# Introduction.

Created by Stanislaw Ulam in 1963, the Ulam Spiral is a

graphical representation of the set of prime numbers.

Ulam Plotted the positive integers on a square spiral and

marked the prime numbers. From this image it was apparent there were

diagonal rows -in both directions- created from what seems like a

disproportionately high number of prime numbers.

As it is human nature to look for patterns mathematicians have spent

decades trying to find any rhyme or reason to this pattern phenomenon.

For some years I’ve thought about the well-known mysteries surrounding

the placement of the prime numbers on that spiral:

• The rows of numbers along the diagonal directions.

• The many different lengths of these rows.

• The many different lengths of the gaps between collinear rows.

All this with the aim of revealing the mathematics governing such placement.

Having seen the mathematics, I simply want to share with others

through the presentation of these in this site,

doing so completely on just this page and the next.

The subsequent interactive page will help to

demonstrate these peculiarities of the spiral.

Pages -4 and -5 contain

advisories pertaining to methods of

populating the spiral in manners that are

most beneficial for exposing internal structures,

with rationales provided.

Also shown:

(6) Tripolar Spiral,

(7) Tetrapolar Spiral,

each featuring remarkably long rows of primes.

In the tripolar spiral the numbers

constituting these rows are located on**every other** arc-segment of the spiral.

this situation is true for all polar spirals with

an **odd** number of poles.

In the tetrapolar spiral the numbers

constituting these rows are located on**every successive** arc-segment of the spiral.

this situation is true for all polar spirals with

an **even** number of poles.

See

(8) Polar Spirals, Operable

for additional polar spirals,

ranging from dipolar through heptapolar.

## Detail -A.

The detail above shows

a row of consecutive 6-folds

from “0” thru “120”.

(“0” is a 6-fold because 0 x 6 = 0.)

## Det. -B.

The Detail above shows

the row from Det. -A.

Shown below and/or above each 6-fold is an

adjacent prime number, if there is one.

“120” is the 1st. 6-fold

without an adjacent prime number.

## Det.-C.

The Detail above shows

the row from Det. -A.

Shown below and/or above each 6-fold is an

adjacent composite number, if there is one.

“120” is the 1st. 6-fold

with 2 adjacent composite numbers.

This website continuously discusses

“Numbers adjacent to the 6-folds”.

That’s a lengthy phrase, making it

more convenient to refer to them as

“The adjacent numbers”,

or, in informal shorthand style:

“The adjacents”.

A single such number then is

“An adjacent”,

using the word “adjacent” as a noun.

The Ulam spiral shown below is

populated by prime numbers.

One can clearly see

its well known mysterious features.

(Every prime number -except “2” & “3”-

is an adjacent.)

The Ulam spiral shown below is

populated by composite adjacents,

such as the ones shown in Det. -C.

One can clearly see

that this spiral has the same

mysterious features

as the spiral to the left.

The spiral above shows

the following populations together:

• The prime adjacents of the spiral per Fig. -1.

• The composite adjacents of the spiral per Fig. -2.

As can be seen:

When the spiral is populated by both groups

there are only uninterrupted endless rows

oriented along both diagonal directions.

Some of these rows have a beginning but no end,

others have neither a beginning nor an end.

Interspersed among these rows is

a regular pattern of isolated numbers,

shown circled in red.

They can be prime or composite.

Thus:

• The irregular gaps between the in-line rows

in the spiral per Fig. -1 are caused by

the absence of the composite adjacents.

• The irregular gaps between the in-line rows

in the spiral per Fig. -2 are caused by

the absence of the prime adjacents.

Thus also:

•The orientation of the uninterrupted endless rows

along both diagonal directions

seen in the spiral per Fig. -3

is preserved in the fragments of these rows

seen in the spirals per Fig’s. -1 & -2.

(The cause for the diagonal directions

of the rows in the spiral per Fig. -3

will be shown on the next page.)

To summarize:

The mysteries associated with the

spirals per Fig’s. -1 & -2 exist only because

the set of prime adjacents by themselves and

the set of composite adjacents by themselves

are ‘incomplete’, being mere subsets of the

complete set comprised of all adjacents.

In the spiral per Fig. -3

any pair of adjacent parallel rows

-along both diagonal directions-

have the same interspace, thus forming

a pattern of squares,

such as the one shown in Fig. -3a.

Each of these squares contain

an isolated number in one of its corners

shown circled in red in Fig. -3a.

These numbers can be conveniently called

“The isolated ones”, or, informally,

“The isolates”.

## Fig. -4.

Fig. -4 shows the spiral

populated exclusively by the 6-folds.

As can be seen:

the 6-folds are grouped in

regularly spaced 90° V-shaped clusters

each consisting of 3 numbers.

These 3-number clusters can be

called “Triads”.

## Fig. -5.

Fig. -5 shows the spiral per Fig. -4

additionally populated by

all the adjacents, clearly showing

the manner of their adjacency.

Note that, generally:

the void areas

have the same peripheries as

the filled areas.