The Ulam Spiral

… and many other spirals 
inspired by it …

1, 2, 3. fig1.2 -01

All prime numbers -except “2” & “3”-
are adjacent to a 6-fold.
By not being adjacent to a 6-fold
the little group consisting of “2” & “3”
constitutes an entirely different species of primes.

For this reason none of the spirals in this site
contain these numbers, since their presence
muddles the geometric pattern.
This can be seen in the spiral at left
where they are included, color-coded purple.

All the spirals in this site, including the one in Fig. -1,
have “0” as the spiral origin instead of “1” because of
numerous advantages itemized in “Spiral Origins”.

All these spirals
include “1” as an entry
since it is an adjacent of “0”,
which is a 6-fold since 0 x 6 = 0.
The presence of “1”
maintains the wholeness of the geometric pattern,
as can be seen by inspecting the spiral at left.

1, 2, 3 fig2-01

The numbers-stack in Fig. -2 also demonstrates
the detriments of including “2” & “3” and
the benefits of including “1” mentioned above.

Researchers wishing to find patterns in
the distribution of prime- or composite adjacents
-in a spiral or elsewhere-
might want to consider these concerns.

Suggested number-definitions:

Prime number:
Any whole number larger than 4
that cannot be evenly divided by
any whole number besides itself
larger than 1.

Quasi-Prime:
1, 2, 3.

Composite number:
Any whole number larger than 3
that can be evenly divided by
one- or more whole numbers besides itself
larger than 1.