Read here about stories that help people understand matters of scale.

"What makes humans unique, perhaps more than anything else, is that we are a linguistically adept story-telling species." - Joseph Carrol

The Relevance of Story to Scale Studies

A tale, myth, or anecdote is always the most effective way to express insights about life and personal transformation. Some quotes that make this point:
  • Humans are a story-telling species (Variously attributed to Alexander Marshack, Milton Erickson)
  • Humans are fundamentally a story-telling species - and hence the importance of 'narrative metaphors' (Greeley)

Stories convey qualitative information, as opposed to quantitative, and scale studies often focus on qualitative issues. Kahneman associates qualitative issues with "system 1", the human immediate, out-of-time cognitive mechanism, as opposed to the quantitative, mathematics-performing and rationally reasoning "system 2."

Stories of Exponential Growth

Rice on a chessboard

According to legend, a courtier presented the Persian king with a beautiful, hand-made chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for 2 n − 1 grains on the nth square demanded over a million grains on the 21st square, more than a million million (aka trillion) on the 41st and there simply was not enough rice in the whole world for the final squares. (From Meadows et al. 1972, p. 29 via Porritt 2005)


The story is also told about the inventor of chess, and grains of wheat or of gold.

The water lily

French children are told a story in which they imagine having a pond with water lily leaves floating on the surface. The lily population doubles in size every day and if left unchecked will smother the pond in 30 days, killing all the other living things in the water. Day after day the plant seems small and so it is decided to leave it to grow until it half-covers the pond, before cutting it back. They are then asked, on what day that will occur. This is revealed to be the 29th day, and then there will be just one day to save the pond. (From Meadows et al. 1972, p. 29 via Porritt 2005)


For variation of this see second half of the chessboard in reference to the point where an exponentially growing factor begins to have a significant economic impact on an organization's overall business strategy.

Stories of People Visiting Other Scales

Goldilocks and the Three Bears

Not too cold, not too hot, just right.

David Christian talks about "Goldilocks conditions".

Gulliver's Travels

Gulliver travels both to Lilliput and Brobdignan.


He appeared as tall as an ordinary spire-steeple; and took about ten yards in each stride, as near as I could guess. I was struck with the utmost fear and astonishment and ran to hide my self in the corn, from whence I saw him at the top of the stile, looking back to the next field on the right hand; and hears him in a voice many degrees louder than a speaking trumpet; but the noise was so high in the air, that at first I certainly thought it was thunder.

There are three orders of magnitude, as each is ten times taller than the other. [2]

Alice in Wonderland

Drink me

Tom Thumb and Angus MacAskill

MacAskill was 7 feet 9 inches tall, and strong. His palms were almost eight inches across and he could hold Tom Thumb who was a mere 3 feet tall, in his hand. – Bonner p. 12

The Incredible Shrinking Man

The Incredible Shrinking Man is a 1950's novel and an 81 min Sci-Fi Thriller released in April 1957. Director: Jack Arnold, Writers: Richard Matheson, Stars: Grant Williams, Randy Stuart and April Kent.

Synopsis: Scott Carey and his wife Louise are sunning themselves on their cabin cruiser, the small craft adrift on a calm sea. While his wife is below deck, a low mist passes over him. Scott, lying in the sun, is sprinkled with glittery particles that quickly evaporate. Later he is accidentally sprayed with an insecticide while driving and, in the next few days, he finds that he has begun to shrink. First just a few inches, so that his clothes no longer fit, then a little more. Soon he is only three feet tall, and a national curiosity. At six inches tall he can only live in a doll's house and even that becomes impossible when his cat breaks in. Scott flees to the cellar, his wife thinks he has been eaten by the cat and the door to the cellar is closed, trapping him in the littered room where, menaced by a giant spider, he struggles to survive

Fantastic Voyage

Fantastic Voyage is a 1960's novel written by Isaac Asimov that was made into a commercial feature film, released in 1966. It was 100 min duration, Director: Richard Fleischer, Writers: Harry Kleiner (screenplay), David Duncan (adaptation), Stars: Stephen Boyd, Raquel Welch and Edmond O'Brien.

