The Ulam Spiral

… and many other spirals 
inspired by it …

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.

Preview Page - Section 1-01

The detail above shows
a row of consecutive 6-folds
from “0” thru “120”.
(“0” is a 6-fold because 0 x 6 = 0.)

Preview Page - Section 1b-01

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.

Preview Page Section 3-01

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”.

Preview Page - Section 2.3-01

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).