd, and missing either by a factor of two is the same misjudgement.

Measured in orders of magnitude, the game is strict. Being twice off is about three tenths of an order of magnitude. Four times off, about six tenths, scores nothing. A whole power of ten off is far past zero. The perfect window, within 5%, is about two hundredths of an order of magnitude. So the game asks you to place things not just on the right rung of Boeke's ladder, but within a small fraction of a rung.

The rounds sit inside that same scale. A round's pair is usually between one and a quarter and two and a half times apart, and in the hardest pairs three to six and a half times: always less than one step of ten. Every round is a small piece of the Eames film, slowed right down.

It is no accident that ratios feel natural. In the 1830s the physiologist Ernst Weber found that the smallest change people can notice in a weight is a roughly fixed fraction of the weight, not a fixed amount: a few grams can be felt on a light weight and are lost entirely on a heavy one. In 1860 Gustav Fechner built that into a law that sensation grows with the logarithm of the stimulus, and a century later S. S. Stevens refined it into a power law. The details differ, but the lesson holds: the senses compare by proportion.

Where they go wrong is in the extremes, and in areas, and in very big numbers, which is the subject of our piece on why we misjudge size.

A few habits make big ratios manageable, on the board and off it.

1. Learn three logarithms. Twice is about 0.3 of a power of ten, three times about 0.5, and five times about 0.7. So ten times is a little over three doublings, and a hundred times is two steps of ten.
2. Chain small steps. You cannot see "forty times" at a glance, but you can see "about six" and then "about seven of those". The reference is your first step; imagine a row of them, then a row of rows.
3. Find the rung first. Before you worry about the exact figure, settle whether the answer is in tens, hundreds or thousands. Most bad guesses are on the wrong rung, not the wrong spot on the right one.
4. Keep a ladder of anchors. One thing you know well at every power of ten, from a credit card to the Sun, turns any new size into a step from something familiar.

Our planets and Sun size comparison puts those habits to work on the biggest ratios in the solar system.

Powers of Ten is easy to find and worth the nine minutes, especially the moment the picnic blanket vanishes into a city. Then play a run of Scale of Things and, before each guess, say the ratio out loud: "three of these", "about half". Thinking in factors is a habit, and the board is a good place to build it.