The Field and Rule
We fill a triangular lattice with prime and composite numbers. Letting 2 occupy the first cell, it has an orientation with a ‘bottom’, a ‘left’ side and a ‘right’ side, figure 1(a). We move to the next cell through 2’s ‘left’ (left spin), and assign it 3, figure 1(b). If we continue to sequentially exit cells through their ‘left’ walls, the numbers will fill a 6 cell hexagon, and continue to ‘spin’ counter-clockwise around an axis forever.
Figure 1. Each number has an orientation within a triangular cell, left, right and bottom. (a) Beginning with 2, the original orientation is set (b) and the number line advances through two’s left wall to place 3 in the next cell. (c) The wall the cell is entered from is the new bottom.
(a) (b) (c)
We now place integers sequentially into the lattice with a simple rule: Each time a prime number is encountered, the spin or ‘wall preference’ is switched. So, from the first cell, exit from 2’s left side. This sets the spin to left and the next cell is 3, a prime, so switches to right. 4 is not prime and continues right. 5 is prime, so switch to left and so on. There are twists and turns until 19 abuts 2, figure 2 (a).
Figure 2. (a) The tessellating field of equilateral triangles fills with numbers, with spin orientation flipping with each prime number encountered, creating 3 minor hexagons, assigned names blue, red and green. (b) as the numbers continue to fill the field, the other minor hexagons from, purple, cyan and yellow.
Proceeding, the number line begins to coil upon itself; 20 lands on 2’s cell, 21 on 3’s cell. Prime number 23 sends the number line left to form the fourth hexagon, purple. As it is not a twin, the clockwise progression (rotation) reverses itself. Twin primes 29 and 31 define the fifth hexagon, cyan. Finally, 37, again not a twin, reverses the rotation of the system, so 47 can define the yellow hexagon, figure 2(b).
Surprisingly, the 24-cell hexagon confines all natural numbers. The reason: no prime numbers occupy a cell with a right or left wall on the t-hexagon’s outer boundary, other than 2 and 3, the initial primes that forced the number line into this complex coil. Without a prime number in the outer set of triangles, the number line does not change to an outward course and remains forever contained in the 24 cells shown above.