## 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.

## 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.

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.

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.