# 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 in 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

disproportionate 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 numbers on that spiral:

• The rows of numbers

along the diagonal directions in unpredictable places.

• The many different lengths of these rows.

• The many different lengths of the gaps between co-linear rows.

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

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

thru the presentation of these in this site,

doing so completely on just this page and the next.

The remaining pages contain material that may be of interest,

including items such as those listed to the right:

Under “Findings”:

Exposition of a group of numbers on the spiral exhibiting a

distinct local vertical symmetry among their positions on the spiral,

affecting a ~4/5ths. portion of these numbers.

Under “Polygonal Spirals”:

“Digonal Spiral”.

(A “Digon” is a polygon with only 2 corners, it exists in spherical geometry.)

Under “Utility (1)”:

“Tetragonal Spiral”, featuring 2 lengthy numbers-sequences.

Alas: they are broken, geometrically, due to the angles of the spiral

but they remain whole algebraically.

Under “Utility”:

“Tetrapolar Spiral”, featuring the same

lengthy numbers-sequences as mentioned above,

but now they are unbroken, geometrically, because the spiral has no angles. (!!!)

(Interestingly:

these 2 lengthy numbers-sequences do not arise in the standard Ulam Spiral,

plotting these sequences in the Ulam spiral produces just chaotic jumbles.)

Under “Further investigations” and the 2 entries following it:

A totally different area of research, but one using the spirals.

The detail above shows

a row of consecutive 6-folds

from “0” thru “120”.

(“0” is a 6-fold because 0 x 6 = 0.)

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.

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.

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.)

These 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

overall 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:

in Fig. -3a it is #295.

They can be conveniently referred to as

“The isolated ones”,

or, informally, as the

“The isolates”.

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 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.

The spiral per fig. -2

-when shown in a more extensive representation-

strongly suggests a local vertical symmetry among

many of the numbers in its left area.

The same is true of

the spiral per Fig. -1,

but it is less noticeable.

This phenomenon is explored in

Findings (1).