Synopsis: Scientist Jan Benes, who knows the secret to keeping soldiers shrunken for an indefinite period, escapes from behind the Iron Curtain with the help of CIA agent Grant. While being transferred, their motorcade is attacked. Benes strikes his head, causing a blood clot to form in his brain. Grant is ordered to accompany a group of scientists as they are miniaturized. The crew has one hour to get in Benes's brain, remove the clot and get out.

Stories About Exceedingly Large Numbers

Lalitavistara Sutra

The Lalitavistara Sutra (a Mahayana Buddhist work) recounts a contest including writing, arithmetic, wrestling and archery, in which the Buddha was pitted against the great mathematician Arjuna and showed off his numerical skills by citing the names of the powers of ten up to 1 'tallakshana', which equals 10^53, but then going on to explain that this is just one of a series of counting systems that can be expanded geometrically. The last number at which he arrived after going through nine successive counting systems was 10^421, that is, a 1 followed by 421 zeros.

The Sand Reckoner

Archimedes (287 BCE- ?) offered to show Gelon, King of Syracuse, how to name quantities greater than the number of grains of sand not only on all the beaches around Syracuse, but on all the beaches of Sicily; and in all the lands of the world, known or unknown; and in the world itself, were it made wholly of sand; and, he says, `I will try to show you by means of geometrical proofs which you will be able to follow, that, of the numbers named by me ... some exceed the number of the grains of sand ... in a mass equal in magnitude to the universe.'
There are some who ... think that no number has been named which is great enough to exceed [the number of grains of sand in every region of the earth]. But I will try to show you ... that, of the numbers named by me ... some exceed ... the number [of grains of sand that would fill up the universe].

Archimedes achieved this with a clever series of multiplications.

Take it, he says, that there are at most 10,000 grains of sand in a heap the size of a poppy-seed; and that a row of 40 poppy-seeds will be as wide as a finger. To keep things simple, picture each seed as a sphere. Since the volumes of spheres are to each other as the cubes of their diameters, this line of 40 seeds becomes the diameter of a sphere with a volume (40)³ = 64,000 times the volume of one seed; and since that one holds 10,000 grains of sand, we're already talking about 64,000 x 10,000, that is, 640,000,000 grains. In our modern notation, that's 4³ x 107 grains. Round 64 up to 100 for convenience, and we'll have 109 grains in a sphere whose diameter is a finger-breadth. Don't worry that all these estimates may be too large: exaggeration, as you'll see, is part of Archimedes' game.

Now 10,000 (104) finger-breadths make a Greek unit of length called a stade (roughly a tenth of our mile). A sphere whose diameter is 104 finger-breadths will have a volume (104)³ = 1012 times the volume of one with a diameter of one finger-breadth, which we know contains 109 grains of sand; so a sphere with a one-stade diameter will contain 1012 x 109 = 1021 grains of sand.

Archimedes next cites Aristarchus of Samos who held that the earth circled the sun and estimated that distance. Archimedes assumes that the ratio of the diameter of the earth relative to that distance is the same as the diameter of the orbit to the size of the universe. This gives him (after modifying Aristarchus' figures) 100,000,000,000,000 or 1014 stadia for the diameter of the universe. Its volume is therefore (1014)³ = 1042 times the volume of the sphere whose diameter is one stade, which held 1021 grains of sand. Hence there would be 1021 x 1042 = 1063 grains of sand in a universe compacted wholly of sand.

`I suppose, King Gelon,' says Archimedes, `that all this will seem incredible to those who haven't studied mathematics, but to a mathematician the proof will be convincing. And it was for this reason that I thought it worth your while to learn it.'

