Cellular Symmetry of the Prime Numbers

 Barry Dalgarno

27 September 2014

Dedicated to my mother Ruth Dalgarno

The structure of minimum voids was originally published on 16 November 2013 under the title Symmetry of the Prime Numbers.

The sequence of the prime numbers is finely tuned from the perspective of symmetry. This was first shown in 2009 (Ref. 1). In that paper, which describes the structure of minimum vertices, the natural numbers from 2 onwards are positioned on a grid, with the primes forming the axis. Positioning is such that each number is connected to the next in a continuous network, with no step ever retracing its previous path. Then in 2013 the structure of minimum voids  was discovered in which each prime number interval has the degree of freedom of not having to rejoin the prime axis. This enabled the formation of a continuous sequence of cells with at least D1 symmetry. Although the internal structure of each cell with respect to the prime intervals is asymmetric, the perimeter is symmetric. 

Cellular symmetry of the prime numbers. A short section from the structure of minimum voids (consecutive prime numbers 2 to 103)

The structure is the result of two simple rules. 

Rule 1

Each cell must have at least D1 mirror reflection symmetry  (for example cells 1 and 2).

Rule 2

The entire structure is the one with minimum voids. A void is a point within a given cell that is not occupied by a non-prime. The greater the number of voids within a cell, the greater the difference between the cell and a square of the same width. For example, the first cell has a single unoccupied lattice point compared to a square of the same width. The fourth cell has two unoccupied lattice points compared to a square of the same width.

As we move along the prime axis, there is a relatively smooth transition from one cell to the next. This harmonious transition is maintained throughout. Within each cell the number of voids is denoted by the delta symbol. The structure has the lowest  possible number of cumulative voids as the number of cells (c) increases. The table shows the number of voids for the first seven cells of the infinite structure. Alpha is the number of occupied lattice points (primes and non-primes) for a given cell. Omega denotes the the number of lattice points that would correspond to a square of the same width. For each cell, the number of voids is given by:

Pc (mean) is the symmetry directed cumulative average of the prime numbers relating to completed symmetry cells with respect to the prime axis. Psi (mean) is the cumulative average geometric entanglement. The symbol psi has been chosen to represent the geometric entanglement for each symmetry cell, so as to distinguish it from the specific type of entanglement described in reference 1, where the symbol xi was used. In reference 1, each prime interval rejoins the prime axis in a continuous network. The type of entanglement described in the current paper is different in that each prime interval can be positioned without the restriction of rejoining the axis. This additional geometric freedom enables the formation of an infinite continuous sequence of consecutive symmetry cells, in which psi denotes the entanglement value within each cell. A value of one refers to a symmetry cell formed from a single interval (see the first cell in the above figure). A value of two refers to a composite symmetry cell formed from the entanglement of two intervals, and so on. There are three possible levels of cell symmetry : D1 , D2  and D4 . The first level is the minimum symmetry of D1 (for example cell 1 which has dihedral reflection symmetry). The next level of symmetry is D2 (for example cell 4 which has the additional mirror reflection. The third level has the maximum possible symmetry of D4 (for example the first completed square that is cell 3 , which has two additional reflections). For brevity, the entire sequence of symmetry cells will be referred to simply as the Structure of Minimum Voids.

Exponential form of the equation :

References

Ruth Dalgarno - A Life Remembered.

https://sites.google.com/view/ruth-dalgarno/home

1. Dalgarno, B. (2009). A New Law of Prime Numbers Based on an Infinite Structure of Sequential Triangular Symmetry from Finely-Tuned Geometric Entanglement of the Prime Intervals.

 https://sites.google.com/site/geometryoftheprimes/