Chance and pattern

Whoever searches long enough finds 7, 12 and 24 everywhere. An experiment with random buildings, the pyramid measurements of the nineteenth century, the strong law of small numbers and four questions that tell real patterns from chance, on this website too.

This chapter belongs to dialectic, but it deserves a place of its own. For whoever deals with number symbolism, and this whole website does, meets a danger more powerful than any fallacy: the human mind is a machine for finding patterns. It finds them even where there are none.

Faces in the clouds

We see faces in clouds, power sockets and the fronts of cars. A widespread explanation for this: a human who recognised a face in the bushes once too often lost little, one who overlooked it once might lose everything. In 1958 the psychiatrist Klaus Conrad coined the word apophenia for the pathological experience of seeing meaningful connections everywhere. Today the word is also used for the ordinary tendency to do so, and that is not a flaw of particular people but a property of all of us.

With numbers it is especially strong. There are only a few small numbers, and each has properties that make it look special. In 1988 the mathematician Richard Guy put this as the “strong law of small numbers”: there aren’t enough small numbers to meet the many demands made of them. That is why patterns among small numbers coincide so often without meaning anything.

An experiment

The following graphic invents a building with six random measurements, length, width, height, gate, window and steps. Then it searches them for “sacred” numbers: 3, 7, 9, 10, 12, 24, 33, 40, 72, 108, 144 and 365, and for ratios that give π, the golden section φ or √2 to within 1 per cent. Only the simplest calculations are allowed: one measurement alone, or two measurements added, subtracted, multiplied or divided.

A building with six random measurements and the 'discoveries' in it. 'Test 1000 buildings' shows how rarely a building remains without a sacred number.

The result is sobering. Almost every random building contains at least one sacred number, about 98 in 100 at least three, and on average there are about eight. And only twelve numbers and three ratios are allowed. Whoever may change units, from metres to cubits or feet, whoever adds diagonals, perimeters and areas, finds far more.

The pyramid and the Washington Monument

This is exactly how pyramid mysticism arose in the nineteenth century. The British astronomer Charles Piazzi Smyth, Astronomer Royal for Scotland, measured the Great Pyramid in 1865 and found in its measurements the circle number, the distance to the sun and the “pyramid inch”, a divine measure he had taken over from John Taylor. His book on it had already appeared in 1864, so the interpretation was fixed before the measuring. The young Flinders Petrie travelled to Giza in 1880 to confirm this theory. He measured the pyramid more accurately than anyone before him and concluded that it does not hold.

The American science writer Martin Gardner later exposed the method with a parody. He showed that with the same tricks one finds the number five everywhere in the Washington Monument. Nobody would conclude from this that its builders meant to encode a secret doctrine of the five.

Three famous cases

That great scientists too succumb to this temptation is shown by three stories. They are more instructive than any mockery of pyramid researchers, for they are about clever people.

The Titius-Bode series. In 1766 the Wittenberg scholar Johann Daniel Titius noticed, and after him the astronomer Johann Elert Bode made it known, that the distances of the planets can be described by a simple rule: take the numbers 0, 3, 6, 12, 24, 48, 96, add 4 to each and divide by 10. Out come 0.4, 0.7, 1.0, 1.6, 2.8, 5.2 and 10.0, and these are almost exactly the distances from Mercury to Saturn, if one sets that of the earth at 1. Only at 2.8 was a planet missing. Then came success. In 1781 William Herschel discovered Uranus at 19.2, the rule said 19.6. And in 1801 the dwarf planet Ceres was found at 2.77, exactly in the gap. The rule seemed confirmed, and astronomers were delighted. Then in 1846 Neptune was found.

The distances of the planets from the sun in astronomical units, the distance of the earth is 1. Hollow circles show the prediction of the Titius-Bode series, filled ones the measured value, the vertical axis is logarithmic. Up to Uranus the two lie close together, for Neptune almost a quarter is missing.

Neptune stands at 30.1, the rule demanded 38.8. To this day nobody knows whether a law lies behind the first hits. Most astronomers regard the series as a mixture of chance and the general fact that planetary orbits must not come too close to each other if they are to remain stable. The lesson: even a pattern that makes a correct prediction can fall apart. The fourth question below must be asked again and again.