When you consider that in the 1940s two persistent New Yorkers estimated that the number of grains of sand on Coney Island came to about 1020; and that present estimates for the total number of much smaller particles in our much larger universe weigh in at between 1072 and 1087, you have to say that Archimedes' estimate wasn't all that bad.
This is a spectacular application of the Greek insight that the world afar can be grasped by analogy to the world at hand. (Robert Kaplan, the Nothing that is, p29ff).
Archimedes did all this math without exponential notation. He worked with number names rather than digits, and the largest of the Greek names was `myriad', for 10,000. This let him speak of a myriad myriads (108), and he then invented a new term, calling any number up to 108 a number of the first order.

He next took a myriad myriads as his unit for numbers of the second order, which therefore go up to 1016 (as we would say — but he didn't); and 1016 as the unit for numbers of the third order (up to 1024), and so on; so that the unimaginably gigantic 1063 is a number somewhere in the eighth order.

But Archimedes doesn't stop there. He then leaves the sand-filled universe behind, diminished itself to a grain of sand, as he piles up order on order, even unto the 108 order, which frighteningly enough contains all the numbers from [(100,000,000).sup.99,999,999 to ([100,000,000).sup.100,000,000].
Is he done? Hardly. All of those orders, up to the one just named, make up the first period. He wrote:

And let the last number of the first period be called a unit of numbers of the first order of the second period. And again, let a myriad myriads of numbers of the first order of the second period be called a unit of numbers of the second order of the second period. Similarly…. And let the process continue up to a myriad-myriad units of the myriad-myriadth order of the myriad-myriadth period.

In our Hindu-Arabic notation, up to 1080,000,000,000,000,000. There is not enough time, from the Big Bang to now, to recite its digits at one a second, since the last number of his first period is 1 followed by 800 million zeroes, and this one has 108 times as many.

John Donne on Sand

Eighteen hundred years after Archimedes, John Donne said in a Lenten sermon:

Men have calculated how many particular graines of sand, would fill up all the vast space between the Earth and the Firmament: and we find, that a few lines of cyphers will designe and expresse that number ... But if every grain of sand were that number, and multiplied again by that number, yet all that, all that inexpressible, inconsiderable number, made up not one minute of this eternity; neither would this curse [of God on the inveterate sinner] be a minute shorter for having been endured so many generations, as there were graines of sand in that number ... How do men bear it, we know not; what passes between God and those men, upon whom the curse of God lieth, in their dark horrours at midnight, they would not have us know ... This is the Anathema Maranatha, accursed till the Lord come; and when the Lord cometh, he cometh not to reverse, nor to alleviate, but to ratifie and aggravate the curse.

Stories of Ignoring the Difference in Other Scales

Cartoon Coyote and the Cliff

In the Loony Tunes world of coyote and road runner, road runner frequently runs straight off a cliff, a logical step for her since, being a bird, she can fly. Coyote runs off the cliff right after her. That’s when the laws of the cartoon world take over. Coyote can keep running in mid-air, until the moment that he becomes aware that he is running in mid-air. When that happens, and we share with him the fact that he has become suddenly, painfully aware that he is in mid-air, he falls.

The story illustrates the problem of error blindness. Measurement that has been repeated feels like any other measurement, so when we use it in a way that is incorrect, there is a phase where we behave like the coyote that has gone off the cliff but has not yet looked down. The person wielding the measurement is already wrong, he is already in trouble, but he feels like he’s on a reliable extension of previous work.

Stories of Optical Illusions

Our senses can be fooled: see optical illusions.

Stories of Slow Slicing

Stories of slow slicing illustrate the slow process by which the behaviors we expect slowly and incrementally cease to be relevant at other scales. if the process is slow enough, we miss the signs that things are going awry, and can make mistakes.
The term "slow slicing" (alternately transliterated Ling Chi or Leng T'che), also translated as the slow process, the lingering death, or death by a thousand cuts, originated as a form of execution used in China from roughly 900 AD until its abolition in 1905. In this form of execution, the condemned person was killed by using a knife to methodically remove portions of the body over an extended period of time. The term língchí derives from a classical description of ascending a mountain slowly.
The phrase "death of a thousand cuts" is often used metaphorically to describe the gradual or incrementalism destruction of something, such as an institution or program, by repeated minor attacks. The term is also used in business management to describe a product or idea that is damaged or destroyed by too many minor changes.

