This is especially true of the Prime shells (red). Any power of a prime number will only have itself as a factor, so shells that are the powers of Prime shells should have exactly the same pattern as the Prime shell. In Figure 17 Reveal shells are black, Prime shells are red, Echoes and Powers of 2 are cyan, all other Echoes are blue, Powers of any number are green, and all other numbers are yellow. The diagram extends to 37, to show 6^2, ie the square echo of the reveal shell of 3. Prime shells, the red ones, are defined as having no numbers excluded except for the zero line and/or the self line, ie Ɵ=0 or 2π (0° or 360°).

Co-Prime Shells (Relatively prime or Mutually prime)

Taking a closer look at Figure 17, we have:
  1. Prime shells in red.
  2. Squares of primes in green.
  3. Reveal shells in black.
  4. Echo shells in blue and cyan.
  5. The shells that are left, the yellow ones, reveal that shells Co-Primes.
In a sense all shells reveal co-primes. When the shell is prime, it is because it is co-prime with all the other numbers below it except 1.
  1. Squares of primes (green shells) follow the same rule as for yellow shells.
  2. Reveal shells are co-prime with only primes up to the value of the square of the prime number following it's highest prime factor.
  3. Echo shells follow the same rules as for Reveal shells.

Shell symmetry and pattern

It is interesting that the shells display symmetry in the x-axis. This is not unusual, as the images are drawn from frequencies described as sin waves, which display symmetry.