The Ulam Spiral

… and many other spirals 
inspired by it …

Introduction

Every prime -except “2” & “3”- is an adjacent of a 6-fold,
with “0” being the first of the 6-folds, given that 0 x 6 = 0.
For that reason it could be advantageous to
investigate by what manner the 6-folds determine
the pattern of their prime adjacents on
the Ulam-spiral and other spirals.

6-folds can be classified in 4 different types as follows:
1):
A type consisting of 6-folds each having
a prime adjacent below it & above it.
The smallest 6-fold in that group is “0”.
(Its prime adjacents are “-1” & “1”.)
6-folds in this group can be
classified by type as follows:
Type – (1,1).
2):
A type consisting of 6-folds each having
a prime adjacent below it only.
The smallest 6-fold in that group is “24”.
6-folds in this group can be
classified by type as follows:
Type – (1,0).
3):
A type consisting of 6-folds each having
a prime adjacent above it only.
The smallest 6-fold in that group is “36”.
6-folds in this group can be
classified by type as follows:
Type – (0,1).
4):
A type consisting of 6-folds each having
no prime adjacent below it & above it.
The smallest 6-fold in that group is “120”.
(Its composite adjacents are
“119”, = 7 x 17 & “121”, = 11 x 11.).
6-folds in this group can be
classified by type as follows:
Type – (0,0).

The manner in which the 6-folds in these 4 groups
are interdistributed among each another
fully suffices to determine the
pattern of their prime adjacents on
the Ulam-spiral and other spirals.

It’s worth investigating to see if
6-folds of each of these 4 types, located a spiral,
are arranged in some kind of array containing
remarkable rows of numbers of various lengths
distributed in some (apparently) random way,
as happens when primes are located on a spiral.

If a mathematical directive is found
causing the emergence of such notable rows
among the 6-folds on a spiral
-assuming that there are such rows-
it would show the way to finding
a mathematical direrctive that
causes the emergence of such remarkable rows
among the primes on a spiral.

Fig. -1 shows
the 6-folds from the
natural numbers-line “0” to “∞”
populating the Ulam-spiral,
with all other intervening numbers omitted.

Further-investigations-Fig1

Fig. -1

A spiral so populated is problematic because
numbers become ‘large’ at a faster rate.
Also, as the numbers become larger
it becomes harder to discern
their nominal place in the line of 6-folds.

To avoid these problems,
the spiral can be populated with
a normal numbers-line, starting with “0”,
as shown in Fig, -2,
with the understanding that
these numbers actually are ‘multiplicands’
to be multiplied by “6” to
generate the desired 6-folds.
Also, the numbers in that spiral
show the ordinal numbers of
the 6-folds generated from them,
keeping in mind that
the 1st. 6-fold is “0”, not “6”.

A spiral populated in such fashion
should always have an usage directive
placed close by it
as seen to the right of the spiral
in Fig. -2.

Further-investigations-Fig2

Spiral entries to be
multiplied by “6”.

Fig. -2

Additionally:
the 6-folds of each of the 4 types
need to be color-coded distinctly from
those of the other types,
in the manner as shown in Fig. -3.

Type – (1,1)
Type – (1,0)
Type – (0,1)
Type – (0,0)

Fig. -3

The spiral per Fig. -2 can then
represent each 6-fold and
its identifying type,
as shown in Fig. -4.

Further-investigations-Fig4

Spiral entries to be
multiplied by “6”.

Fig. -4

The next 3 pages show the
Ulam-, tetragonal- & tetrapolar spirals
populated by 6-folds of all 4 types in
the manner as shown in Fig. -4.

Also shown in additional details:
spirals populated by
groups of related types &
by each type separately.

To see these spirals
click on “Spirals For 6-folds”.