The distribution of primes – a question of perspective
Why the tetractys should have anything to do with the primes, why professional mathematicians think little of the idea, and why it is still worth insisting on.
For professional mathematicians, linking the Pythagorean tetractys with the distribution of the primes is not up for discussion. Yet precisely this link is at the heart of the ideas presented here.
Numbers are quantities
What do we actually mean by numbers? Numerals are symbols for quantities, dependent on a freely chosen place-value system, in our case the decimal system. Whoever wants to explore the nature of number must lead the symbols back to their natural state, to point, line, surface and solid, to the figurate numbers. The Pythagoreans already saw this. And the primes in particular, the holy grail of number theory, are pure geometry: thirteen points remain a prime set however they are written. Anyone interested in primes without knowing the geometry behind them is therefore at a great disadvantage (see Number theory + geometry = philosophy).
The most perfect symmetry is circle and sphere. This also explains why the number π appears in Euler’s formulas about the primes, which lead to the Riemann hypothesis. The number theory section therefore looks at point intervals on the number line and in the coordinate system, the number and its divisors. The geometry section looks at them in space: simplexes, sphere packings, Platonic solids. Every new point on the circle of a simplex is a leap into the next dimension, and simplexes have a direct bearing on the primes.
The simplex: each new point on the circle is joined to all others, a leap into the next dimension.
The obvious objection
One objection comes up again and again: common encryption methods rely on breaking large numbers into prime factors. Anyone who found a structure in the primes could break many keys. So many people would have an interest in this that it would long since have happened and been published.
The answer: the problem of the distribution of primes is to a large extent a problem of communication.
Many discoverers, little exchange
On this view, the geometric connections are well known to a small group of people but are not widely publicised. The number-theoretical connections are already contained in the sieve of Eratosthenes. So far the sieve has been used only as a tool for sorting out non-primes. Its significance has not been recognised. There are variants of the sieve that are useless for sieving but reveal much more about the character of numbers.
Several private scholars have each seen part of this order:
- Felix Stoffel published a classification of the primes in 2010, based on a grid of 30, into “four prime temperaments”. In doing so he found a variant of the sieve. Every correct system of prime gaps must contain the sieve, or it would be wrong.
- Peter Plichta developed the prime number cross and put the number 24 at its centre.
- A retired chief design engineer of a printing-press company also discovered a grid of 30 containing all prime positions.
What is overlooked: these grids are “decimally coded”. That only becomes visible once you leave the decimal system and take the geometry of the simplexes as a guide. If amateur researchers were more open to one another and worked together more, the puzzle would long since have been solved. Professional mathematicians, on the other hand, risk their reputations if they stray from the axioms of their discipline.
The architecture between the primes
The real substance of the objection lies elsewhere. Mathematics looks at the primes themselves. The numbers in between it notices only in passing. But to look at the primes alone is to blank out most of the picture.
The thesis: the distribution of the primes is governed by a system consisting of an infinite sequence of ever larger grids. These grids are formed by the numbers divisible by three and by the “empty” prime positions. Because all numbers are made of prime factors, the primes are regarded as the cause of all others. But an irregular distribution cannot be caused by the primes themselves. Symmetry is the supreme law of nature and physics, as a balance of forces. There are symmetries among the numbers too, and the fact that primes so often appear in pairs is one of them. If the primes are irregularly distributed, they cannot form the axes of symmetry themselves.
The image for this: the primes are not the building but what remains between its walls.
Searching at the wrong end
All too often the search is conducted at the wrong end of the number line. Finding ever larger primes has become a sport, but what do these giants reveal about the primes? Whoever wants to understand the combinatorics of numbers must go to the source, to the first numbers from which all others follow. This is not about prize money or medals, no one will climb Mount Olympus of mathematics single-handed. The glory could in any case be given to a group that lived some 2,500 years ago and today mostly figures in the history of mathematics as a band of mystical enthusiasts: the Pythagoreans. What counts is the insight of people who think for themselves.
Why is the link between number theory and geometry not taught in schools? For pupils who struggle with mathematics, the geometry of simplexes in particular would be a vivid way into otherwise abstract numbers. And the lambdoma, known almost only in music theory and harmonics, is really indispensable for understanding the primes (see The lambdoma), not to mention its musical aspects, which lead straight into physics.
Primes in nature
A two-part German documentary from 2010, Die Code-Knacker (“The code breakers”, 3sat), tells of the Riemann hypothesis, which has haunted many mathematicians. Leonhard Euler already discovered the connection between the primes and π, that is the geometry of the circle (see The number π). The second part shows that the distribution of primes also matters in physics: the spacings of the zeros of the Riemann zeta function follow the same statistics as the energy levels of heavy atomic nuclei, and mathematicians and physicists are working together to get to the bottom of it.
And a final thought: if a solution has been sought in vain for centuries, would a different approach not be in order?
Continue with The order of the primes.