11.3 — The Scale of Everything

From the smallest length that means anything to the largest thing that can be seen, in powers of ten — and the useful discovery is where a human being sits in that range.

Going down

What are the sizes, from a person downward?

Each step here is roughly a factor of a thousand, which is three powers of ten.

1 metre — a person, near enough.

1 millimetre, 10⁻³ m — a grain of sand, the thickness of a coin, the smallest thing an unaided eye resolves comfortably.

1 micrometre, 10⁻⁶ m — a bacterium. A human cell is about 10 to 20 of these. Visible light has a wavelength of 0.4 to 0.7 micrometres, which is why an optical microscope cannot resolve anything much smaller than this: you cannot see detail finer than the wavelength you are looking with.

1 nanometre, 10⁻⁹ m — a small molecule. A DNA double helix is 2 nanometres across. A single atom is about 0.1 to 0.5 nanometres. Transistors on a modern chip are described in nanometres, though the number is now a marketing label rather than a physical dimension.

1 picometre, 10⁻¹² m — the scale of atomic structure. A hydrogen atom's radius is about 53 picometres.

1 femtometre, 10⁻¹⁵ m — the nucleus. A proton is about 0.84 femtometres across.

Here is the fact worth stopping on. An atom is about 100,000 times wider than its nucleus. If a nucleus were a marble on the centre spot of a football stadium, the electrons would be at the outer walls, and everything between would be empty. All the mass is in the marble.

Below that: quarks and electrons have no measured size and are treated as point-like down to about 10⁻¹⁸ metres, which is the limit of current experiment.

And at 10⁻³⁵ metres is the Planck length, which is not a smallest object but the scale at which the current description of space itself is expected to fail, because gravity and quantum effects become comparable there and no theory covers both.

What does "atoms are mostly empty" actually mean?

That the volume is unoccupied by mass, not that there is nothing there.

The space is filled by the electrons' probability distribution and by electromagnetic fields. You do not fall through a chair because the electrons in your body and in the chair repel each other electromagnetically, and because a quantum rule — the exclusion principle — forbids the electrons from occupying the same states.

You have never actually touched anything. The sensation of touch is electromagnetic repulsion at a separation of a fraction of a nanometre, which is exactly what your nerves are reporting. The word "touch" describes the repulsion, so this is a fact about physics rather than a paradox.

Going up

What are the sizes, from a person upward?

1 kilometre, 10³ m — a long walk, a tall building, the width of a large park.

1,000 km, 10⁶ m — the size of a country. The Earth's diameter is 12,742 km.

1 million km, 10⁹ m — the Earth-moon distance is 384,000 km, so this is a few times that. The sun's diameter is 1.39 million km, about 109 Earths across.

150 million km is the Earth-sun distance, defined as 1 astronomical unit. Light takes 8 minutes 20 seconds to cross it.

The solar system: Neptune orbits at about 30 AU, and light takes about 4 hours to reach it. The heliopause, where the sun's wind gives way to interstellar space, is around 120 AU — which Voyager 1 crossed in 2012 after 35 years of travel.

1 light year = 9.46 × 10¹⁵ m. The nearest star system is 4.25 light years away.

The Milky Way is about 100,000 light years across and contains something in the region of 100 to 400 billion stars. We are about 26,000 light years from the centre, in a minor spiral arm, and the sun takes roughly 230 million years to complete one orbit — so it has gone round about 20 times since it formed.

The Local Group, our galaxy cluster, is about 10 million light years across and holds around 80 galaxies, dominated by the Milky Way and Andromeda.

The observable universe is about 93 billion light years across.

Why is the observable universe 93 billion light years across if it is 13.8 billion years old?

Because space expanded while the light was travelling, and this is the most common point of confusion in cosmology.

The oldest light reaching us left its source about 13.8 billion years ago. But the source has not been sitting still. The expansion of space has carried it further away during the journey, so the object that emitted that light is now about 46 billion light years away — giving a diameter of about 93 billion.

Nothing is moving through space faster than light. Space itself is expanding, which is not motion through space and is not subject to that limit. Sufficiently distant galaxies are receding faster than light and their light will never reach us, which sets a horizon.

Time

What does the timeline look like at the same scale?

