Explore. The cost of a world
Chapter 2 prices a world in operations, in heat, and in waiting, from the limits physics places on any machine. Change its assumptions here and the figures follow. At the chapter’s own settings the model gives every figure the chapter states, and each is shown beside the sentence it comes from.
Your settings
What the Sun’s lifetime output buys
| World | Forgetful machine at room temperature | Forgetful machine in the cold of space | Reversible machine near the best case |
|---|---|---|---|
| The lake | 400 years | 40,000 years | 59,000 years |
| The boy’s brain at molecular grain | 330 trillion years | 3.3 × 10to the power 16 years | 54 billion years |
| A peopled world | 57 years | 5,700 years | 22,000 years |
| Earth’s oceans | 9.4 seconds | 16 minutes | 1.6 years |
The chapter’s figures
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The lake at room temperature
The lake’s 3 × 10 to the power 54 operations per second then release roughly 10 to the power 34 watts of heat.
Chapter 2, Ceilings and Costs, ¶ 22
The model gives 3.3 × 10to the power 54 operations each second, 9.6 × 10to the power 33 watts, and the output of 25 million suns.
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The boy’s brain at molecular grain
The boy’s brain at the same grain, about 4 × 10 to the power 42 operations each second, would release around 10 to the power 22 watts, hundreds of millions of times all the power humanity now uses.
Chapter 2, Ceilings and Costs, ¶ 22
The model gives 4 × 10to the power 42 operations each second, 1.2 × 10to the power 22 watts, and 580 million times humanity’s power use.
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A mind at a functional grain
At a functional grain, by contrast, a mind computed at Landauer’s limit would need at most a few thousandths of a watt, and the boy’s own brain, like ours, runs on about twenty.
Chapter 2, Ceilings and Costs, ¶ 22
The model gives 2.9 × 10to the power −3 watts.
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The sequential floor
Each femtosecond of the lake’s time asks for some ten operations in sequence, so logic switching in a trillionth of a second would run the lake ten thousand times slower than it lives, however much hardware was built.
Chapter 2, Ceilings and Costs, ¶ 23
The model gives 10,000 times slower than it lives.
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The lake for as long as the Sun shines
At that price alone, such a machine in the cold of space, about three degrees above absolute zero, could carry the lake for some forty thousand of its years on the Sun’s lifetime output, and at room temperature for about four hundred.
Chapter 2, Ceilings and Costs, ¶ 23
The model gives 40,000 years of its own time.
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Growing Earth’s oceans on a forgetful machine
Carrying Earth’s oceans at molecular grain for the three and a half billion years between the origin of life and the first minds would cost a forgetful machine, even at Landauer’s limit in the cold of space, about a hundred trillion times the Sun’s lifetime output, the starlight of thousands of galaxies.
Chapter 2, Ceilings and Costs, ¶ 25
The model gives 120 trillion times the Sun’s lifetime output.
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Growing them on a reversible machine
A reversible machine could pay for the same history from one star’s output, hoarded and spent slowly, but only by waiting more than three billion billion times the present age of the universe, and only if its switching lost no more than a speculative best case that no known law forbids and none guarantees.
Chapter 2, Ceilings and Costs, ¶ 25
The model gives a wait of 3.4 × 10to the power 18 times the present age of the universe.
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Keeping a peopled world
A forgetful machine at Landauer’s limit in the cold of space would keep such a world for nearly six thousand years of its own time on the Sun’s lifetime output.
The model gives 5,700 years of its own time.
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Keeping it on a reversible machine
Reversible machines spending the same output within the Sun’s life would keep it for several centuries to a millennium or more on the most frugal logic yet measured, and for about twenty thousand years near the best case. Each of those twenty thousand years would take half a million of ours, on speculative molecular machinery as massive as a mountain range.
The model gives 22,000 years of its own time, each taking 450,000 years outside.
How the model counts
- Water holds about 3.3 × 1028 molecules in each cubic meter. At molecular grain the machine takes a step every femtosecond, with a hundred operations for each molecule at each step and some ten operations in sequence, as the chapter assumes.
- A forgetful machine erases about one bit with each operation, at a cost of Boltzmann’s constant times the temperature times the natural logarithm of two.
- On a reversible machine sized to the world, the energy multiplied by the waiting is fixed for a given task. The model takes the speculative best case as an energy multiplied by switching time of about 3.3 × 10−34 joule-seconds for each operation, half of Planck’s constant, which reproduces the chapter’s figures.
- The Sun gives out about 3.8 × 1026 watts and is taken to shine for ten billion years. Humanity’s power use is taken as about twenty trillion watts, and the age of the universe as 13.8 billion years.
- The conservatism lies in how the work is counted and in the kinds of machine allowed, since each operation is already priced at the best physics allows. Whatever superintelligence saves would have to come from doing far less work or from kinds of machine the model leaves out.