the subject of my artistic and mathematical work is the distribution of the prime numbers.
It is the prevailing opinion in the mathematical community that the prime numbers are distributed in a way that does not allow any rule to be found, there seems to be no system to make a prediction for a next prime number´s position.
The mathematician Bernhard Riemann has given with his famous conjecture precise tools to calculate the quantitative decrease of the density of the prime numbers under given higher numbers.
But never the less the prime numbers still have the reputation of being chaotically positioned.
What was missing so far is a structural pattern of the prime numbers distribution.
And I promise you, I have found it:
It is a beautiful symmetry-pattern.
I hope my lecture, my art work and in addition a computer program will convince you.
This following tabular compilation of numbers consists of prime numbers with the exception that in the central vertical column there are numbers that exist between the twin-prime-numbers below 360, like 6 between 5 and 7, 12 between 11 and 13 and so on up to 348 between 347 and 349.
These numbers between the twins are the keys to the PNDP (prime-number- distribution-pattern)
These numbers between the twins are 6 or 6n and the twins are 6 or 6n +1 and 6n -1. These numbers between the twin-prime-numbers function as symmetry numbers, because the twins are to be found in symmetric positions to these.
What can easily be comprehended is the fact that beyond these twins there are many, many further prime-numbers standing symmetrically in the same mirror-like distances to the numbers between the twins.
Here an example:
5 7 13 17 18 19 23 29 31
13 11 5 1 1 5 11 13
PNS ( prime numbers)
D (distances to the center)
T (twin-primes next to center: SN)
NHT (next higher twin-prime-pair)
PNF (prime-number family)
1 7 13 17 19 23 29 30 31 37 41 43 47 53 59
29 23 17 13 11 7 1 1 7 11 13 17 23 29
These numbers, like the twins, standing pairwise in equal distances to the symmetry-numbers I call sib-pair-prime-numbers and all these sib-pairs including the twins and the number in the center I call a prime-number -family (PNF) named after the symmetry-number in the center, in the case above PNF 30.
The next prime-number-family is easily to be found:
The NHT (next higher twin-prime-pair) of the PNF30 is 41 and 43.
We put the SN 42 into the center and look for or let a computer-program find all the prime numbers standing symmetrically to this particular SN42.
Here it is:
1 5 11 13 17 23 31 37 41 42 43 47 53 61 67 71 73 79 83
The NHT (next higher twin prime pair) of the PNF42 is 71 and 73.
Again we put the SN 72 into the center and look for or let a computer find all the prime numbers standing symmetrically to the particular SN72.
The PNF72 has the following members:
5 ,7 ,13, 17, 31, 37 ,41 ,43, 47 ,61, 71, 72 ,73 ,83 ,97 ,101 ,103 ,107 ,113 ,127,131,137 ,139
67 65 59 55 41 35 31 29 25 11 1 1 11 25 29 31 35 41 55 59 65 67
We can easily build up the next PNF by putting 102 into the symmetry center and so on and so on.
In this following painting the center ist 348. The symmetry of the larger getting families is obvious.
Here I would like to present to you a painting measuring 144 x 144 cm, having the number 72 in the center, unfortunately to large to be shown here in Linz.
1,44 m x 1,44 m , acrylic on cavas
Each prime number between 0 and 144 has got its geometrical place in this painting.
In the blue center of the painting stands the no 72 (not a prime-number itself)
neighbored by the two twin-prime-numbers 71 and 72 and as well surrounded further
outside by more prime-numbers that stand pairwise in equal distances to the
symmetry-number 72 in the center.
These numbers in allusion to the twins that are siblings as well I call sib pairs.
They are visualized by further blue squares with inscribed circles.
All these numbers framing the blue center create a prime-number family named
after the center 72. (PNF72)
Looking up from the center of the painting we find more prime-number-family-
centers that are neighbored by twins:
60 (59 und 61, red-violet), 42 (41 und 43, red), 30 (29 und 31orange),
18 (17 und 19, yellow), 12 (11 und 13, yellow-green), 6 (5 und 7, green),
4 (3 und 5, blue), 2 (1 und 3, violet).
All these numbers are framed by twins.
Each symmetry-number with its neighboring twins represents centers of prime-
number-families that are colored spectral like a rainbow.
They overlap each other but due to their transparency can be identified.
In this painting the number string is in the vertical axis of the painting.
In the beginning of my occupation wit the distribution of the prime-numbers
I investigated those up to 360 in the circle of 360 degrees.
So the number string was limited and round like a circle.
And I found these symmetries with the help of compasses.
Later I stretched the number string into a horizontal line and visualized
these symmetries with half-circles like that:
In my attempts to find order, rules, symmetries or other regularities in the distribution of the prime-numbers I found it very helpful to limit these researches to the circle of 360 degrees.
I found :
twelve trigones. five squares two hexagons
And I discovered that each of the twins and the numbers between them
has at least one neighboring in the angle of 42 degrees and most obvious there rules an enormous and beautiful symmetry.
Last not least I would like to present the two paintings I have in the show as originals.
Again the symmetry-number 2,4,6,12,18,30,42,60
72,102,108, 138, 150 and 180:
Colored axes divide the circle in halves and curved lines reach for the symmetrically positioned prime-numbers
This symmetrical round painting has left and right 23 prime numbers at the rectangular corners of Thales-triangles.
All the opposite prime-numbers with corners of same colors add up to 360.