13.8 billion years — the Big Bang. 9.2 billion years ago — the solar system forms; the Earth at about 4.54 billion years ago. 3.8 to 4.1 billion years ago — earliest evidence of life. 2.4 billion years ago — the Great Oxidation Event, when photosynthesising bacteria filled the atmosphere with oxygen and poisoned most of the existing life on the planet. 540 million years ago — the Cambrian explosion, when most animal body plans appear. 66 million years ago — the asteroid, per Volume VIII. 300,000 years ago — anatomically modern humans. 12,000 years ago — agriculture. 5,000 years ago — writing. 500 years ago — printing in Europe. 200 years ago — the railway. 80 years ago — the computer. 35 years ago — the web.

What is the standard way of feeling this?

Compress the age of the universe into one year.

The Big Bang is 1 January. The Milky Way forms in May. The solar system forms on 1 September. Life appears in late September. The Cambrian explosion is 15 December. The dinosaurs die on 30 December.

Human beings appear at about 11:52 PM on 31 December. Agriculture starts at 11:59:32. The whole of recorded history occupies the last 12 seconds. Printing, industry, electricity, the computer and the internet fall inside the final second.

Carl Sagan's version of this in Cosmos remains the clearest presentation of the point, which is not that humans are insignificant but that everything humans have ever done happened in an interval too short to draw on the chart.

Big numbers

How do you get any sense of a billion?

By converting it into time, which is the only reliable method.

A million seconds is 11.6 days.A billion seconds is 31.7 years.A trillion seconds is 31,700 years — longer than agriculture has existed.

That single comparison does more work than any amount of writing out zeros, and it should be applied every time a large financial figure appears in the news. A billion is not a big million. It is a thousand millions, and the difference between a millionaire and a billionaire is the difference between eleven days and thirty-one years.

What are the biggest named numbers?

A googol is 10¹⁰⁰, named by a nine-year-old, per 1.6. There are about 10⁸⁰ atoms in the observable universe, so a googol exceeds the number of atoms in existence by a factor of 10²⁰.

A googolplex is 10 to the power of a googol. It cannot be written out in the physical universe — writing it in ordinary digits would require more space than exists, even at one digit per atom.

Beyond that, mathematicians use numbers that cannot be expressed in exponent notation at all. Graham's number, which arose as an upper bound in a combinatorics problem, is so large that its notation itself requires a specially defined system of operations, and it was for a period the largest number ever used in a serious proof.

And yet Graham's number is finite, and infinitely smaller than any infinity — which is the point Cantor established in 3.1.

And the very small side?

A probability of 10⁻⁶ is one in a million — roughly the chance of being struck by lightning in a given year in many countries.

10⁻¹² — one in a trillion. Beyond the reach of any intuition, and the level at which physicists start to describe something as effectively impossible.

The most precisely tested prediction in physics, the electron's magnetic moment in quantum electrodynamics, agrees with measurement to about 10⁻¹² — twelve decimal places, per 3.2.

Where you are in it

Are humans large or small?

Remarkably close to the middle, on a logarithmic scale, and this is a genuine and slightly startling result.

The smallest meaningful length is the Planck length, 10⁻³⁵ metres. The largest is the observable universe, about 10²⁷ metres. That is a range of 62 orders of magnitude, and its midpoint on a logarithmic scale is 10⁻⁴ metres — a tenth of a millimetre.

A human at 1 metre is about four orders of magnitude above that midpoint, which on a 62-order range is close to the centre. We are roughly as many powers of ten larger than an atom as we are smaller than a star.

The same holds for time. The shortest meaningful interval is the Planck time, about 10⁻⁴³ seconds; the age of the universe is about 10¹⁸ seconds. A human lifetime, about 10⁹ seconds, sits well up that range but nowhere near either end.

The honest conclusion is not that we are central. It is that we sit at the scale where chemistry works — where molecules are stable, where liquid water exists, where structures can be complex without being crushed by gravity or shaken apart by thermal motion. We are the size we are because that is the only size a thing like us could be.

What comes next

The last page of this Part comes back to earth and to money — why buffets can be profitable, why watches cost what they do, why retirement happens at sixty, and how advertising decides what to put on a billboard.