Eddington’s 137. The British astrophysicist Arthur Eddington, one of the most important of his time, believed he could derive the constants of nature by pure reasoning. In 1929 he proved, as he thought, that the so-called fine-structure constant, a number describing the strength of the electric force, must be exactly 1/136. When measurements gave rather 1/137, he found a reason to add one. Mockers thereupon called him “Arthur Adding-One”. Today the value is known to twelve places, 1/137.035999…, and it is no whole number.

The Bible code. In 1994 three Israeli researchers published a study in a respected journal according to which the names of famous rabbis and their dates were hidden in the Hebrew text of the Book of Genesis, if one reads only every so-many-th letter. In 1997 the journalist Michael Drosnin turned this into the bestseller The Bible Code and claimed that the assassination of Yitzhak Rabin had been foretold in it too. Drosnin challenged his critics to find something similar in Moby Dick. The mathematician Brendan McKay did exactly that and found in Melville’s novel, by the same method, “predictions” of the assassinations of Indira Gandhi, Martin Luther King and others. In 1999 McKay and colleagues showed in the same journal that the original study had enough leeway in choosing the forms of the names to produce its result. It is the second of the four questions: how many possibilities were tried?

Four questions for a pattern

How does one tell a real pattern from chance? Four questions help:

Was the pattern claimed beforehand or found afterwards? Whoever searches first and explains afterwards always finds something.
How many possibilities were tried? Whoever tries ten operations on pairs from twenty measurements has almost two thousand attempts, and some always hit.
Is there a reason why it must be so? A proof or a mechanism turns a coincidence into a rule.
Does the pattern hold for new cases? A real pattern continues, a chance one falls apart.

This website must face these questions too. Some things stand up to them. That ten is the only number for which the angle sums of all figures of a simplex come to exactly as many full circles as it has vertices is no find made by trial and error but a theorem with a proof (see Angles). That all primes from 5 on lie next to a multiple of 6 is proven as well (see Arithmetic with remainders). Other things, for instance readings of the 24 elders of Revelation or of the twelve disciples, are interpretations. They can be stimulating, but they do not pass the four questions the way a theorem does. The pages of this website try to make that clear in each case. The reader should still test it for himself.

Does this mean there are no hidden truths? No. It only means that they do not hide in number games that any die produces as well. What makes a real secret is described on the page What a secret is.

Exercise

Take your date of birth, your phone number or the house number and postcode of your home. Search them for sacred numbers, with the same rules as in the graphic above: single digits and numbers, as well as sums, differences and products. Give yourself ten minutes. You will find something.

Then ask yourself: what does that say about you? The honest answer is: nothing. And precisely this experience, finding something that means nothing, is the most important vaccination for anyone who interprets numbers.

Sources and further reading

Sources

Klaus Conrad, Die beginnende Schizophrenie (Beginning schizophrenia), Stuttgart 1958 (the term apophenia)
Richard K. Guy, The Strong Law of Small Numbers, The American Mathematical Monthly 95, 1988, pp. 697 to 712
Charles Piazzi Smyth, Our Inheritance in the Great Pyramid, London 1864
W. M. Flinders Petrie, The Pyramids and Temples of Gizeh, London 1883
Martin Gardner, Fads and Fallacies in the Name of Science, New York 1957, ch. 15
Johann Elert Bode, Anleitung zur Kenntniß des gestirnten Himmels, 2nd ed. Hamburg 1772
Arthur S. Eddington, The Charge of an Electron, Proceedings of the Royal Society A 122, 1929, pp. 358 to 369
Doron Witztum, Eliyahu Rips and Yoav Rosenberg, Equidistant Letter Sequences in the Book of Genesis, Statistical Science 9, 1994, pp. 429 to 438
Brendan McKay, Dror Bar-Natan, Maya Bar-Hillel and Gil Kalai, Solving the Bible Code Puzzle, Statistical Science 14, 1999, pp. 150 to 173