Boiling Frog

The boiling frog story is a widespread anecdote describing a frog slowly being boiled alive. The premise is that if a frog is placed in boiling water, it will jump out, but if it is placed in cold water that is slowly heated, it will not perceive the danger and will be cooked to death. The story is often used as a metaphor for the inability of people to react to significant changes that occur gradually. According to contemporary biologists the premise of the story is not literally true; an actual frog submerged and gradually heated will jump out.

Camel’s Nose

The camel's nose is a metaphor for a situation where permitting some small undesirable situation will allow gradual and unavoidable worsening. According to Geoffrey Nunberg, the image entered the English langauge in the middle of the 19th century. An early example is a fable printed in 1858 in which an Arab miller allows a camel to stick its nose into his bedroom, then other parts of its body, until the camel is entirely inside and refuses to leave. Lydia Sigourney wrote another version, a widely reprinted poem for children, in which the camel enters a shop because the workman does not forbid it at any stage.
Once in his shop a workman wrought
With languid hand and listless thought
When through the open window’s space
Behold! – A Camel thrust his face.
“My nose is cold,” he meekly cried,
Oh let me warm it by thy side.”

Since no denial word was said,
In came the nose, in came the head
As sure as sermon follows text
The long excursive neck came next,
And then, as falls the threatening storm
In leap’d the whole ungainly form.

Aghast, the owner gazed around
And on the rude invader frowned
Convinced as closer still he pressed
There was no room for such a guest,
Yet more astonished, heard him say,
“If inconvenienced, go your way,
For in this place, I choose to stay.”

Oh youthful hearts, to gladness born,
Treat not this Arab lore with scorn
To evil habit’s earliest wile
Lend neither ear nor glance nor smile,
Choke the dark fountain ere it flows,
Nor even admit the Camel’s Nose.

Stories of the Different Rules At Different Scales

Poker Wild Card

The difference in the rules between poker wild card and study poker is a metaphor for the different rules at different scales: same reality, same "number of cards"--different results.

The overall rules remain the same, the contrast is with stud poker where you work with what you have, as opposed to draw poker where you can pick a new element from the deck:
In stud poker, the player gets five cards and must play the game with these five cards. The player cannot add new cards to the hand he or she has been given. This situation resembles degrees of freedom within a given scale; new possibilities are are not added to the mix and development takes place only with the force ratios and emergent entities on hand. Extending withoutward a scale or intending withinward a scale is more like draw poker. In draw poker players get five cards and can choose to discard cards, replacing them with new cards from the deck, trying to improve their hand. In contrast to degrees of freedom within a scale, new degrees of freedom are discovered and the very composition of the permutations is altered; novelty can be generated, by going back to the deck and getting the behavior equivalent of new cards. As in poker, however, the other rules of the game remain the same.

Stories of False Taxonomies

When a superscale survey proceeds, it feels like it is a logical and consistent sequence; but in fact, reality changes so drastically at different scales that it is possible that the sequence, though in order from A to Z, may be a sequence of nonsense. These stories help make that point.

Borge's Chinese Taxonomy of Animals

Borge's taxonomy is a famous literary illustration of the arbitrary nature of 'relatedness'. Since parthood relations are so important to the idea of scale, Borge's taxonomy is a good reminder that even the most consistent classification scheme is probably the result of some irrational, human association and is not inherent in the reality under consideration.

The Borges chinese taxonomy divides animals into:
(a) belonging to the Emperor,
(b) embalmed,
(c) tamed,
(d) sucking pigs,
(e) sirens,
(f) fabulous,
(g) stray dogs,
(h) included in the present classification,
(i) frenzied,
(j) innumerable,
(k) drawn with a very fine camelhair brush,
(l) et cetera,
(m) having just broken the water pitcher,
(n) that from a long way look like flies.

Links and References

[2] Bonner p.34.

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